A167700 Number of partitions of n into distinct odd squares.
1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0
Examples
a(50) = #{49+1} = 1;
a(130) = #{121+9, 81+49} = 2.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
- Vaclav Kotesovec, Graph - The asymptotic ratio
- Index entries for sequences related to sums of squares.
Programs
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Haskell
a167700 = p a016754_list where p _ 0 = 1 p (q:qs) m = if m < q then 0 else p qs (m - q) + p qs m -- Reinhard Zumkeller, Mar 15 2014
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Mathematica
nmax = 100; CoefficientList[Series[Product[1 + x^((2*k-1)^2), {k, 1, Floor[Sqrt[nmax]/2] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 18 2017 *)
Formula
a(n) = f(n,1,8) with f(x,y,z) = if x
G.f.: Product_{k>=0} (1 + x^((2*k+1)^2)). - Ilya Gutkovskiy, Jan 11 2017
a(n) ~ exp(3 * 2^(-7/3) * Pi^(1/3) * (sqrt(2)-1)^(2/3) * Zeta(3/2)^(2/3) * n^(1/3)) * (sqrt(2)-1)^(1/3) * Zeta(3/2)^(1/3) / (2^(7/6) * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Sep 18 2017
A290275 Numbers that are the sum of distinct odd positive squares.
1, 9, 10, 25, 26, 34, 35, 49, 50, 58, 59, 74, 75, 81, 82, 83, 84, 90, 91, 106, 107, 115, 116, 121, 122, 130, 131, 139, 140, 146, 147, 155, 156, 164, 165, 169, 170, 171, 178, 179, 180, 194, 195, 196, 202, 203, 204, 205, 211, 212, 218, 219, 225, 226, 227, 228, 234, 235, 236, 237, 243, 244, 250, 251, 252, 253
Offset: 1
Keywords
Comments
Complement of A167703.
1922 is the largest of positive integers not in this sequence.
Examples
139 is in the sequence because 139 = 9 + 49 + 81 = 3^2 + 7^2 + 9^2.
Links
Programs
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Mathematica
max = 253; f[x_] := Product[1 + x^(2 k + 1)^2, {k, 0, 10}]; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, max}]] // Rest
Comments