cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A027441 a(n) = (n^4 + n)/2 (Row sums of an n X n X n magic cube, when it exists).

Original entry on oeis.org

0, 1, 9, 42, 130, 315, 651, 1204, 2052, 3285, 5005, 7326, 10374, 14287, 19215, 25320, 32776, 41769, 52497, 65170, 80010, 97251, 117139, 139932, 165900, 195325, 228501, 265734, 307342, 353655, 405015, 461776, 524304, 592977, 668185, 750330, 839826, 937099
Offset: 0

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Author

Eric W. Weisstein

Keywords

Comments

Starting with offset 1 = binomial transform of (1, 8, 25, 30, 12, 0, 0, 0, ...). - Gary W. Adamson, May 20 2009

Links

Crossrefs

Subsequence of A057590.
Cf. A002061, A139562, A179268.

Programs

Formula

O.g.f.: x*(1+4*x+7*x^2)/(1-x)^5. - R. J. Mathar, Feb 13 2008
a(n) = Sum_{k=n..n^2} k; for n>0: a(n) = A037270(n) - A000217(n-1). - Reinhard Zumkeller, Jul 06 2010
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Aug 13 2014
a(n) = A071232(n) / n^2, for n > 0. - Wesley Ivan Hurt, Aug 13 2014
a(n) = A002061(n)*A000217(n). - Anton Zakharov, Dec 16 2016
a(n) = (n+1)*(a(n-1)/(n-1) + n*(n-1)), a(0)=0, a(1)=1. - Vladimir Kruchinin, Oct 10 2018

Extensions

More terms from Wesley Ivan Hurt, Aug 13 2014

A168124 a(n) = n^2*(n^6+1)/2.

Original entry on oeis.org

0, 1, 130, 3285, 32776, 195325, 839826, 2882425, 8388640, 21523401, 50000050, 107179501, 214990920, 407865445, 737894626, 1281445425, 2147483776, 3487878865, 5509980450, 8491781701, 12800000200, 18911429901, 27437937010, 39155492905, 55037657376
Offset: 0

Views

Author

N. J. A. Sloane, Dec 11 2009

Keywords

Links

Crossrefs

Subsequence of A027441 and A057590.

Programs

  • Magma
    [n^2*(n^6+1)/2: n in [0..25]]; // Vincenzo Librandi, Jul 14 2016
  • Mathematica
    Table[n^2*(n^6 + 1)/2, {n, 0, 50}] (* G. C. Greubel, Jul 13 2016 *)

Formula

O.g.f.: x*(1 + x)*(1 + 120*x + 2031*x^2 + 5776*x^3 + 2031*x^4 + 120*x^5 + x^6)/(1 - x)^9. - Bruno Berselli, Jul 14 2016
E.g.f.: x*(2 + 128*x + 966*x^2 + 1701*x^3 + 1050*x^4 + 266*x^5 + 28*x^6 + x^7)*exp(x)/2. - Bruno Berselli, Jul 14 2016
a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - Wesley Ivan Hurt, Jun 09 2023

A361263 Numbers of the form k*(k^5 +- 1)/2.

Original entry on oeis.org

0, 1, 31, 33, 363, 366, 2046, 2050, 7810, 7815, 23325, 23331, 58821, 58828, 131068, 131076, 265716, 265725, 499995, 500005, 885775, 885786, 1492986, 1492998, 2413398, 2413411, 3764761, 3764775, 5695305, 5695320, 8388600, 8388616, 12068776, 12068793, 17006103, 17006121, 23522931, 23522950
Offset: 1

Views

Author

Thomas Scheuerle, Mar 06 2023

Keywords

Comments

Integer solutions of x + y = (x - y)^6. If x = a(n) then y = a(n - (-1)^n).

Links

Crossrefs

Cf. A006003, A021003, A027441, A057587, A057590, A135503, A168029.
Cf. A167963 (bisection).

Programs

  • Maple
    map(k -> (k*(k^5-1)/2, k*(k^5+1)/2), [$1..100]);
  • PARI
    concat(0, Vec(x^2*(1+30*x-4*x^2+150*x^3+6*x^4+150*x^5-4*x^6+30*x^7+x^8)/((1-x)^7*(1+x)^6) + O(x^100)))
  • Python
    def A361263(n): return (k:=n+1>>1)*(k**5+1-((n&1)<<1))>>1 # Chai Wah Wu, Mar 22 2023

Formula

G.f.: x^2*(1+30*x-4*x^2+150*x^3+6*x^4+150*x^5-4*x^6+30*x^7+x^8) / ((1-x)^7*(1+x)^6).
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - 15*a(n-4) + 15*a(n-5) + 20*a(n-6) - 20*a(n-7) - 15*a(n-8) + 15*a(n-9) + 6*a(n-10) - 6*a(n-11) - a(n-12) + a(n-13).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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