A281706 Number of sets of three positive numbers <= n whose sum is a square.

0, 0, 0, 0, 1, 2, 3, 5, 8, 11, 15, 20, 26, 33, 40, 48, 57, 67, 80, 93, 107, 121, 136, 153, 172, 193, 214, 236, 259, 284, 311, 340, 371, 402, 433, 466, 501, 538, 577, 618, 661, 705, 751, 798, 847, 897, 949, 1002, 1057, 1115, 1176, 1239, 1303, 1369, 1436, 1505

Offset: 0

Author

Alois P. Heinz and Robert G. Wilson v, Jan 28 2017

Keywords

• nonn
• easy

Comments

Inspired by A278329.

a(n) >= a(n-1), first differences are nondecreasing.

Examples

			a(1) .. a(3) = 0;
a(4) = 1 since 2+3+4 = 9;
a(5) = 2 since 2+3+4 = 1+3+5 = 9;
a(6) = 3 since 2+3+4 = 1+3+5 = 1+2+6 = 9;
a(7) = 5 since a(6) = 3 plus 3+6+7 = 4+5+7 = 16;
a(8) = 8 since a(7) = 5 plus 1+7+8 = 2+6+8 = 3+5+8 = 16; etc.
		

Links

• Alois P. Heinz and Robert G. Wilson v, Table of n, a(n) for n = 0..20000

Crossrefs

Cf. A176615, A278329.

Column k=3 of A281871.

Programs

Maple

g:= (n, t)-> max(0, iquo(n-1, 2)-max(1, n-t)+1):
b:= n-> add(g(k^2-n, n-1), k=ceil(sqrt(n+3))..floor(sqrt(3*n-3))):
a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+b(n)) end:
seq(a(n), n=0..100);

Mathematica

f[n_] := Length@ Select[Plus @@@ Subsets[ Range@ n, {3}], Mod[ Sqrt[#], 1] == 0 &]; Array[f, 65] (* or *)
f = Compile[{{n, _Integer}}, Block[{a = 1, b = 2, c = 3, cnt = 0}, While[a < b, b = a +1; While[b < c, c = b +1; While[c < n +1, If[ Mod[ Sqrt[a + b + c], 1] == 0, cnt++]; c++]; b++]; a++]; cnt]]; Array[f, 65] (* or *)
g = Compile[{{n, Integer}}, If[n < 4, 0, Block[{a = 1, b = 2, cnt = f[n -1]}, While[a < b, b = a +1; While[b < n, If[ Mod[ Sqrt[a + b + n], 1] == 0, cnt++]; b++]; a++]; cnt]]]; f[n] := f[n] = g[n]; Array[f, 100]