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A008805 - OEIS
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A008805 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A008805" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fA008805">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="A008805 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A008805 </div> <div class=seqname> Triangular numbers repeated. </div> </div> <div class=scorerefs> 72 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 1, 3, 3, 6, 6, 10, 10, 15, 15, 21, 21, 28, 28, 36, 36, 45, 45, 55, 55, 66, 66, 78, 78, 91, 91, 105, 105, 120, 120, 136, 136, 153, 153, 171, 171, 190, 190, 210, 210, 231, 231, 253, 253, 276, 276, 300, 300, 325, 325, 351, 351, 378, 378, 406, 406, 435, 435</div> <div class=seqdatalinks> (<a href="/A008805/list">list</a>; <a href="/A008805/graph">graph</a>; <a href="/search?q=A008805+-id:A008805">refs</a>; <a href="/A008805/listen">listen</a>; <a href="/history?seq=A008805">history</a>; <a href="/search?q=id:A008805&fmt=text">text</a>; <a href="/A008805/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,3</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Number of choices for nonnegative integers x,y,z such that x and y are even and x + y + z = n.</div> <div class=sectline>Diagonal sums of <a href="/A002260" title="Triangle read by rows: T(n,k) = k for n >= 1, k = 1..n.">A002260</a>, when arranged as a number triangle. - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, Feb 28 2003</div> <div class=sectline>a(n) = number of partitions of n+4 such that the differences between greatest and smallest parts are 2: a(n-4) = <a href="/A097364" title="Triangle read by rows, 0 <= k < n: T(n,k) = number of partitions of n such that the differences between greatest and smalles...">A097364</a>(n,2) for n>3. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Aug 09 2004</div> <div class=sectline>For n >= i, i=4,5, a(n-i) is the number of incongruent two-color bracelets of n beads, i from them are black (cf. <a href="/A005232" title="Expansion of (1-x+x^2)/((1-x)^2*(1-x^2)*(1-x^4)).">A005232</a>, <a href="/A032279" title="Number of bracelets (turnover necklaces) of n beads of 2 colors, 5 of them black.">A032279</a>), having a diameter of symmetry. - <a href="/wiki/User:Vladimir_Shevelev">Vladimir Shevelev</a>, May 03 2011</div> <div class=sectline>Prefixing <a href="/A008805" title="Triangular numbers repeated.">A008805</a> by 0,0,0,0 gives the sequence c(0), c(1), ... defined by c(n)=number of (w,x,y) such that w = 2x+2y, where w,x,y are all in {1,...,n}; see <a href="/A211422" title="Number of ordered triples (w,x,y) with all terms in {-n,...,0,...,n} and w^2 + x*y = 0.">A211422</a>. - <a href="/wiki/User:Clark_Kimberling">Clark Kimberling</a>, Apr 15 2012</div> <div class=sectline>Partial sums of positive terms of <a href="/A142150" title="The nonnegative integers interleaved with 0's.">A142150</a>. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jul 07 2012</div> <div class=sectline>The sum of the first parts of the nondecreasing partitions of n+2 into exactly two parts, n >= 0. - <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Jun 08 2013</div> <div class=sectline>Number of the distinct symmetric pentagons in a regular n-gon, see illustration for some small n in links. - <a href="/wiki/User:Kival_Ngaokrajang">Kival Ngaokrajang</a>, Jun 25 2013</div> <div class=sectline>a(n) is the number of nonnegative integer solutions to the equation x + y + z = n such that x + y <= z. For example, a(4) = 6 because we have 0+0+4 = 0+1+3 = 0+2+2 = 1+0+3 = 1+1+2 = 2+0+2. - <a href="/wiki/User:Geoffrey_Critzer">Geoffrey Critzer</a>, Jul 09 2013</div> <div class=sectline>a(n) is the number of distinct opening moves in n X n tic-tac-toe. - <a href="/wiki/User:I._J._Kennedy">I. J. Kennedy</a>, Sep 04 2013</div> <div class=sectline>a(n) is the number of symmetry-allowed, linearly-independent terms at n-th order in the series expansion of the T2 X t2 vibronic perturbation matrix, H(Q) (cf. Opalka & Domcke). - <a