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.

A271870 Convolution of nonzero hexagonal numbers (A000384) with themselves.

Original entry on oeis.org

1, 12, 66, 236, 651, 1512, 3108, 5832, 10197, 16852, 26598, 40404, 59423, 85008, 118728, 162384, 218025, 287964, 374794, 481404, 610995, 767096, 953580, 1174680, 1435005, 1739556, 2093742, 2503396, 2974791, 3514656, 4130192, 4829088, 5619537, 6510252, 7510482, 8630028
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 20 2016

Keywords

Links

Crossrefs

Cf. A000384.
Cf. similar sequences of the convolution of k-gonal numbers with themselves listed in A271662.

Programs

  • Magma
    [(n+1)*(n+2)*(n+3)*(4*n^2+6*n+5)/30 : n in [0..40]]; // Wesley Ivan Hurt, Apr 20 2016
  • Maple
    A271870:=n->(n+1)*(n+2)*(n+3)*(4*n^2+6*n+5)/30: seq(A271870(n), n=0..50); # Wesley Ivan Hurt, Apr 20 2016
  • Mathematica
    LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 12, 66, 236, 651, 1512}, 36]
Table[(n + 1) (n + 2) (n + 3) ((4 n^2 + 6 n + 5)/30), {n, 0, 35}]
  • PARI
    a(n)=binomial(n+3,3)*(4*n^2 + 6*n + 5)/5 \\ Charles R Greathouse IV, Jul 26 2016

Formula

O.g.f.: (1 + 3*x)^2/(1 - x)^6.
E.g.f.: (30 + 330*x + 645*x^2 + 365*x^3 + 70*x^4 + 4*x^5)*exp(x)/30.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) = (n + 1)*(n + 2)*(n + 3)*(4*n^2 + 6*n + 5)/30.

Extensions

a(35)=8630028 corrected by Georg Fischer, Apr 03 2019

A272103 Convolution of nonzero heptagonal numbers (A000566) with themselves.

Original entry on oeis.org

1, 14, 85, 320, 910, 2156, 4494, 8520, 15015, 24970, 39611, 60424, 89180, 127960, 179180, 245616, 330429, 437190, 569905, 733040, 931546, 1170884, 1457050, 1796600, 2196675, 2665026, 3210039, 3840760, 4566920, 5398960, 6348056, 7426144, 8645945, 10020990, 11565645, 13295136
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 20 2016

Keywords

Links

Crossrefs

Cf. A000566.
Cf. similar sequences of the convolution of k-gonal numbers with themselves listed in A271662.

Programs

  • Mathematica
    LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 14, 85, 320, 910, 2156}, 36]
Table[(n + 1) (n + 2) (n + 3) ((5 n^2 + 5 n + 4)/24), {n, 0, 35}]

Formula

O.g.f.: (1 + 4*x)^2/(1 - x)^6.
E.g.f.: (24 + 312*x + 696*x^2 + 424*x^3 + 85*x^4 + 5*x^5)*exp(x)/24.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) = (n + 1)*(n + 2)*(n + 3)*(5*n^2 + 5*n + 4)/24.

A276159 Convolution of nonzero octagonal numbers (A000567) with themselves.

Original entry on oeis.org

1, 16, 106, 416, 1211, 2912, 6132, 11712, 20757, 34672, 55198, 84448, 124943, 179648, 252008, 345984, 466089, 617424, 805714, 1037344, 1319395, 1659680, 2066780, 2550080, 3119805, 3787056, 4563846, 5463136, 6498871, 7686016, 9040592, 10579712, 12321617, 14285712, 16492602, 18964128
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 05 2016

Keywords

Links

Crossrefs

Cf. A000567.
Cf. similar sequences of the convolution of k-gonal numbers with themselves listed in A271662.

Programs

  • Mathematica
    LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 16, 106, 416, 1211, 2912}, 36]
Table[(n + 1) (n + 2) (n + 3) ((9 n^2 + 6 n + 5)/30), {n, 0, 35}]

Formula

O.g.f.: (1 + 5*x)^2/(1 - x)^6.
E.g.f.: (30 + 450*x + 1125*x^2 + 725*x^3 + 150*x^4 + 9*x^5)*exp(x)/30.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) = (n + 1)*(n + 2)*(n + 3)*(9*n^2 + 6*n + 5)/30.
Showing 1-3 of 3 results.

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