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.

A061000 a(n) = Sum_{k=1..n} A060999(k) * k^3.

Original entry on oeis.org

0, 8, 89, 537, 2287, 7471, 20505, 49689, 108738, 219738, 416726, 748502, 1284570, 2121490, 3387115, 5250795, 7933293, 11712429, 16938987, 24050987, 33580556, 46177140, 62626924, 83860588, 110985588, 145311516, 188358237, 241899165
Offset: 1

Views

Author

N. J. A. Sloane, May 15 2001

Keywords

Comments

Original name: x.v where x = first n terms of A060999, v = [1,8,27,...,n^3].

Examples

			a(3) = 0*1 + 1*8 + 3*27 = 89.

Crossrefs

Cf. A000578, A060999.

Programs

  • Mathematica
    nn=30;With[{c=Range[nn]^3,d=Table[Floor[(n+1)^3/9+1/2],{n,0,nn-1}]},Table[Take[c,n].Take[d,n],{n,nn}]] (* Harvey P. Dale, Jan 20 2013 *)

Formula

a(n) = Sum_{k=1..n} A060999(k) * A000578(k). - Sean A. Irvine, Jan 13 2023

A061001 Partial sums of the squares of the terms of A060999.

Original entry on oeis.org

0, 1, 10, 59, 255, 831, 2275, 5524, 12085, 24406, 46310, 83174, 142710, 235735, 376360, 583385, 881501, 1301405, 1882049, 2672370, 3731211, 5130700, 6958604, 9317900, 12331596, 16145805, 20928774, 26877495, 34221595, 43221595
Offset: 1

Views

Author

N. J. A. Sloane, May 15 2001

Keywords

Crossrefs

Cf. A060999.

Formula

a(n) = Sum_{k=1..n} A060999(k)^2. - Sean A. Irvine, Jan 13 2023

Extensions

Title clarified by Sean A. Irvine, Jan 13 2023

A213396 Number of (w,x,y) with all terms in {0,...,n} and 2*w < |x+y-w|.

Original entry on oeis.org

0, 3, 9, 21, 42, 72, 114, 171, 243, 333, 444, 576, 732, 915, 1125, 1365, 1638, 1944, 2286, 2667, 3087, 3549, 4056, 4608, 5208, 5859, 6561, 7317, 8130, 9000, 9930, 10923, 11979, 13101, 14292, 15552, 16884, 18291, 19773, 21333, 22974, 24696
Offset: 0

Views

Author

Clark Kimberling, Jun 12 2012

Keywords

Comments

For a guide to related sequences, see A212959.
Also, integer values of (m^3+1)/3 for m>0. - Bruno Berselli, Jan 19 2013

Links

Crossrefs

Cf. A060999, A212959, A213397.

Programs

  • Mathematica
    t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[2 w < Abs[x + y - w], s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]]; m = Map[t[#] &, Range[0, 60]]
CoefficientList[ Series[(3 (x + x^3))/((-1 + x)^4 (1 + x + x^2)), {x, 0, 41}], x] (* or *)
LinearRecurrence[{3, -3, 2, -3, 3, -1}, {0, 3, 9, 21, 42, 72}, 41] (* Robert G. Wilson v, Dec 22 2017 *)
  • PARI
    x='x+O('x^99); concat([0], Vec(3*x*(1+x^2)/((1-x)^4*(1+x+x^2)))) \\ Altug Alkan, Dec 22 2017

Formula

a(n) = (n+2)*(n+1)*n/3 + floor((n-1)/3) + 1.
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6).
G.f.: 3*x*(1 + x^2)/((1 - x)^4*(1 + x + x^2)).
a(n) + A213397(n) = (n+1)^3.
a(n) = 3*A060999(n). - Bruno Berselli, Dec 22 2017

Extensions

Corrected the title. Robert G. Wilson v, Dec 22 2017
Showing 1-3 of 3 results.

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

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