A167661 Number of partitions of n into odd squares.
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 11, 11, 12, 12, 13, 13, 13, 13, 14, 15, 15, 16, 16, 17, 17, 17, 17, 18, 19, 19, 20, 20, 21, 21, 22, 23, 24, 25, 25, 26, 26, 28, 28, 29, 30, 31
Offset: 0
Keywords
Examples
a(10)=#{9+1,1+1+1+1+1+1+1+1+1+1}=2;
a(20)=#{9+9+1+1,9+1+1+1+1+1+1+1+1+1+1+1,20x1}=3;
a(30)=#{25+1+1+1+1+1,9+9+9+1+1+1,9+9+12x1,9+21x1,30x1}=5.
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from R. Zumkeller)
- Index entries for sequences related to sums of squares.
Programs
-
Maple
g := 1/mul(1-x^((2*i-1)^2), i = 1 .. 150): gser := series(g, x = 0, 105): seq(coeff(gser, x, n), n = 0 .. 100);
-
Mathematica
nmax = 100; CoefficientList[Series[Product[1/(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.: G = 1/Product_{i>=1}(1-x^{(2i-1)^2}). - Emeric Deutsch , Jan 26 2016
a(n) ~ exp(3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 4) * Zeta(3/2)^(1/3) / (4 * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Sep 18 2017
A167662 Records for partitions into odd squares, cf. A167661.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 44, 45, 46, 47, 48, 49, 51, 52, 55, 56, 57, 58, 59, 60, 62, 63, 66, 67, 68, 69, 71, 72, 73, 76, 78, 81, 82, 83, 84, 87, 89, 90, 93
Offset: 1
Keywords
Crossrefs
Cf. A167701.
A167702 Where records occur for partitions into distinct odd squares, cf. A167700.
0, 130, 251, 299, 420, 588, 645, 660, 741, 885, 1005, 1045, 1174, 1221, 1245, 1270, 1366, 1390, 1485, 1486, 1510, 1606, 1630, 1726, 1750, 1774, 1846, 1966, 2014, 2110, 2135, 2255, 2303, 2350, 2375, 2470, 2471, 2495, 2591, 2615, 2639, 2711, 2735, 2807
Offset: 1
Comments