A280542 -id:A280542 - cp's OEIS frontend

A006456 Number of compositions (ordered partitions) of n into squares.

1, 1, 1, 1, 2, 3, 4, 5, 7, 11, 16, 22, 30, 43, 62, 88, 124, 175, 249, 354, 502, 710, 1006, 1427, 2024, 2870, 4068, 5767, 8176, 11593, 16436, 23301, 33033, 46832, 66398, 94137, 133462, 189211, 268252, 380315, 539192, 764433, 1083764, 1536498, 2178364

Offset: 0

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..5000 (terms 0..500 from T. D. Noe)
Jan Bohman, Carl-Erik Fröberg, Hans Riesel, Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301.
J. Bohman et al., Partitions in squares, Nordisk Tidskr. Informationsbehandling (BIT) 19 (1979), 297-301. (Annotated scanned copy)
N. Robbins, On compositions whose parts are polygonal numbers, Annales Univ. Sci. Budapest., Sect. Comp. 43 (2014) 239-243. See p. 242.
Index entries for sequences related to sums of squares

Cf. A280542.

Row sums of A337165.

a:= proc(n) option remember; `if`(n<0, 0,
      `if`(n=0, 1, add(a(n-j^2), j=1..isqrt(n))))
    end:
seq(a(n), n=0..44);  # Alois P. Heinz, Jul 26 2025

a[n_]:=a[n]=If[n==0, 1, Sum[a[n - k], {k, Select[Range[n], IntegerQ[Sqrt[#]] &]}]]; Table[a[n], {n,0,  100}] (* Indranil Ghosh, Jul 28 2017, after David W. Wilson's formula *)

N=66;  x='x+O('x^N);
Vec( 1/( 1 - sum(k=1,1+sqrtint(N), x^(k^2) ) ) )
/* Joerg Arndt, Sep 30 2012 */

from gmpy2 import is_square
class Memoize:
    def _init_(self, func):
        self.func=func
        self.cache={}
    def _call_(self, arg):
        if arg not in self.cache:
            self.cache[arg] = self.func(arg)
        return self.cache[arg]
@Memoize
def a(n): return 1 if n==0 else sum([a(n - k) for k in range(1, n + 1) if is_square(k)])
print([a(n) for n in range(101)]) # Indranil Ghosh, Jul 28 2017, after David W. Wilson's formula

a(0) = 1; a(n) = Sum_{1 <= k^2 <= n} a(n-k^2), if n > 0. - David W. Wilson
G.f.: 1/(1-x-x^4-x^9-....) - Jon Perry, Jul 04 2004
a(n) ~ c * d^n, where d is the root of the equation EllipticTheta(3, 0, 1/d) = 3, d = 1.41774254618138831428829091099000662953179532057717725688..., c = 0.46542113389379672452973940263069782869244877335179331541... - Vaclav Kotesovec, May 01 2014, updated Jan 05 2017
G.f.: 2/(3 - theta_3(q)), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018

Name corrected by Bob Selcoe, Feb 12 2014

A280863 Expansion of 1/(1 - Sum_{k>=0} x^((2*k+1)^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 19, 24, 30, 37, 45, 55, 66, 79, 95, 115, 140, 171, 209, 255, 312, 381, 464, 564, 685, 832, 1011, 1229, 1494, 1818, 2214, 2697, 3285, 4000, 4869, 5926, 7211, 8772, 10670, 12980, 15793, 19219, 23391, 28470, 34653, 42179, 51336, 62475, 76025, 92510

Offset: 0

Ilya Gutkovskiy, Jan 09 2017

nonn

Number of compositions (ordered partitions) of n into odd squares (A016754).

			a(12) = 5 because we have [9, 1, 1, 1], [1, 9, 1, 1], [1, 1, 9, 1], [1, 1, 1, 9] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Cf. A000290, A006456, A016754, A167661, A167700, A280542.

nmax = 63; CoefficientList[Series[1/(1 - Sum[x^(2 k + 1)^2, {k, 0, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=0} x^((2*k+1)^2)).

A298640 Number of compositions (ordered partitions) of n^2 into squares > 1.

Original entry on oeis.org

1, 0, 1, 1, 2, 8, 12, 129, 874, 9630, 167001, 3043147, 72844510, 2423789655, 106665874384, 6156805673648, 470151743582651, 47558937432498729, 6363358599941131580, 1126147544855148769425, 263646401550138303553708, 81649922556593759124887197

Offset: 0

Ilya Gutkovskiy, Jan 24 2018

nonn

			a(5) = 8 because we have [25], [16, 9], [9, 16], [9, 4, 4, 4, 4], [4, 9, 4, 4, 4], [4, 4, 9, 4, 4], [4, 4, 4, 9, 4] and [4, 4, 4, 4, 9].

Cf. A000290, A006456, A078134, A224366, A280542, A298642.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j^2), j=2..isqrt(n)))
    end:
a:= n-> b(n^2):
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 05 2018

b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j^2], {j, 2, Floor @ Sqrt[n]}]];
a[n_] := b[n^2];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 21 2018, after Alois P. Heinz *)

a(n) = [x^(n^2)] 1/(1 - Sum_{k>=2} x^(k^2)).

a(n) = A280542(A000290(n)).

