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A005894 - 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>A005894 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A005894" 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=%2fA005894">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="A005894 - 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> A005894 </div> <div class=seqname> Centered tetrahedral numbers. <br><font size=-1>(Formerly M3850)</font> </div> </div> <div class=scorerefs> 71 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 5, 15, 35, 69, 121, 195, 295, 425, 589, 791, 1035, 1325, 1665, 2059, 2511, 3025, 3605, 4255, 4979, 5781, 6665, 7635, 8695, 9849, 11101, 12455, 13915, 15485, 17169, 18971, 20895, 22945, 25125, 27439, 29891, 32485, 35225, 38115</div> <div class=seqdatalinks> (<a href="/A005894/list">list</a>; <a href="/A005894/graph">graph</a>; <a href="/search?q=A005894+-id:A005894">refs</a>; <a href="/A005894/listen">listen</a>; <a href="/history?seq=A005894">history</a>; <a href="/search?q=id:A005894&fmt=text">text</a>; <a href="/A005894/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,2</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Binomial transform of (1,4,6,4,0,0,0,...). - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, Jul 01 2003</div> <div class=sectline>If X is an n-set and Y a fixed 4-subset of X then a(n-4) is equal to the number of 4-subsets of X intersecting Y. - <a href="/wiki/User:Milan_Janjic">Milan Janjic</a>, Jul 30 2007</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>T. D. Noe, <a href="/A005894/b005894.txt">Table of n, a(n) for n = 0..1000</a></div> <div class=sectline>Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a></div> <div class=sectline>T. P. Martin, <a href="https://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Rep., 273 (1996), 199-241, eq. (10).</div> <div class=sectline>Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de s茅ries g茅n茅ratrices et quelques conjectures</a>, Dissertation, Universit茅 du Qu茅bec 脿 Montr茅al, 1992; arXiv:0911.4975 [math.NT], 2009.</div> <div class=sectline>Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992</div> <div class=sectline>Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.</div> <div class=sectline>B. K. Teo and N. J. A. Sloane, <a href="http://neilsloane.com/doc/magic1/magic1.html">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985), 4545-4558.</div> <div class=sectline><a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -6, 4, -1).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = (2*n + 1)*(n^2 + n + 3)/3.</div> <div class=sectline>G.f.: (1+x)*(1+x^2)/(1-x)^4.</div> <div class=sectline>a(n) = C(n, 0) + 4*C(n, 1) + 6*C(n, 2) + 4*C(n, 3). - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, Jul 01 2003</div> <div class=sectline>a(n) is the sum of 4 consecutive tetrahedral (or pyramidal) numbers: C(n+3,3) = (n+1)*(n+2)*(n+3)/6 = <a href="/A000292" title="Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.">A000292</a>(n). a(n) = <a href="/A000292" title="Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.">A000292</a>(n-3) + <a href="/A000292" title="Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.">A000292</a>(n-2) + <a href="/A000292" title="Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.">A000292</a>(n-1) + <a href="/A000292" title="Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.">A000292</a>(n). - <a href="/wiki/User:Alexander_Adamchuk">Alexander Adamchuk</a>, May 20 2006</div> <div class=sectline>a(n) = binomial(n+3,n) + binomial(n+2,n-1) + binomial(n+1,n-2) + binomial(n,n-3). (modified by <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Nov 30 2017)</div> <div class=sectline>a(n) = a(n-1) + 2*n^2 + 2, n>=1 (first differences <a href="/A005893" title="Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).">A005893</a>). - <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Mar 27 2011</div> <div class=sectline>a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=5, a(2)=15, a(3)=35. - <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Nov 03 2011</div> <div class=sectline>E.g.f.: (3 + 12*x + 9*x^2 + 2*x^3)*exp(x)/3. - <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Nov 30 2017</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline><a href="/A005894" title="Centered tetrahedral numbers.">A005894</a>:=(z+1)*(1+z**2)/(z-1)**4; # <a href="/wiki/User:Simon_Plouffe">Simon Plouffe</a> in his 1992 