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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.

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A258582 a(n) = n*(2*n + 1)*(4*n + 1)/3.

Original entry on oeis.org

0, 5, 30, 91, 204, 385, 650, 1015, 1496, 2109, 2870, 3795, 4900, 6201, 7714, 9455, 11440, 13685, 16206, 19019, 22140, 25585, 29370, 33511, 38024, 42925, 48230, 53955, 60116, 66729, 73810, 81375, 89440, 98021, 107134, 116795, 127020, 137825, 149226, 161239, 173880
Offset: 0

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Author

Ilya Gutkovskiy, Nov 06 2015

Keywords

Comments

First bisection of the square pyramidal numbers (A000330).

Links

Crossrefs

Cf. A000330, A001477, A005408, A016813, A053126 (partial sums), A100157.

Programs

  • Magma
    [n*(2*n+1)*(4*n+1)/3: n in [0..50]]; // Wesley Ivan Hurt, Nov 17 2015
  • Maple
    A258582:=n->n*(2*n + 1)*(4*n + 1)/3: seq(A258582(n), n=0..50); # Wesley Ivan Hurt, Nov 17 2015
  • Mathematica
    Table[(1/3) n (2 n + 1) (4 n + 1), {n, 0, 45}]
  • PARI
    vector(100, n, n--; n*(2*n+1)*(4*n+1)/3) \\ Altug Alkan, Nov 06 2015
  • PARI
    concat(0, Vec((5*x + 10*x^2 + x^3)/(1 - x)^4 + O(x^50))) \\ Altug Alkan, Nov 06 2015

Formula

G.f.: x*(5 + 10*x + x^2)/(1 - x)^4.
a(n) = A000330(2*n).
Sum_{n>0} 1/a(n) = 3*(6 - Pi - 4*log(2)) = 0.25745587...
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3. - Vincenzo Librandi, Nov 18 2015
a(n) = A006918(4*n-1) = A053307(4*n-1) = A228706(4*n-1) for n>0. - Bruno Berselli, Nov 18 2015
a(n) = Sum_{k=1..2*n} k^2 (see the first comment). E.g.f. exp(x)*(5*x+ 20*x^2/2+16*x^3/3!). - Wolfdieter Lang, Mar 13 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2) + 6*sqrt(2)*log(1+sqrt(2)) + 3*(sqrt(2)-1/2)*Pi - 18. - Amiram Eldar, Sep 17 2022

A228707 G.f.: (1-3*x+5*x^2-5*x^3+5*x^4-5*x^5+5*x^6-3*x^7+x^8)/((1-x)^4*(1+x^4)*(1+x^2)^2).

Original entry on oeis.org

1, 1, 1, 3, 6, 8, 10, 16, 24, 29, 35, 47, 61, 72, 84, 104, 127, 145, 165, 195, 228, 256, 286, 328, 374, 413, 455, 511, 571, 624, 680, 752, 829, 897, 969, 1059, 1154, 1240, 1330, 1440, 1556, 1661, 1771, 1903, 2041, 2168, 2300, 2456, 2619, 2769
Offset: 0

Views

Author

N. J. A. Sloane, Sep 06 2013

Keywords

Links

Crossrefs

Cf. A032279, A228706.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x+5*x^2-5*x^3+5*x^4-5*x^5+5*x^6-3*x^7+x^8)/((1-x)^4*(1+x^4)*(1+x^2)^2))); // Vincenzo Librandi, Sep 07 2013
  • Mathematica
    CoefficientList[Series[(1 - 3 x + 5 x^2 - 5 x^3 + 5 x^4 - 5 x^5 + 5 x^6 - 3 x^7 + x^8) / ((1 - x)^4 (1 + x^4) (1 + x^2)^2), {x, 0, 50}],x] (* Vincenzo Librandi, Sep 07 2013 *)

Formula

G.f.: (1-x+x^2)*(1-2 *x+2*x^2-x^3+2*x^4-2*x^5+x^6)/((1+x^2)^2*(1-x)^4*(1+x^4)).
Showing 1-2 of 2 results.

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

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