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A214661 - 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>A214661 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A214661" 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=%2fA214661">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="A214661 - OEIS"></a> </div> <div class="motdbox"> <div class="motd"> <p>Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).</p> </div> <div class="donate"> <div id="donate-button-container"> <div id="donate-button"></div> <script src="https://www.paypalobjects.com/donate/sdk/donate-sdk.js" charset="UTF-8"></script> <script> PayPal.Donation.Button({ env:'production', hosted_button_id:'SVPGSDDCJ734A', image: { src:'https://www.paypalobjects.com/en_US/i/btn/btn_donateCC_LG.gif', alt:'Donate with PayPal button', title:'PayPal - The safer, easier way to pay online!', } }).render('#donate-button'); </script> </div> <a href="https://oeisf.org/donate/"> <strong>Other ways to Give</strong> </a> </div> </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> A214661 </div> <div class=seqname> Odd numbers obtained by transposing the left half of <a href="/A176271" title="The odd numbers as a triangle read by rows.">A176271</a> into rows of a triangle: T(n,k) = <a href="/A176271" title="The odd numbers as a triangle read by rows.">A176271</a>(n - 1 + k, k), 1 <= k <= n. </div> </div> <div class=scorerefs> 9 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 3, 9, 7, 15, 25, 13, 23, 35, 49, 21, 33, 47, 63, 81, 31, 45, 61, 79, 99, 121, 43, 59, 77, 97, 119, 143, 169, 57, 75, 95, 117, 141, 167, 195, 225, 73, 93, 115, 139, 165, 193, 223, 255, 289, 91, 113, 137, 163, 191, 221, 253, 287, 323, 361, 111, 135, 161, 189, 219, 251, 285, 321, 359, 399, 441</div> <div class=seqdatalinks> (<a href="/A214661/list">list</a>; <a href="/A214661/table">table</a>; <a href="/A214661/graph">graph</a>; <a href="/search?q=A214661+-id:A214661">refs</a>; <a href="/A214661/listen">listen</a>; <a href="/history?seq=A214661">history</a>; <a href="/search?q=id:A214661&fmt=text">text</a>; <a href="/A214661/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,2</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Reinhard Zumkeller, <a href="/A214661/b214661.txt">Rows n = 1..150 of triangle, flattened</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>T(n, k) = (n+k)^2 - 3*n - k + 1.</div> <div class=sectline>T(n,k) = <a href="/A176271" title="The odd numbers as a triangle read by rows.">A176271</a>(n+k-1, k).</div> <div class=sectline>T(n, k) = <a href="/A214604" title="Odd numbers by transposing the right half of A176271, triangle read by rows: T(n,k) = A176271(n - 1 + k, n), 1 <= k <= n.">A214604</a>(n,k) - 2*<a href="/A025581" title="Triangle read by rows: T(n, k) = n-k, for 0 <= k <= n.">A025581</a>(n,k).</div> <div class=sectline>T(n, k) = 2*<a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>(<a href="/A094727" title="Triangle read by rows: T(n,k) = n + k, 0 <= k < n, n >= 1.">A094727</a>(n,k)) - <a href="/A214604" title="Odd numbers by transposing the right half of A176271, triangle read by rows: T(n,k) = A176271(n - 1 + k, n), 1 <= k <= n.">A214604</a>(n,k).</div> <div class=sectline>T(2*n-1, n) = <a href="/A214675" title="a(n) = 9*n^2 - 13*n + 5.">A214675</a>() (main diagonal).</div> <div class=sectline>T(n,1) = <a href="/A002061" title="Central polygonal numbers: a(n) = n^2 - n + 1.">A002061</a>(n).</div> <div class=sectline>T(n,n) = <a href="/A016754" title="Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.">A016754</a>(n-1).</div> <div class=sectline>Sum_{k=1..n} T(n, k) = <a href="/A051673" title="Cubic star numbers: a(n) = n^3 + 4*Sum_{i=0..n-1} i^2.">A051673</a>(n) (row sums).</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>. Take the first n elements of the n-th diagonal (northwest to</div> <div class=sectline>. southeast) of the triangle on the left side</div> <div class=sectline>. and write this as n-th row on the triangle of the right side.</div> <div class=sectline>. 