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
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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);
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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
A338482 Least number of centered triangular numbers that sum to n.
1, 2, 3, 1, 2, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 5, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5
Offset: 1
Keywords
Comments
It appears that a(n) = 3 for n == 0 (mod 3), 1 <= a(n) <= 4 for n == 1 (mod 3), and 2 <= a(n) <= 5 for n == 2 (mod 3). - Robert Israel, Nov 13 2020
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
- Eric Weisstein's World of Mathematics, Centered Triangular Number
- Index entries for sequences related to centered polygonal numbers
Programs
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Maple
f:= proc(n) option remember; local r,i; r:= sqrt(24*n-15)/6+1/2; if r::integer then return 1 fi; 1+min(seq(procname(n-(3*i*(i-1)/2+1)),i=1..floor(r))) end proc: map(f, [$1..200]); # Robert Israel, Nov 13 2020 -
Mathematica
f[n_] := f[n] = Module[{r}, r = Sqrt[24n-15]/6+1/2; If[IntegerQ[r], Return[1]]; 1+Min[Table[f[n-(3i*(i-1)/2+1)], {i, 1, Floor[r]}]]]; Map[f, Range[200]] (* Jean-François Alcover, Sep 16 2022, after Robert Israel *)
A338484 Least number of centered square numbers needed to represent n.
1, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 2, 3, 4, 5
Offset: 1
Keywords
Links
- Eric Weisstein's World of Mathematics, Centered Square Number
- Index entries for sequences related to centered polygonal numbers
A338491 Least number of centered pentagonal numbers needed to represent n.
1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6, 2, 3, 4, 5, 6, 7, 3, 4, 5, 6
Offset: 1
Keywords
Links
- Eric Weisstein's World of Mathematics, Centered Pentagonal Number
- Index entries for sequences related to centered polygonal numbers
A338492 Least number of centered heptagonal numbers needed to represent n.
1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 2, 3, 4, 5, 6, 7, 8, 9, 3
Offset: 1
Keywords
Links
- Eric Weisstein's World of Mathematics, Centered Polygonal Numbers
- Index entries for sequences related to centered polygonal numbers
A338497 Least number of odd cubes needed to represent n.
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, 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, 27, 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, 27, 28, 3
Offset: 1
Comments