dissertation</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[(2n+1)(n^2+n+3)/3, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 5, 15, 35}, 40] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Nov 03 2011 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) a(n)=(2*n+1)*(n^2+n+3)/3 \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Sep 24 2015</div> <div class=sectline>(Magma) [(2*n+1)*(n^2+n+3)/3: n in [0..30]]; // <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Nov 30 2017</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>(1/12)*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives <a href="/A049480" title="a(n) = (2*n-1)*(n^2 -n +6)/6.">A049480</a>, <a href="/A005894" title="Centered tetrahedral numbers.">A005894</a>, <a href="/A063488" title="a(n) = (2*n-1)*(n^2 -n +2)/2.">A063488</a>, <a href="/A001845" title="Centered octahedral numbers (crystal ball sequence for cubic lattice).">A001845</a>, <a href="/A063489" title="a(n) = (2*n-1)*(5*n^2-5*n+6)/6.">A063489</a>, <a href="/A005898" title="Centered cube numbers: n^3 + (n+1)^3.">A005898</a>, <a href="/A063490" title="a(n) = (2*n - 1)*(7*n^2 - 7*n + 6)/6.">A063490</a>, <a href="/A057813" title="a(n) = (2*n+1)*(4*n^2+4*n+3)/3.">A057813</a>, <a href="/A063491" title="a(n) = (2*n - 1)*(3*n^2 - 3*n + 2)/2.">A063491</a>, <a href="/A005902" title="Centered icosahedral (or cuboctahedral) numbers, also crystal ball sequence for f.c.c. lattice.">A005902</a>, <a href="/A063492" title="a(n) = (2*n - 1)*(11*n^2 - 11*n + 6)/6.">A063492</a>, <a href="/A005917" title="Rhombic dodecahedral numbers: a(n) = n^4 - (n - 1)^4.">A005917</a>, <a href="/A063493" title="a(n) = (2*n-1)*(13*n^2-13*n+6)/6.">A063493</a>, <a href="/A063494" title="a(n) = (2*n - 1)*(7*n^2 - 7*n + 3)/3.">A063494</a>, <a href="/A063495" title="a(n) = (2*n-1)*(5*n^2-5*n+2)/2.">A063495</a>, <a href="/A063496" title="a(n) = (2*n - 1)*(8*n^2 - 8*n + 3)/3.">A063496</a>.</div> <div class=sectline>Cf. <a href="/A000292" title="Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.">A000292</a>.</div> <div class=sectline>The 28 uniform 3D tilings: cab: <a href="/A299266" title="Coordination sequence for "cab" 3D uniform tiling formed from octahedra and truncated cubes.">A299266</a>, <a href="/A299267" title="Partial sums of A299266.">A299267</a>; crs: <a href="/A299268" title="Coordination sequence for "crs" 3D uniform tiling formed from tetrahedra and truncated tetrahedra.">A299268</a>, <a href="/A299269" title="Partial sums of A299268.">A299269</a>; fcu: <a href="/A005901" title="Number of points on surface of cuboctahedron (or icosahedron): a(0) = 1; for n > 0, a(n) = 10n^2 + 2. Also coordination sequ...">A005901</a>, <a href="/A005902" title="Centered icosahedral (or cuboctahedral) numbers, also crystal ball sequence for f.c.c. lattice.">A005902</a>; fee: <a href="/A299259" title="Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 4.8.8 2D tiling (cf. A008576).">A299259</a>, <a href="/A299265" title="Partial sums of A299259.">A299265</a>; flu-e: <a href="/A299272" title="Coordination sequence for "flu" 3D uniform tiling formed from tetrahedra, rhombicuboctahedra, and cubes.">A299272</a>, <a href="/A299273" title="Partial sums of A299272.">A299273</a>; fst: <a href="/A299258" title="Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 4.6.12 2D tiling (cf. A072154).">A299258</a>, <a href="/A299264" title="Partial sums of A299258.">A299264</a>; hal: <a href="/A299274" title="Coordination sequence for "hal" 3D uniform tiling.">A299274</a>, <a href="/A299275" title="Partial sums of A299274.">A299275</a>; hcp: <a href="/A007899" title="Coordination sequence for hexagonal close-packing.">A007899</a>, <a href="/A007202" title="Crystal ball sequence for hexagonal close-packing.">A007202</a>; hex: <a href="/A005897" title="a(n) = 6*n^2 + 2 for n > 0, a(0)=1.">A005897</a>, <a href="/A005898" title="Centered cube numbers: n^3 + (n+1)^3.">A005898</a>; kag: <a href="/A299256" title="Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.6.3.6 2D tiling (cf. A008579).">A299256</a>, <a href="/A299262" title="Partial sums of A299256.">A299262</a>; lta: <a href="/A008137" title="Coordination sequence T1 for Zeolite Code LTA and RHO.">A008137</a>, <a href="/A299276" title="Partial sums of A008137.">A299276</a>; pcu: <a href="/A005899" title="Number of points on surface of octahedron; also coordination sequence for cubic lattice: a(0) = 1; for n > 0, a(n) = 4n^2 + 2.">A005899</a>, <a href="/A001845" title="Centered octahedral numbers (crystal ball sequence for cubic lattice).">A001845</a>; pcu-i: <a href="/A299277" title="Coordination