A025434 -id:A025434 - 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).

A340947 Number of ways to write n as an ordered sum of 10 squares of positive integers.

Original entry on oeis.org

1, 0, 0, 10, 0, 0, 45, 0, 10, 120, 0, 90, 210, 0, 360, 262, 45, 840, 300, 360, 1260, 480, 1260, 1350, 1015, 2520, 1560, 2200, 3150, 2880, 4186, 2880, 5430, 6240, 3780, 8300, 7080, 7920, 11160, 7320, 13257, 14640, 10600, 16470, 18570, 18240, 19620, 22230, 25135, 27720, 28020, 28480, 38160

Offset: 10

Ilya Gutkovskiy, Jan 31 2021

nonn

David A. Corneth, Table of n, a(n) for n = 10..10009
Index entries for sequences related to sums of squares

Cf. A000144, A000290, A010052, A025434, A045852, A063691, A063725, A063730, A340481, A340905, A340906, A340915, A340946.

Column k=10 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, 10):
seq(a(n), n=10..62);  # Alois P. Heinz, Jan 31 2021

nmax = 62; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^10/1024, {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: (theta_3(x) - 1)^10 / 1024, where theta_3() is the Jacobi theta function.

A341001 Number of partitions of n into 10 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, 0, 0, 0, 0, 1, 0, 0, 0, 0, 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, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 1

Offset: 385

Ilya Gutkovskiy, Feb 02 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 385..20000
Index entries for sequences related to sums of squares

Cf. A000144, A000290, A010052, A025434, A025441, A025442, A025443, A025444, A045852, A340947, A340988, A340998, A340999, A341000.

Column k=10 of A341040.

A346807 Numbers that are the sum of ten squares in eight or more ways.

Original entry on oeis.org

58, 61, 64, 66, 67, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123

Offset: 1

David Consiglio, Jr., Aug 04 2021

nonn

			61 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 7^2
   = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 3^2 + 3^2 + 6^2
   = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 5^2 + 5^2
   = 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 4^2 + 5^2
   = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 3^2 + 3^2 + 3^2 + 5^2
   = 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 5^2
   = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 + 2^2 + 2^2 + 4^2 + 4^2 + 4^2
   = 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 4^2 + 4^2
   = 1^2 + 1^2 + 2^2 + 2^2 + 2^2 + 2^2 + 3^2 + 3^2 + 3^2 + 4^2
   = 1^2 + 1^2 + 1^2 + 2^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2
so 61 is a term.

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A025434, A345505, A345556, A346806.

from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x**2 for x in range(1, 1000)]
for pos in cwr(power_terms, 10):
    tot = sum(pos)
    keep[tot] += 1
    rets = sorted([k for k, v in keep.items() if v >= 8])
    for x in range(len(rets)):
        print(rets[x])

a(n) = n + 64 for n >= 8 (conjectured). - Chai Wah Wu, Dec 05 2023

A323891 a(n) is the number of partitions of 72*n + 42 into 10 odd squares.

Original entry on oeis.org

2, 9, 22, 41, 68, 106, 154, 212, 285, 368, 477, 598, 741, 898, 1076, 1286, 1524, 1785, 2068, 2379, 2741, 3131, 3554, 4002, 4497, 5044, 5644, 6274, 6939, 7653, 8445, 9295, 10186, 11117, 12113, 13192, 14355, 15556, 16807, 18147, 19570, 21089, 22673, 24300, 26029, 27865, 29821, 31822, 33894, 36088

Offset: 0

Marius A. Burtea, Feb 12 2019

nonn

			For n=0, 72*0+42 = 42 = 25+9+1+1+1+1+1+1+1+1 = 9+9+9+9+1+1+1+1+1+1, so a(0)=2.
For n=1, 72*1+42 = 114 = 81+25+1+1+1+1+1+1+1+1 = 81+9+9+9+1+1+1+1+1+1 = 49+49+9+1+1+1+1+1+1+1 = 49+25+25+9+1+1+1+1+1+1 = 49+25+9+9+9+9+1+1+1+1 = 49+9+9+9+9+9+9+9+1+1 = 25+25+25+25+9+1+1+1+1+1 = 25+25+25+9+9+9+9+1+1+1 = 25+25+9+9+9+9+9+9+9+1, so a(1)=9.

Laurențiu Panaitopol, Alexandru Gica, Arithmetic problems and number theory, Ed. Gil, Zalău, (2006), ch. 14, p. 85, pr. 32. (in Romanian).

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A000041, A001156, A016754, A025425, A025434, A033461, A035294, A078406, A090677, A167661, A167700.

[#RestrictedPartitions(72*n+42, 10, {(2*d+1)^2:d in [0..100]}): n in [0..100]];

S:= proc(n, k, m)
   option remember;
   local p,j;
   if k = 0 then if n = 0 then return 1 else return 0 fi
   elif m < 1 then return 0
   elif n < k then return 0
   elif n > k*m^2 then return 0
   fi;
   if m^2 > n then
     p:= floor(sqrt(n));
     if p::even then p:= p-1 fi;
     return procname(n, k, p)
   fi;
   add(procname(n-j*m^2,k-j,m-2), j=0..n/m^2)
end proc:
seq(S(72*n+42, 10, 72*n+42), n=0..100); # Robert Israel, Feb 24 2019

a[n_] := IntegerPartitions[72n+42, {10}, Select[ Range[1, 72n+42, 2], IntegerQ@Sqrt@#&]] // Length;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 19 2022 *)

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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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A340947 Number of ways to write n as an ordered sum of 10 squares of positive integers.

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A341001 Number of partitions of n into 10 distinct nonzero squares.

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A346807 Numbers that are the sum of ten squares in eight or more ways.

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A323891 a(n) is the number of partitions of 72*n + 42 into 10 odd squares.

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