A025432 -id:A025432 - cp's OEIS frontend

A243148 Triangle read by rows: T(n,k) = number of partitions of n into k nonzero squares; n >= 0, 0 <= k <= n.

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1

Offset: 0

Alois P. Heinz, May 30 2014

			T(20,5) = 2 = #{ (16,1,1,1,1), (4,4,4,4,4) } since 20 = 4^2 + 4 * 1^2 = 5 * 2^2.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 0, 1;
  0, 1, 0, 0, 1;
  0, 0, 1, 0, 0, 1;
  0, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 0, 1, 0, 0, 1;
  0, 0, 1, 0, 0, 1, 0, 0, 1;
  0, 1, 0, 1, 0, 0, 1, 0, 0, 1;
  0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1;
  0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1;
  (...)

Alois P. Heinz, Rows n = 0..200, flattened

Columns k = 0..10 give: A000007, A010052 (for n>0), A025426, A025427, A025428, A025429, A025430, A025431, A025432, A025433, A025434.
Row sums give A001156.
T(2n,n) gives A111178.
T(n^2,n) gives A319435.
Cf. A000290, A276559, A281541, A292520, A341040.
T(n,k) = 1 for n in A025284, A025321, A025357, A294675, A295670, A295797 (for k = 2..7, respectively).

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(i^2>n, 0, b(n-i^2, i, t-1))))
    end:
T:= (n, k)-> b(n, isqrt(n), k):
seq(seq(T(n, k), k=0..n), n=0..14);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+(s-> `if`(s>n, 0, expand(x*b(n-s, i))))(i^2)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, isqrt(n))):
seq(T(n), n=0..14);  # Alois P. Heinz, Oct 30 2021

b[n_, i_, k_, t_] := b[n, i, k, t] = If[n == 0, If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i-1, k, t] + If[i^2 > n, 0, b[n-i^2, i, k, t-1]]]]; T[n_, k_] := b[n, Sqrt[n] // Floor, k, k]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jun 06 2014, after Alois P. Heinz *)
T[n_, k_] := Count[PowersRepresentations[n, k, 2], r_ /; FreeQ[r, 0]]; T[0, 0] = 1; Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 19 2016 *)

T(n,k,L=n)=if(n>k*L^2, 0, k>n-3, k==n, k<2, issquare(n,&n) && n<=L*k, k>n-6, n-k==3, L=min(L,sqrtint(n-k+1)); sum(r=0,min(n\L^2,k-1),T(n-r*L^2,k-r,L-1), n==k*L^2)) \\ M. F. Hasler, Aug 03 2020

T(n,k) = [x^n y^k] 1/Product_{j>=1} (1-y*x^A000290(j)).
Sum_{k=1..n} k * T(n,k) = A281541(n).
Sum_{k=1..n} n * T(n,k) = A276559(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A292520(n).

A340915 Number of ways to write n as an ordered sum of 8 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 8, 0, 0, 28, 0, 8, 56, 0, 56, 70, 0, 168, 64, 28, 280, 84, 168, 280, 176, 420, 224, 345, 560, 392, 616, 420, 848, 924, 336, 1246, 1064, 868, 1464, 988, 1680, 1820, 1120, 1904, 2464, 1932, 1904, 2870, 2752, 2772, 2912, 2892, 4256, 3640, 3248, 4480, 5040, 4760, 3696, 6120

Offset: 8

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..20000
Index entries for sequences related to sums of squares

Cf. A000143, A000290, A010052, A025432, A045850, A063691, A063725, A063730, A340481, A340905, A340906, A340946, A340947.

Column k=8 of A337165.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add((s->
      `if`(s>n, 0, b(n-s, t-1)))(j^2), j=1..isqrt(n))))
    end:
a:= n-> b(n, 8):
seq(a(n), n=8..64);  # Alois P. Heinz, Jan 31 2021

nmax = 64; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^8/256, {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: (theta_3(x) - 1)^8 / 256, where theta_3() is the Jacobi theta function.

A340999 Number of partitions of n into 8 distinct nonzero squares.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 3, 0, 0, 1

Offset: 204

Ilya Gutkovskiy, Feb 02 2021

nonn

Index entries for sequences related to sums of squares

Cf. A000143, A000290, A010052, A025432, A025441, A025442, A025443, A025444, A045850, A224983, A340915, A340988, A340998, A341000, A341001.

Column k=8 of A341040.

A287167 Smallest number with exactly n representations as a sum of 8 nonzero squares or 0 if no such number exists.

Original entry on oeis.org

8, 23, 35, 32, 46, 58, 72, 56, 62, 70, 71, 79, 80, 83, 88, 89, 91, 86, 103, 94, 109, 104, 107, 112, 113, 110, 122, 119, 126, 121, 118, 144, 0, 128, 131, 136, 137, 153, 143, 139, 149, 134, 0, 0, 142, 152, 164, 154

Offset: 1

Ilya Gutkovskiy, May 20 2017

nonn

			a(1) = 8 because 8 is the smallest number with exactly 1 representation as a sum of 8 nonzero squares: 8 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2;
a(2) = 23 because 23 is the smallest number with exactly 2 representations as a sum of 8 nonzero squares: 23 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 4^2 = 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2, etc.

Eric Weisstein's World of Mathematics, Square Number
Index entries for sequences related to sums of squares

Cf. A016032, A025414, A025416, A025432, A080654, A287165, A287166.

A025432(a(n)) = n for a(n) > 0.

A295930 Largest number with exactly n representations as a sum of eight nonnegative squares.

Original entry on oeis.org

3, 7, 15, 14, 19, 24, 27, 28, 33, 35, 39

Offset: 1

Robert Price, Nov 29 2017

It appears that a(12) does not exist.

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.

Cf. A025423, A025432, A294690.

cp's OEIS Frontend

A243148 Triangle read by rows: T(n,k) = number of partitions of n into k nonzero squares; n >= 0, 0 <= k <= n.

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A340915 Number of ways to write n as an ordered sum of 8 squares of positive integers.

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A340999 Number of partitions of n into 8 distinct nonzero squares.

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A287167 Smallest number with exactly n representations as a sum of 8 nonzero squares or 0 if no such number exists.

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A295930 Largest number with exactly n representations as a sum of eight nonnegative squares.

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