sequence for "pcu-i" 3D uniform tiling.">A299277</a>, <a href="/A299278" title="Partial sums of A299277.">A299278</a>; reo: <a href="/A299279" title="Coordination sequence for "reo" 3D uniform tiling.">A299279</a>, <a href="/A299280" title="Partial sums of A299279.">A299280</a>; reo-e: <a href="/A299281" title="Coordination sequence for "reo-e" 3D uniform tiling.">A299281</a>, <a href="/A299282" title="Partial sums of A299281.">A299282</a>; rho: <a href="/A008137" title="Coordination sequence T1 for Zeolite Code LTA and RHO.">A008137</a>, <a href="/A299276" title="Partial sums of A008137.">A299276</a>; sod: <a href="/A005893" title="Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).">A005893</a>, <a href="/A005894" title="Centered tetrahedral numbers.">A005894</a>; sve: <a href="/A299255" title="Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.3.4.3.4 2D tiling (cf. A219529).">A299255</a>, <a href="/A299261" title="Partial sums of A299255.">A299261</a>; svh: <a href="/A299283" title="Coordination sequence for "svh" 3D uniform tiling.">A299283</a>, <a href="/A299284" title="Partial sums of A299283.">A299284</a>; svj: <a href="/A299254" title="Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3^4.6 2D tiling (cf. A250120).">A299254</a>, <a href="/A299260" title="Partial sums of A299254.">A299260</a>; svk: <a href="/A010001" title="a(0) = 1, a(n) = 5*n^2 + 2 for n>0.">A010001</a>, <a href="/A063489" title="a(n) = (2*n-1)*(5*n^2-5*n+6)/6.">A063489</a>; tca: <a href="/A299285" title="Coordination sequence for "tea" 3D uniform tiling.">A299285</a>, <a href="/A299286" title="Partial sums of A299285.">A299286</a>; tcd: <a href="/A299287" title="Coordination sequence for "tcd" 3D uniform tiling.">A299287</a>, <a href="/A299288" title="Partial sums of A299287.">A299288</a>; tfs: <a href="/A005899" title="Number of points on surface of octahedron; also coordination sequence for cubic lattice: a(0) = 1; for n > 0, a(n) = 4n^2 + 2.">A005899</a>, <a href="/A001845" title="Centered octahedral numbers (crystal ball sequence for cubic lattice).">A001845</a>; tsi: <a href="/A299289" title="Coordination sequence for "tsi" 3D uniform tiling.">A299289</a>, <a href="/A299290" title="Partial sums of A299289.">A299290</a>; ttw: <a href="/A299257" title="Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.12.12 2D tiling (cf. A250122).">A299257</a>, <a href="/A299263" title="Partial sums of A299257.">A299263</a>; ubt: <a href="/A299291" title="Coordination sequence for "ubt" 3D uniform tiling.">A299291</a>, <a href="/A299292" title="Partial sums of A299291.">A299292</a>; bnn: <a href="/A007899" title="Coordination sequence for hexagonal close-packing.">A007899</a>, <a href="/A007202" title="Crystal ball sequence for hexagonal close-packing.">A007202</a>. See the Proserpio link in <a href="/A299266" title="Coordination sequence for "cab" 3D uniform tiling formed from octahedra and truncated cubes.">A299266</a> for overview.</div> <div class=sectline>Sequence in context: <a href="/A061829" title="Multiples of 5 having only odd digits.">A061829</a> <a href="/A063382" title="a(1) = 5, a(n) = concatenation of two closest factors of a(n-1) whose product equals a(n-1) or if a(n-1) is a prime then the...">A063382</a> <a href="/A069983" title="Number of 5-gonal compositions of n into positive parts.">A069983</a> * <a href="/A374711" title="Number of distinct sums i^3 + j^3 + k^3 + l^3 for 0<=i<=j<=k<=l<=n.">A374711</a> <a href="/A015622" title="Quadruples of different integers from [ 1,n ] with no global factor.">A015622</a> <a href="/A346761" title="a(n) = Sum_{d|n} mu(n/d) * binomial(d,4).">A346761</a></div> <div class=sectline>Adjacent sequences: <a href="/A005891" title="Centered pentagonal numbers: (5n^2+5n+2)/2; crystal ball sequence for 3.3.3.4.4. planar net.">A005891</a> <a href="/A005892" title="Truncated square numbers: 7*n^2 + 4*n + 1.">A005892</a> <a href="/A005893" title="Number of points on surface of tetrahedron; coordination sequence for sodalite net (equals 2*n^2+2 for n > 0).">A005893</a> * <a href="/A005895" title="Weighted count of partitions with distinct parts.">A005895</a> <a href="/A005896" title="Weighted count of partitions with odd parts.">A005896</a> <a href="/A005897" title="a(n) = 6*n^2 + 2 for n > 0, a(0)=1.">A005897</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>,<span title="an exceptionally nice sequence">nice</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 April 11 08:48 EDT 2025. 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