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(13) counts the 4 partitions [1, 1, 1, 1, 1, 1, 1, 3, 3] with 9 = 3^2 parts and 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 3^2 + 3^2 = 5^2, [1, 4, 4, 4] with 2^2 parts and 1^2 + 4^2 + 4^2 + 4^2 = 7^2, [2, 2, 4, 5] with 4 = 2^2 parts and 2^2 + 2^2 + 4^2 + 5^2 = 7^2, [13] with 1 = 1^2 part and 13^2 = 13^2.
# first Maple program to calculate the sequence: A375731:=proc(n) local a,i,j; a:=0; for i in combinat:-partition(n) do if issqr(numelems(i)) and issqr(add(i[j]^2,j=1..nops(i))) then a:=a+1 fi od; return a end proc; seq(A375731(n),n=0..63); # second Maple program to calculate the partitions: A375731part:=proc(n) local L,i,j;L:=[]; for i in combinat:-partition(n) do if issqr(numelems(i)) and issqr(add(i[j]^2,j=1..nops(i))) then L:=[op(L),i] fi od; return op(L) end proc; A375731part(13);
a(n) = my(nb=0); forpart(p=n, if (issquare(#p) && issquare(norml2(Vec(p))), nb++)); nb; \\ Michel Marcus, Aug 30 2024
G.f. = x^12 + 3*x^17 + 3*x^22 + 3*x^24 + x^27 + 6*x^29 + 3*x^33 + 3*x^34 + 3*x^36 + ... a(17) = 3 because we have [9, 4, 4], [4, 9, 4] and [4, 4, 9].
nmax = 105; CoefficientList[Series[Sum[x^k^2, {k, 2, nmax}]^3, {x, 0, nmax}], x]
CoefficientList[Series[(-1 - 2 x + EllipticTheta[3, 0, x])^3/8, {x, 0, 105}], x]
a(6) = 15 because we have [4, 1, 1], [1, 4, 1], [1, 1, 4], [1, 1, 1, 1, 1, 1] and 3 + 3 + 3 + 6 = 15.
b:= proc(n) option remember; `if`(n=0, [1, 0], add(
(p-> p+[0, p[1]])(b(n-j^2)), j=1..isqrt(n)))
end:
a:= n-> b(n)[2]:
seq(a(n), n=1..45); # Alois P. Heinz, Aug 07 2019
nmax = 45; Rest[CoefficientList[Series[Sum[x^i^2, {i, 1, nmax}]/(1 - Sum[x^j^2, {j, 1, nmax}])^2, {x, 0, nmax}], x]]
nmax = 45; Rest[CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)/(2 (1 + (1 - EllipticTheta[3, 0, x])/2)^2), {x, 0, nmax}], x]]
Table[SeriesCoefficient[1/(1 + (1/2) n (1 - EllipticTheta[3, 0, x])), {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[1/(1 - n Sum[x^k^2, {k, 1, n}]), {x, 0, n}], {n, 0, 20}]
a(35) = 6 because we have [25, 9, 1], [25, 1, 9], [9, 25, 1], [9, 1, 25], [1, 25, 9] and [1, 9, 25].
N:= 200: # for a(0)..a(N) G:= mul(1+t*x^(i^2),i=1..floor(sqrt(N)),2): F:= proc(n) local R, k, v; R:= coeff(G,x,n); add(k!*coeff(R,t,k),k=1..degree(R,t)) end proc: F(0):= 1: map(F, [$0..N]); # Robert Israel, Feb 03 2020
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k^2] * a[n - k^2], {k, 1, Floor@Sqrt[n]}]; Array[a, 23, 0] (* Amiram Eldar, Apr 30 2022 *)
my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, sqrtint(N), x^k^2/(k^2)!))))
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, sqrtint(i), binomial(i, j^2)*v[i-j^2+1])); v;
a[0] = 1; a[n_] := a[n] = Sum[(k^2 - 1)! * Binomial[n, k^2] * a[n - k^2], {k, 1, Floor@Sqrt[n]}]; Array[a, 22, 0] (* Amiram Eldar, Apr 30 2022 *)
my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, sqrtint(N), x^k^2/(k^2)))))
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, sqrtint(i), (j^2-1)!*binomial(i, j^2)*v[i-j^2+1])); v;
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