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

Author

Anthony C Robin, Jan 17 2005

Keywords

• nonn

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).

Crossrefs

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

Column k=3 of A277239.

Programs

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 *)

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

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)

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