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 1-2 of 2 results.

A061925 a(n) = ceiling(n^2/2) + 1.

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 19, 26, 33, 42, 51, 62, 73, 86, 99, 114, 129, 146, 163, 182, 201, 222, 243, 266, 289, 314, 339, 366, 393, 422, 451, 482, 513, 546, 579, 614, 649, 686, 723, 762, 801, 842, 883, 926, 969, 1014, 1059, 1106, 1153, 1202, 1251, 1302, 1353, 1406
Offset: 0

Views

Author

Henry Bottomley, May 17 2001

Keywords

Comments

a(n+1) gives index of the first occurrence of n in A100795. - Amarnath Murthy, Dec 05 2004
First term in each group in A074148. - Amarnath Murthy, Aug 28 2002
From Christian Barrientos, Jan 01 2021: (Start)
For n >= 3, a(n) is the number of square polyominoes with at least 2n - 2 cells whose bounding box has size 2 X n.
For n = 3, there are 6 square polyominoes with a bounding box of size 2 X 3:
_ _ _ _ _
|||_| |||_| |||_| |||_| |||_| |||_
|||_| ||| || || || || |||
(End)
Except for a(2), a(n) agrees with the lower matching number of the (n+1) X (n+1) bishop graph up to at least n = 13. - Eric W. Weisstein, Dec 23 2024

Links

Crossrefs

Cf. A100795, A074147-A074149.

Programs

Formula

a(n) = a(n-1) + 2*floor((n-1)/2) + 1 = A061926(3, k) = 2*A002620(n+1) - (n-1) = A000982(n) + 1.
a(2*n) = a(2*n-1) + 2*n - 1 = 2*n^2 + 1 = A058331(n).
a(2*n+1) = a(2*n) + 2*n + 1 = 2*(n^2 + n + 1) = A051890(n+1).
a(n) = floor((n^2+3)/2). - Gary Detlefs, Feb 13 2010
From R. J. Mathar, Feb 19 2010: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1-x^2+2*x^3)/((1+x) * (1-x)^3). (End)
a(n) = (2*n^2 - (-1)^n + 5)/4. - Bruno Berselli, Sep 29 2011
a(n) = A007590(n+1) - n + 1. - Wesley Ivan Hurt, Jul 15 2013
a(n) + a(n+1) = A027688(n). a(n+1) - a(n) = A109613(n). - R. J. Mathar, Jul 20 2013
E.g.f.: ((2 + x + x^2)*cosh(x) + (3 + x + x^2)*sinh(x))/2. - Stefano Spezia, May 07 2021

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 09 2007

A061927 a(n) = n*(n+1)*(2*n+1)*(n^2+n+3)/30.

Original entry on oeis.org

0, 1, 9, 42, 138, 363, 819, 1652, 3060, 5301, 8701, 13662, 20670, 30303, 43239, 60264, 82280, 110313, 145521, 189202, 242802, 307923, 386331, 479964, 590940, 721565, 874341, 1051974, 1257382, 1493703, 1764303, 2072784, 2422992, 2819025
Offset: 0

Views

Author

Henry Bottomley, May 17 2001

Keywords

Comments

Also number of magic labelings of the cubical graph of magic sum n-1 [Ahmed]. - R. J. Mathar, Jan 25 2007
If Y_i (i=1,2,3) are 2-blocks of a (n+3)-set X then a(n-4) is the number of 8-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Oct 28 2007
The cube graph is also the prism graph I X C_4, so this is related to the number of magic labelings of other prism & related graphs. - David J. Seal, Sep 13 2017

Links

Crossrefs

Cf. A006325, A019298, A244497, A244873, A289992, A292281, partial sums of A014820, A006975 (binomial transform shifted left).

Programs

  • Mathematica
    Table[n (n + 1) (2 n + 1) (n^2 + n + 3)/30, {n, 0, 33}] (* or *)
CoefficientList[Series[x (1 + x)^3/(-1 + x)^6, {x, 0, 33}], x] (* Michael De Vlieger, Sep 15 2017 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,9,42,138,363},40] (* Harvey P. Dale, Apr 18 2018 *)
  • PARI
    a(n) = { n*(n + 1)*(2*n + 1)*(n^2 + n + 3)/30 } \\ Harry J. Smith, Jul 29 2009

Formula

a(n) = a(n-1) + A014820(n) = A061926(9, n).
G.f.: x*(1+x)^3/(-1+x)^6 = 20/(-1+x)^5 + 1/(-1+x)^2 + 7/(-1+x)^3 + 18/(-1+x)^4 + 8/(-1+x)^6. - R. J. Mathar, Nov 18 2007
Showing 1-2 of 2 results.

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