A281155 Expansion of (Sum_{k>=2} x^(k^2))^3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 0, 3, 0, 3, 0, 0, 1, 0, 6, 0, 0, 0, 3, 3, 0, 3, 0, 6, 0, 0, 3, 0, 3, 3, 6, 0, 0, 1, 6, 6, 0, 0, 0, 6, 0, 6, 6, 0, 3, 0, 6, 6, 0, 0, 6, 3, 3, 3, 6, 6, 0, 3, 0, 6, 1, 3, 12, 6, 0, 0, 6, 3, 6, 6, 0, 3, 0, 3, 15, 6, 0, 0, 6, 12, 0, 3, 3, 6, 6, 0, 12, 3, 0, 6, 6

Offset: 0

Ilya Gutkovskiy, Jan 16 2017

nonn

Number of ways to write n as an ordered sum of 3 squares > 1.

			G.f. = x^12 + 3*x^17 + 3*x^22 + 3*x^24 + x^27 + 6*x^29 + 3*x^33 + 3*x^34 + 3*x^36 + ...
a(17) = 3 because we have [9, 4, 4], [4, 9, 4] and [4, 4, 9].

Index entries for sequences related to sums of squares

Cf. A000290, A005875, A002102, A006456, A063691, A078134, A280542.

nmax = 105; CoefficientList[Series[Sum[x^k^2, {k, 2, nmax}]^3, {x, 0, nmax}], x]
CoefficientList[Series[(-1 - 2 x + EllipticTheta[3, 0, x])^3/8, {x, 0, 105}], x]

G.f.: (Sum_{k>=2} x^(k^2))^3.

G.f.: (1/8)*(-1 - 2*x + theta_3(0,x))^3, where theta_3 is the 3rd Jacobi theta function.

A281154 Expansion of (Sum_{k>=2} x^(k^2))^2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 1, 0, 2, 0, 0, 0, 2

Offset: 0

Ilya Gutkovskiy, Jan 16 2017

nonn

Number of ways to write n as an ordered sum of 2 squares > 1.

			G.f. = x^8 + 2*x^13 + x^18 + 2*x^20 + 2*x^25 + 2*x^29 + x^32 + 2*x^34 + 2*x^40 + ...
a(13) = 2 because we have [9, 4] and [4, 9].

Index entries for sequences related to sums of squares

Cf. A000290, A000925, A004018, A006456, A063725, A078134, A085989, A280542.

nmax = 105; CoefficientList[Series[Sum[x^k^2, {k, 2, nmax}]^2, {x, 0, nmax}], x]
CoefficientList[Series[(1 + 2 x - EllipticTheta[3, 0, x])^2/4, {x, 0, 105}], x]

G.f.: (Sum_{k>=2} x^(k^2))^2.

G.f.: (1/4)*(1 + 2*x - theta_3(0,x))^2, where theta_3 is the 3rd Jacobi theta function.

A303908 Expansion of 1/(2 + x - theta_2(sqrt(x))/(2*x^(1/8))), where theta_2() is the Jacobi theta function.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 5, 2, 0, 9, 5, 0, 15, 10, 1, 27, 20, 3, 46, 40, 9, 80, 78, 22, 139, 152, 51, 242, 290, 114, 427, 550, 247, 753, 1034, 525, 1340, 1933, 1092, 2396, 3602, 2237, 4312, 6685, 4519, 7813, 12380, 9027, 14239, 22877, 17866, 26110, 42214, 35072, 48123, 77829, 68379

Offset: 0

Ilya Gutkovskiy, May 02 2018

nonn

Number of compositions (ordered partitions) of n into triangular numbers > 1.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to compositions

Cf. A000217, A023361, A212804, A280542, A303668, A303906, A303907.

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
      add(a(n-j*(j+1)/2), j=2..isqrt(2*n))))
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, May 02 2018

nmax = 62; CoefficientList[Series[1/(2 + x - EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8))), {x, 0, nmax}], x]
nmax = 62; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1)/2), {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[SquaresR[1, 8 k + 1] a[n - k], {k, 2, n}]/2; Table[a[n], {n, 0, 62}]

G.f.: 1/(1 - Sum_{k>=2} x^(k*(k+1)/2)).

A331983 Number of compositions (ordered partitions) of n into distinct squares > 1.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 8, 0, 0, 0, 0, 2, 0, 1, 0, 6, 0, 2, 2, 0, 0, 0, 8, 0, 0, 0, 7, 6, 0, 2, 2, 24, 0, 6, 0, 2, 0, 0, 8, 6, 0, 1, 32, 0, 0, 2, 6, 6, 0, 0, 2, 32, 0, 0, 12, 30, 0, 2

Offset: 0

Ilya Gutkovskiy, Feb 03 2020

nonn

			a(25) = 3 because we have [25], [16, 9] and [9, 16].