href="/wiki/User:Bradley_Klee">Bradley Klee</a>, Jul 20 2015</div> <div class=sectline>a(n-1) also gives the number of D_4 (dihedral group of order 4) orbits of an n X n square grid with squares coming in either of two colors and only one square has one of the colors. - <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Oct 03 2016</div> <div class=sectline>Also, this sequence is the third column in the triangle of the coefficients of the sum of two consecutive Fibonacci polynomials F(n+1, x) and F(n, x) (n>=0) in ascending powers of x. - <a href="/wiki/User:Mohammad_K._Azarian">Mohammad K. Azarian</a>, Jul 18 2018</div> <div class=sectline>In an n-person symmetric matching pennies game (a zero-sum normal-form game) with n > 2 symmetric and indistinguishable players, each with two strategies (viz. heads or tails), a(n-3) is the number of distinct subsets of players that must play the same strategy to avoid incurring losses (single pure Nash equilibrium in the reduced game). The total number of distinct partitions is <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(n-1). - <a href="/wiki/User:Ambrosio_Valencia-Romero">Ambrosio Valencia-Romero</a>, Apr 17 2022</div> <div class=sectline>a(n) is the number of connected bipartite graphs with n+1 edges and a stable set of cardinality 2. - <a href="/wiki/User:Christian_Barrientos">Christian Barrientos</a>, Jun 15 2022</div> <div class=sectline>a(n) is the number of 132-avoiding odd Grassmannian permutations of size n+2. - <a href="/wiki/User:Juan_B._Gil">Juan B. Gil</a>, Mar 10 2023</div> <div class=sectline>Consider a regular n-gon with all diagonals drawn. Define a "layer" to be the set of all regions sharing an edge with the exterior. Removing a layer creates another layer. Count the layers, removing them until none remain. The number of layers is a(n-2). See illustration. - <a href="/wiki/User:Christopher_Scussel">Christopher Scussel</a>, Nov 07 2023</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>H. D. Brunk, An Introduction to Mathematical Statistics, Ginn, Boston, 1960; p. 360.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Vincenzo Librandi, <a href="/A008805/b008805.txt">Table of n, a(n) for n = 0..3000</a></div> <div class=sectline>G. E. Andrews, M. Beck, and N. Robbins, <a href="http://arxiv.org/abs/1406.3374">Partitions with fixed differences between largest and smallest parts</a>, arXiv preprint arXiv:1406.3374 [math.NT], 2014.</div> <div class=sectline>P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 46.</div> <div class=sectline>Juan B. Gil and Jessica A. Tomasko, <a href="https://arxiv.org/abs/2207.12617">Pattern-avoiding even and odd Grassmannian permutations</a>, arXiv:2207.12617 [math.CO], 2022.</div> <div class=sectline>Jia Huang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Huang/huang8.html">Partially Palindromic Compositions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 19.</div> <div class=sectline>Kival Ngaokrajang, <a href="/A008805/a008805.jpg">The distinct symmetric 5-gons in a regular n-gon for n = 6..13</a></div> <div class=sectline>D. Opalka and W. Domcke, <a href="http://dx.doi.org/10.1063/1.3382912">High-order expansion of T2xt2 Jahn-Teller potential energy surfaces in tetrahedral molecules</a>, J. Chem. Phys., 132, 154108 (2010).</div> <div class=sectline>Christopher Scussel, <a href="/A008805/a008805.pdf">Illustration of layers in regular n-gons with all diagonals drawn</a></div> <div class=sectline>Vladimir Shevelev, <a href="https://arxiv.org/abs/0710.1370">A problem of enumeration of two-color bracelets with several variations</a>, arXiv:0710.1370 [math.CO], 2007-2011.</div> <div class=sectline><a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a></div> <div class=sectline><a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).</div> <div class=sectline><a href="/index/Mo#Molien">Index entries for Molien series</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>G.f.: 1/((1-x)*(1-x^2)^2) = 1/((1+x)^2*(1-x)^3).</div> <div class=sectline>E.g.f.: (exp(x)*(2*x^2 +12*x+ 11) - exp(-x)*(2*x -5))/16.