1: 1 1</div> <div class=sectline>. 2: 3 _ 3 9</div> <div class=sectline>. 3: 7 9 __ 7 15 25</div> <div class=sectline>. 4: 13 15 __ __ 13 23 35 49</div> <div class=sectline>. 5: 21 23 25 __ __ 21 33 47 63 ..</div> <div class=sectline>. 6: 31 33 35 __ __ __ 31 45 61 .. .. ..</div> <div class=sectline>. 7: 43 45 47 49 __ __ __ 43 59 .. .. .. .. ..</div> <div class=sectline>. 8: 57 59 61 63 __ __ __ __ 57 .. .. .. .. .. .. .. .</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[(n+k)^2-3*n-k+1, {n, 15}, {k, n}]//Flatten (* <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Mar 10 2024 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(Haskell)</div> <div class=sectline>import Data.List (transpose)</div> <div class=sectline>a214661 n k = a214661_tabl !! (n-1) !! (k-1)</div> <div class=sectline>a214661_row n = a214661_tabl !! (n-1)</div> <div class=sectline>a214661_tabl = zipWith take [1..] $ transpose $ map reverse a176271_tabl</div> <div class=sectline>(Magma) [(n+k)^2-3*n-k+1: k in [1..n], n in [1..15]]; // <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Mar 10 2024</div> <div class=sectline>(SageMath) flatten([[(n+k)^2-3*n-k+1 for k in range(1, n+1)] for n in range(1, 16)]) // <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Mar 10 2024</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000290" title="The squares: a(n) = n^2.">A000290</a>, <a href="/A002061" title="Central polygonal numbers: a(n) = n^2 - n + 1.">A002061</a>, <a href="/A016754" title="Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.">A016754</a>, <a href="/A025581" title="Triangle read by rows: T(n, k) = n-k, for 0 <= k <= n.">A025581</a>, <a href="/A094727" title="Triangle read by rows: T(n,k) = n + k, 0 <= k < n, n >= 1.">A094727</a>, <a href="/A176271" title="The odd numbers as a triangle read by rows.">A176271</a>, <a href="/A214604" title="Odd numbers by transposing the right half of A176271, triangle read by rows: T(n,k) = A176271(n - 1 + k, n), 1 <= k <= n.">A214604</a>.</div> <div class=sectline>Cf. <a href="/A051673" title="Cubic star numbers: a(n) = n^3 + 4*Sum_{i=0..n-1} i^2.">A051673</a> (row sums), <a href="/A214675" title="a(n) = 9*n^2 - 13*n + 5.">A214675</a> (main diagonal).</div> <div class=sectline>Sequence in context: <a href="/A267363" title="a(0) = 0; a(odd n) = 2n+1; a(even n) = a(n/2 - a(n-1)).">A267363</a> <a href="/A146179" title="Digit sums of Cullen numbers.">A146179</a> <a href="/A294734" title="Lexicographically earliest sequence of distinct positive odd numbers such that, for any n > 1, n divides a(n-1) + a(n).">A294734</a> * <a href="/A178414" title="Least odd number in the Collatz (3x+1) preimage of odd numbers not a multiple of 3.">A178414</a> <a href="/A220654" title="Smallest number that can be written in binary representation in exactly n ways as concatenation of palindromes.">A220654</a> <a href="/A302158" title="Number of nX3 0..1 arrays with every element equal to 0, 1 or 3 horizontally, diagonally or antidiagonally adjacent elements...">A302158</a></div> <div class=sectline>Adjacent sequences: <a href="/A214658" title="Numbers n such that 24^n - 23^n is prime or a strong pseudoprime.">A214658</a> <a href="/A214659" title="a(n) = n*(7*n^2 - 3*n - 1)/3.">A214659</a> <a href="/A214660" title="a(n) = 9*n^2 - 11*n + 3.">A214660</a> * <a href="/A214662" title="Greatest prime divisor of 1 + 2^2 + 3^3 + ... + n^n.">A214662</a> <a href="/A214663" title="Number of permutations of 1..n for which the partial sums of signed displacements do not exceed 2.">A214663</a> <a href="/A214664" title="The x-coordinates of prime numbers in an Ulam spiral oriented counterclockwise with first step east.">A214664</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="typically a triangle of numbers, made into a sequence by reading it row by row"><a href="/A214661/table">tabl</a></span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Jul 25 2012</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 December 3 15:56 EST 2024. Contains 378391 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>