Cf. A000290, A006456, A032020, A032021, A032022, A033461, A078134, A280129, A280542, A331844, A331918.

b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`(i*(i+1)*(2*i+1)/6-1n, 0, b(n-i^2, i-1, p+1))+b(n, i-1, p)))
    end:
a:= n-> b(n, isqrt(n), 0):
seq(a(n), n=0..87);  # Alois P. Heinz, Feb 03 2020

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i(i+1)(2i+1)/6 - 1 < n, 0, If[i^2 > n, 0, b[n - i^2, i - 1, p + 1]] + b[n, i - 1, p]]];
a[n_] := b[n, Floor@Sqrt[n], 0];
a /@ Range[0, 87] (* Jean-François Alcover, Nov 26 2020, after Alois P. Heinz *)

A363738 Number of ordered partitions of n into cubes > 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 1, 0, 0, 5, 0, 0, 3, 0, 2, 0, 0, 6, 0, 0, 6, 0, 3, 0, 0, 7, 0, 0, 10, 0, 4, 1, 0, 8, 0, 0, 15, 0, 5, 4, 0, 11, 0, 0, 21, 0, 6, 10, 0, 16, 0, 0, 28, 0, 7, 20, 0, 23

Offset: 0

Seiichi Manyama, Jun 18 2023

nonn

This sequence is different from A278929.

			a(43) = 3 because we have [27, 8, 8], [8, 27, 8] and [8, 8, 27].

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000578, A010057, A023358, A278929, A280542.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, ispower(j, 3)*v[i-j+1])); v;

G.f.: 1/(1 - Sum_{k>=2} x^(k^3)).

a(0) = 1; a(n) = Sum_{k=2..n} A010057(k) * a(n-k).

A294071 Number of ordered ways of writing n^2 as a sum of n squares > 1.

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 6, 7, 288, 262, 13702, 69531, 610567, 5356091, 51724960, 521956086, 5467658641, 59931636545, 690518644584, 8100858045744, 99142980567486, 1246972499954475, 16142015005905558, 215722810653380845, 2955759897694815985, 41614888439136252691

Offset: 0

Ilya Gutkovskiy, Feb 07 2018

nonn

			a(5) = 5 because we have [9, 4, 4, 4, 4], [4, 9, 4, 4, 4], [4, 4, 9, 4, 4], [4, 4, 4, 9, 4] and [4, 4, 4, 4, 9].

Index entries for sequences related to sums of squares

Cf. A000290, A037444, A066535, A078134, A092362, A232173, A280129, A280542, A281154, A281155, A298329, A298330, A298640, A298642.

Table[SeriesCoefficient[((-1 - 2 x + EllipticTheta[3, 0, x])/2)^n, {x, 0, n^2}], {n, 0, 25}]

a(n) = [x^(n^2)] (Sum_{k>=2} x^(k^2))^n.

A304633 Expansion of 2/((1 - x)(3 + 2x - theta_3(x))), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 7, 7, 7, 9, 12, 13, 13, 16, 20, 23, 23, 27, 35, 41, 42, 47, 61, 71, 75, 82, 104, 124, 134, 146, 178, 217, 237, 258, 307, 377, 419, 456, 535, 651, 739, 804, 933, 1126, 1300, 1422, 1629, 1955, 2275, 2513, 2846, 3397, 3972, 4435, 4990, 5904

Offset: 0

Ilya Gutkovskiy, May 15 2018

nonn

Partial sums of A280542.

Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Index entries for sequences related to compositions
Index entries for sequences related to sums of squares

Cf. A000290, A001156, A006456, A010052, A248801, A280542, A302833, A303667.

nmax = 62; CoefficientList[Series[2/((1 - x) (3 + 2 x - EllipticTheta[3, 0, x])), {x, 0, nmax}], x]
nmax = 62; CoefficientList[Series[1/((1 - x) (1 - Sum[x^k^2, {k, 2, nmax}])), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Boole[IntegerQ[k^(1/2)] && k != 1] a[n - k], {k, 1, n}]; Accumulate[Table[a[n], {n, 0, 62}]]

G.f.: 1/((1 - x)*(1 - Sum_{k>=2} x^(k^2))).

cp's OEIS Frontend

A006456 Number of compositions (ordered partitions) of n into squares.

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A280863 Expansion of 1/(1 - Sum_{k>=0} x^((2*k+1)^2)).

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A298640 Number of compositions (ordered partitions) of n^2 into squares > 1.

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A281155 Expansion of (Sum_{k>=2} x^(k^2))^3.

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A281154 Expansion of (Sum_{k>=2} x^(k^2))^2.

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A303908 Expansion of 1/(2 + x - theta_2(sqrt(x))/(2*x^(1/8))), where theta_2() is the Jacobi theta function.

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A331983 Number of compositions (ordered partitions) of n into distinct squares > 1.

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A304633 Expansion of 2/((1 - x)(3 + 2x - theta_3(x))), where theta_3() is the Jacobi theta function.