This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(5) = 2 because we have [25] and [16, 9].
a(5) = 3 because we have [25], [16, 9] and [9, 16].
b:= proc(n, i, p) option remember;
`if`(i*(i+1)*(2*i+1)/6n, 0, b(n-i^2, i-1, p+1))+b(n, i-1, p)))
end:
a:= n-> b(n^2, n, 0):
seq(a(n), n=0..35); # Alois P. Heinz, Jan 30 2020
b[n_, i_, p_] := b[n, i, p] = If[i(i+1)(2i+1)/6 < n, 0, If[n == 0, p!, If[i^2 > n, 0, b[n - i^2, i - 1, p + 1]] + b[n, i - 1, p]]]; a[n_] := b[n^2, n, 0]; a /@ Range[0, 35] (* Jean-François Alcover, Nov 08 2020, after Alois P. Heinz *)
nmax = 95; CoefficientList[Series[Product[1 + x^(k (k + 1)/2), {k, 2, nmax}], {x, 0, nmax}], x]
nmax = 82; CoefficientList[Series[Product[1/(1 + x^k^2), {k, 2, Floor[Sqrt[nmax]] + 1}], {x, 0, nmax}], x]
nmax = 92; CoefficientList[Series[Product[1 - x^k^2, {k, 2, Floor[Sqrt[nmax]] + 1}], {x, 0, nmax}], x]
a(25) = 3 because we have [25], [16, 9] and [9, 16].
b:= proc(n, i, p) option remember;
`if`(n=0, p!, `if`(i*(i+1)*(2*i+1)/6-1n, 0, b(n-i^2, i-1, p+1))+b(n, i-1, p)))
end:
a:= n-> b(n, isqrt(n), 0):
seq(a(n), n=0..87); # Alois P. Heinz, Feb 03 2020
b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i(i+1)(2i+1)/6 - 1 < n, 0, If[i^2 > n, 0, b[n - i^2, i - 1, p + 1]] + b[n, i - 1, p]]]; a[n_] := b[n, Floor@Sqrt[n], 0]; a /@ Range[0, 87] (* Jean-François Alcover, Nov 26 2020, after Alois P. Heinz *)
a(5) = 5 because we have [9, 4, 4, 4, 4], [4, 9, 4, 4, 4], [4, 4, 9, 4, 4], [4, 4, 4, 9, 4] and [4, 4, 4, 4, 9].
Table[SeriesCoefficient[((-1 - 2 x + EllipticTheta[3, 0, x])/2)^n, {x, 0, n^2}], {n, 0, 25}]
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