A101427 Number of different cuboids with volume (pq)^n, where p,q are distinct prime numbers.

1, 2, 8, 19, 42, 78, 139, 224, 350, 517, 744, 1032, 1405, 1862, 2432, 3115, 3942, 4914, 6067, 7400, 8954, 10729, 12768, 15072, 17689, 20618, 23912, 27571, 31650, 36150, 41131, 46592, 52598, 59149, 66312, 74088, 82549, 91694, 101600, 112267, 123774

Offset

0,2

Comments

Subsequence of A034836, which gives the number of cuboids for volume n.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Geoffrey B. Campbell, Vector Partition Identities for 2D, 3D and nD Lattices, arXiv:2302.01091 [math.CO], 2023.

Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,3,-1,-2,1).

Formula

If n is a multiple of 3, a(n) = ((n+2)^2*(n+1)^2 + 12*(floor(n/2)+1)^2+8)/24, otherwise a(n) = ((n+2)^2*(n+1)^2 + 12*(floor(n/2)+1)^2)/24. - Frederic Solbes, Mar 18 2014

G.f.: -(x^6+3*x^4+4*x^3+3*x^2+1)/((x^2+x+1)*(x+1)^2*(x-1)^5). - Colin Barker, Mar 27 2014

From Daniel Mondot, Sep 20 2016: (Start)

a(n) = a(n-1) + 2*a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 2*a(n-6) + a(n-7) - a(n-8) + 12, n>=8.

a(n) = 4*a(n-6) - 6*a(n-12) + 4*a(n-18) - a(n-24) + 1296, n>=24. (End)

Mathematica

a[n_] := Switch[Mod[n, 6], 0, n+1, 1|5, 3n/4 + 7/24, 2|4, n+2/3, 3, 3n/4 + 5/8] + n^4/24 + n^3/4 + 2n^2/3; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 06 2016, after Frederic Solbes' formula *)

Prog

(PARI) a(n) = if (n % 3, ((n+2)^2*(n+1)^2 + 12*(n\2+1)^2)/24, ((n+2)^2*(n+1)^2 + 12*(n\2+1)^2+8)/24); \\ Michel Marcus, Mar 18 2014

Crossrefs

Cf. A034836, A101423, A101424, A101425, A101426.

Column k=3 of A277239.

Sequence in context: A240279 A327728 A000158 * A286269 A126877 A107769

Adjacent sequences: A101424 A101425 A101426 * A101428 A101429 A101430

Keyword

nonn

Author

Anthony C Robin, Jan 17 2005

Extensions

Extended by Ray Chandler, Dec 17 2008

Edited by Ray Chandler, Dec 19 2008

a(0) = 1 prepended by Daniel Mondot, Sep 20 2016

Status

approved