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.

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A256859 a(n) = n*(n + 1)*(n + 2)*(n^2 - n + 4)/24.

Original entry on oeis.org

1, 6, 25, 80, 210, 476, 966, 1800, 3135, 5170, 8151, 12376, 18200, 26040, 36380, 49776, 66861, 88350, 115045, 147840, 187726, 235796, 293250, 361400, 441675, 535626, 644931, 771400, 916980, 1083760, 1273976, 1490016, 1734425, 2009910, 2319345, 2665776, 3052426
Offset: 1

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Author

Luciano Ancora, Apr 14 2015

Keywords

Comments

This is the case k = n of b(n,k) = n*(n+1)*(n+2)*(k*(n-1)+4)/24, where b(n,k) is the n-th hypersolid number in 4 dimensions generated from an arithmetical progression with the first term 1 and common difference k. Therefore, the sequence is the main diagonal of the Table 3 in Sardelis et al. paper (see Links field).

Links

Crossrefs

Cf. A000389, A261721.
Cf. similar sequences of the form binomial(n+k-2,k-1)+n*binomial(n+k-2,k): A006000 (k=2); A257055 (k=3); this sequence (k=4); A256860 (k=5); A256861 (k=6).

Programs

  • Magma
    [n*(n + 1)*(n + 2)*(n^2 - n + 4)/24: n in [1..30]]; // G. C. Greubel, Nov 23 2017
  • Mathematica
    Table[n (n + 1) (n + 2) (n^2 - n + 4)/24, {n, 40}]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,6,25,80,210,476},40] (* Harvey P. Dale, Mar 19 2022 *)
  • PARI
    vector(40, n, n*(n+1)*(n+2)*(n^2-n+4)/24) \\ Bruno Berselli, Apr 15 2015

Formula

G.f.: x*(1 + 4*x^2)/(1 - x)^6.
a(n) = 4*A000389(n+2) + A000389(n+4). - Bruno Berselli, Apr 15 2015
E.g.f.: (24*x + 48*x^2 + 40*x^3 + 12*x^4 + x^5)*exp(x)/24. - G. C. Greubel, Nov 23 2017
a(n) = A261721(n,n-1). - Alois P. Heinz, Apr 15 2020

A333932 a(n) is the least integer that is 4-dimensional pyramidal in exactly n ways.

Original entry on oeis.org

5, 15, 35, 140, 1820, 11375, 820820, 19019000, 10790015600, 1568726956160, 7278234628665, 7271181889157550
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 10 2020

Keywords

Comments

a(n) has exactly n representations as an 4-dimensional pyramidal number P(m, k) = binomial(k + 2, 3)*(k*(m - 2) - m + 6) / 4, with m > 2, k > 1.

Examples

			a(3) = 35 because 35 is the least integer which is 4-dimensional pyramidal in 3 ways (35 = P(3, 4) = P(7, 3) = P(33, 2)).

Links

Crossrefs

Cf. A063778, A080852, A261721, A333916.

Extensions

a(9) from Giovanni Resta, Apr 11 2020
a(9) corrected and a(10)-a(12) from Bert Dobbelaere, Apr 14 2020
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