A191832 Number of solutions to the Diophantine equation x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x6 = n, with all xi >= 1.

0, 0, 0, 0, 1, 2, 7, 10, 22, 29, 51, 61, 99, 115, 163, 192, 262, 287, 385, 428, 528, 600, 730, 780, 963, 1054, 1202, 1337, 1545, 1646, 1908, 2059, 2269, 2516, 2770, 2933, 3298, 3568, 3792, 4142, 4493, 4786, 5183, 5562, 5831, 6423, 6745, 7140, 7639, 8231, 8479, 9216, 9603, 10260, 10663, 11488, 11752, 12838, 13100, 13887

Offset: 1

Author

N. J. A. Sloane, Jun 17 2011

Keywords

• nonn

Comments

Related to "Liouville's Last Theorem".

Links

• Robert Israel, Table of n, a(n) for n = 1..1000
• George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See L_5(n).

Crossrefs

Cf. A000005, A000203, A002133, A055507, A191822, A191829, A191831.

Programs

Maple

with(numtheory);
D00:=n->add(tau(j)*tau(n-j),j=1..n-1);
D01:=n->add(tau(j)*sigma(n-j),j=1..n-1);
D000:=proc(n) local t1,i,j;
t1:=0;
for i from 1 to n-1 do
for j from 1 to n-1 do
if (i+j < n) then t1 := t1+numtheory:-tau(i)*numtheory:-tau(j)*numtheory:-tau(n-i-j); fi;
od; od;
t1;
end;
L5:=n->D000(n)/6+D00(n)+D01(n)/2+(2*n-1/6)*tau(n)-11*sigma[2](n)/6;
[seq(L5(n),n=1..60)];
# Alternate:
g:= proc(n,k,j) option remember;
     if n < k-1 then 0
     elif k = 2 then
        if n mod j = 0 then 1 else 0 fi
     else
        add(procname(n-j*x,k-1,x), x=1 .. floor((n-k+2)/j))
     fi
end proc:
f:= n -> add(g(n,6,j),j=1..n-4);
seq(f(n),n=1..100); # Robert Israel, Dec 02 2015

Mathematica

g[n_, k_, j_] := g[n, k, j] = If[n < k - 1, 0, If[k == 2, If[ Mod[n, j] == 0, 1, 0], Sum[g[n - j x, k - 1, x], {x, 1, Floor[(n - k + 2)/j]}]]];
f[n_] := Sum[g[n, 6, j], {j, 1, n - 4}];
Array[f, 100] (* Jean-François Alcover, Sep 25 2020, after Robert Israel *)