</div> <div class=sectline>a(-n) = a(-5+n).</div> <div class=sectline>a(n) = binomial(floor(n/2)+2, 2). - <a href="/wiki/User:Vladimir_Shevelev">Vladimir Shevelev</a>, May 03 2011</div> <div class=sectline>From <a href="/wiki/User:Paul_Barry">Paul Barry</a>, May 31 2003: (Start)</div> <div class=sectline>a(n) = ((2*n +5)*(-1)^n + (2*n^2 +10*n +11))/16.</div> <div class=sectline>a(n) = Sum_{k=0..n} ((k+2)*(1+(-1)^k))/4. (End)</div> <div class=sectline>From <a href="/wiki/User:Paul_Barry">Paul Barry</a>, Apr 16 2005: (Start)</div> <div class=sectline>a(n) = Sum_{k=0..n} floor((k+2)/2)*(1-(-1)^(n+k-1))/2.</div> <div class=sectline>a(n) = Sum_{k=0..floor(n/2)} floor((n-2k+2)/2). (End)</div> <div class=sectline>A signed version is given by Sum_{k=0..n} (-1)^k*floor(k^2/4). - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, Aug 19 2003</div> <div class=sectline>a(n) = <a href="/A108299" title="Triangle read by rows, 0 <= k <= n: T(n,k) = binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2].">A108299</a>(n-2,n)*(-1)^floor((n+1)/2) for n>1. - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jun 01 2005</div> <div class=sectline>a(n) = <a href="/A004125" title="Sum of remainders of n mod k, for k = 1, 2, 3, ..., n.">A004125</a>(n+3) - <a href="/A049798" title="a(n) = (1/2)*Sum_{k = 1..n} T(n,k), array T as in A049800.">A049798</a>(n+2). - <a href="/wiki/User:Carl_Najafi">Carl Najafi</a>, Jan 31 2013</div> <div class=sectline>a(n) = Sum_{i=1..floor((n+2)/2)} i. - <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Jun 08 2013</div> <div class=sectline>a(n) = (1/2)*floor((n+2)/2)*(floor((n+2)/2)+1). - <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Jun 08 2013</div> <div class=sectline>From <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Apr 22 2015: (Start)</div> <div class=sectline>a(n) = a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) +a(n-5).</div> <div class=sectline>a(n) = (2*n +3 +(-1)^n)*(2*n +7 +(-1)^n)/32. (End)</div> <div class=sectline>a(n-1) = <a href="/A054252" title="Triangle T(n,k) of n X n binary matrices with k=0..n^2 ones under action of dihedral group of the square D_4.">A054252</a>(n,1) = <a href="/A054252" title="Triangle T(n,k) of n X n binary matrices with k=0..n^2 ones under action of dihedral group of the square D_4.">A054252</a>(n^2-1), n >= 1. See a Oct 03 2016 comment above. - <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Oct 03 2016</div> <div class=sectline>a(n) = <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(<a href="/A008619" title="Positive integers repeated.">A008619</a>(n)). - <a href="/wiki/User:Guenther_Schrack">Guenther Schrack</a>, Sep 12 2018</div> <div class=sectline>From <a href="/wiki/User:Ambrosio_Valencia-Romero">Ambrosio Valencia-Romero</a>, Apr 17 2022: (Start)</div> <div class=sectline>a(n) = a(n-1) if n odd, a(n) = a(n-1) + (n+2)/2 if n is even, for n > 0, a(0) = 1.</div> <div class=sectline>a(n) = (n+1)*(n+3)/8 if n odd, a(n) = (n+2)*(n+4)/8 if n is even, for n >= 0.</div> <div class=sectline>a(n) = <a href="/A002620" title="Quarter-squares: a(n) = floor(n/2)*ceiling(n/2). Equivalently, a(n) = floor(n^2/4).">A002620</a>(n+2) - a(n-1), for n > 0, a(0) = 1.</div> <div class=sectline>a(n) = <a href="/A142150" title="The nonnegative integers interleaved with 0's.">A142150</a>(n+2) + a(n-1), for n > 0, a(0) = 1.</div> <div class=sectline>a(n) = <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(n+3)/2 - <a href="/A135276" title="a(0)=0, a(1)=1; for n>1, a(n) = a(n-1) + n^0 if n odd, a(n) = a(n-1) + n^1 if n is even.">A135276</a>(n+3)/2. (End)</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>a(5) = 6, since (5) + 2 = 7 has three nondecreasing partitions with exactly 2 parts: (1,6),(2,5),(3,4). The sum of the first parts of these partitions = 1 + 2 + 3 = 6. - <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Jun 08 2013</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline><a href="/A008805" title="Triangular numbers repeated.">A008805</a>:=n->(2*n+3+(-1)^n)*(2*n+7+(-1)^n)/32: seq(<a