A191822 Number of solutions to the Diophantine equation x1*x2 + x2*x3 + x3*x4 + x4*x5 = n, with all xi >= 1.

0, 0, 0, 1, 2, 6, 8, 16, 20, 32, 36, 58, 58, 86, 92, 125, 122, 178, 164, 228, 224, 286, 268, 382, 330, 436, 424, 534, 474, 660, 556, 740, 692, 840, 752, 1043, 846, 1094, 1032, 1276, 1078, 1476, 1204, 1582, 1458, 1710, 1480, 2070, 1628, 2096, 1924, 2332, 1946, 2652, 2148, 2770, 2480, 2908, 2480, 3512

Offset: 1

Author

N. J. A. Sloane, Jun 17 2011

Keywords

• nonn

Comments

Related to "Liouville's Last Theorem".

Examples

			G.f.: x^4 + 2 x^5 + 6 x^6 + 8 x^7 + 16 x^8 + 20 x^9 + 32 x^10 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000
• George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See L_4(n).
• E. T. Bell, The form wx+xy+yz+zu, Bull. Amer. Math. Soc., 42 (1936), 377-380.

Crossrefs

Cf. A000005, A002133, A065608, A191832.

Cf. A001157, A038040, A055507.

Cf. A189835.

Programs

Maple

with(numtheory);
D00:=n->add(tau(j)*tau(n-j),j=1..n-1);
L4:=n->sigma[2](n)-n*sigma[0](n)-D00(n);
[seq(L4(n),n=1..60)];

Mathematica

a[ n_] := Length @ FindInstance[{x1 > 0, x2 > 0, x3 > 0, x4 > 0, x5 > 0, n == x1 x2 + x2 x3 + x3 x4 + x4 x5}, {x1, x2, x3, x4, x5}, Integers, 10^9]; (* Michael Somos, Nov 12 2016 *)

Formula

a(n) = sigma_2(n) - n*sigma_0(n) - A055507(n-1).