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
A167663 Where records occur for partitions into odd squares, cf. A167661.
0, 9, 18, 25, 27, 34, 36, 43, 45, 49, 50, 52, 54, 58, 59, 61, 63, 67, 68, 70, 72, 74, 75, 76, 77, 79, 81, 83, 84, 85, 86, 88, 90, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 111, 112, 113, 115, 116, 117, 118, 119, 120, 121, 122, 123
Offset: 1
Keywords
A167701 Records for partitions into distinct odd squares, cf. A167700.
1, 2, 3, 4, 6, 7, 8, 10, 11, 15, 18, 22, 23, 24, 26, 31, 32, 35, 37, 41, 46, 51, 53, 57, 60, 66, 75, 84, 89, 104, 113, 119, 122, 130, 142, 150, 162, 173, 176, 193, 202, 203, 223, 229, 236, 256, 272, 304, 305, 332, 341, 350, 372, 394, 404, 409, 428, 440, 461, 464, 467
Offset: 1
Keywords
Crossrefs
Cf. A167662.
Comments