href="/A008805" title="Triangular numbers repeated.">A008805</a>(n), n=0..50); # <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Apr 22 2015</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>CoefficientList[Series[1/(1-x^2)^2/(1-x), {x, 0, 50}], x]</div> <div class=sectline>Table[Binomial[Floor[n/2] + 2, 2], {n, 0, 57}] (* <a href="/wiki/User:Michael_De_Vlieger">Michael De Vlieger</a>, Oct 03 2016 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) a(n)=(n\2+2)*(n\2+1)/2</div> <div class=sectline>(Haskell)</div> <div class=sectline>import Data.List (transpose)</div> <div class=sectline>a008805 = a000217 . (`div` 2) . (+ 1)</div> <div class=sectline>a008805_list = drop 2 $ concat $ transpose [a000217_list, a000217_list]</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Feb 01 2013</div> <div class=sectline>(Magma) [(2*n+3+(-1)^n)*(2*n+7+(-1)^n)/32 : n in [0..50]]; // <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Apr 22 2015</div> <div class=sectline>(Sage) [(2*n +3 +(-1)^n)*(2*n +7 +(-1)^n)/32 for n in (0..60)] # <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Sep 12 2019</div> <div class=sectline>(GAP) List([0..60], n-> (2*n +3 +(-1)^n)*(2*n +7 +(-1)^n)/32); # <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Sep 12 2019</div> <div class=sectline>(Python)</div> <div class=sectline>def <a href="/A008805" title="Triangular numbers repeated.">A008805</a>(n): return (m:=(n>>1)+1)*(m+1)>>1 # <a href="/wiki/User:Chai_Wah_Wu">Chai Wah Wu</a>, Oct 20 2023</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>, <a href="/A002260" title="Triangle read by rows: T(n,k) = k for n >= 1, k = 1..n.">A002260</a>, <a href="/A002620" title="Quarter-squares: a(n) = floor(n/2)*ceiling(n/2). Equivalently, a(n) = floor(n^2/4).">A002620</a>, <a href="/A006918" title="a(n) = binomial(n+3, 3)/4 for odd n, n*(n+2)*(n+4)/24 for even n.">A006918</a> (partial sums), <a href="/A054252" title="Triangle T(n,k) of n X n binary matrices with k=0..n^2 ones under action of dihedral group of the square D_4.">A054252</a>, <a href="/A135276" title="a(0)=0, a(1)=1; for n>1, a(n) = a(n-1) + n^0 if n odd, a(n) = a(n-1) + n^1 if n is even.">A135276</a>, <a href="/A142150" title="The nonnegative integers interleaved with 0's.">A142150</a>, <a href="/A158920" title="Binomial transform of A008805 (triangular numbers with repeats).">A158920</a> (binomial trans.).</div> <div class=sectline>Sequence in context: <a href="/A079551" title="a(n) = Sum_{primes p <= n} d(p-1), where d() = A000005.">A079551</a> <a href="/A182843" title="Number of composite integers greater than or equal to n whose proper divisors are all less than n.">A182843</a> <a href="/A358558" title="a(n) is the number of pairs (k,m) of positive integers with 1 <= k < m <= n such that gcd(k,m) = 2^t, t > 0.">A358558</a> * <a href="/A188270" title="Number of nondecreasing strings of numbers (x(i), i=1..n) in -2..2 with sum x(i)^3 equal to 0.">A188270</a> <a href="/A026925" title="Number of partitions of n into an odd number of parts, the greatest being 5; also, a(n+9) = number of partitions of n+4 into...">A026925</a> <a href="/A343481" title="a(n) is the sum of all digits of n in every prime base 2 <= p <= n.">A343481</a></div> <div class=sectline>Adjacent sequences: <a href="/A008802" title="Molien series for group [2,9]+ = 229.">A008802</a> <a href="/A008803" title="Molien series for group [2,10]+ = 2 2 10.">A008803</a> <a href="/A008804" title="Expansion of 1/((1-x)^2*(1-x^2)*(1-x^4)).">A008804</a> * <a href="/A008806" title="Expansion of (1+x^3)/((1-x^2)^2*(1-x^3)).">A008806</a> <a href="/A008807" title="Expansion of (1+x^5)/((1-x^2)^2*(1-x^5)).">A008807</a> <a href="/A008808" title="Expansion of (1+x^7)/((1-x^2)^2*(1-x^7)).">A008808</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span>,<span title="it is very easy to produce terms of sequence">easy</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a></div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified November 28 16:58 EST 2024. 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