A006456 -id:A006456 - cp's OEIS frontend

A337165 Number T(n,k) of compositions of n into k nonzero squares; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 1, 0, 0, 5, 0, 0, 1, 0, 1, 0, 3, 0, 0, 6, 0, 0, 1, 0, 0, 2, 0, 6, 0, 0, 7, 0, 0, 1, 0, 0, 0, 3, 0, 10, 0, 0, 8, 0, 0, 1, 0, 0, 0, 1, 4, 0, 15, 0, 0, 9, 0, 0, 1

Offset: 0

Alois P. Heinz, Feb 03 2021

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 0, 1;
  0, 1, 0, 0, 1;
  0, 0, 2, 0, 0,  1;
  0, 0, 0, 3, 0,  0,  1;
  0, 0, 0, 0, 4,  0,  0, 1;
  0, 0, 1, 0, 0,  5,  0, 0, 1;
  0, 1, 0, 3, 0,  0,  6, 0, 0, 1;
  0, 0, 2, 0, 6,  0,  0, 7, 0, 0, 1;
  0, 0, 0, 3, 0, 10,  0, 0, 8, 0, 0, 1;
  0, 0, 0, 1, 4,  0, 15, 0, 0, 9, 0, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..350, flattened

Columns k=0-10 give: A000007, A010052, A063725, A063691, A063730, A340481, A340905, A340906, A340915, A340946, A340947.
Row sums give A006456.
T(2n,n) gives A338464.
Main diagonal gives A000012.
Cf. A000290, A281704, A317665, A341040.

b:= proc(n) option remember; `if`(n=0, 1, add((s->
     `if`(s>n, 0, expand(x*b(n-s))))(j^2), j=1..isqrt(n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..14);

b[n_] := b[n] = If[n == 0, 1, Sum[With[{s = j^2},
     If[s>n, 0, Expand[x*b[n - s]]]], {j, 1, Sqrt[n]}]];
T[n_] := CoefficientList[b[n], x];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Feb 07 2021, after Alois P. Heinz *)

G.f. of column k: (Sum_{j>=1} x^(j^2))^k.
Sum_{k=0..n} k * T(n,k) = A281704(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A317665(n).

A331844 Number of compositions (ordered partitions) of n into distinct squares.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 0, 2, 6, 0, 1, 2, 0, 0, 2, 6, 0, 0, 0, 3, 8, 0, 0, 8, 30, 0, 0, 0, 2, 6, 1, 2, 6, 24, 2, 8, 6, 0, 0, 8, 30, 0, 0, 7, 32, 24, 2, 8, 30, 120, 6, 24, 2, 6, 0, 8, 36, 24, 1, 34, 150, 0, 2, 12, 30, 24, 0, 2, 38, 150, 0, 12, 78, 144, 2

Offset: 0

Ilya Gutkovskiy, Jan 29 2020

nonn

			a(14) = 6 because we have [9,4,1], [9,1,4], [4,9,1], [4,1,9], [1,9,4] and [1,4,9].

Cf. A000290, A006456, A032020, A032021, A032022, A033461, A218396, A219107, A331843, A331845, A331846, A331847.

b:= proc(n, i, p) option remember;
      `if`(i*(i+1)*(2*i+1)/6n, 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..82);  # Alois P. Heinz, Jan 30 2020

b[n_, i_, p_] := b[n, i, p] = If[i(i+1)(2i+1)/6 < n, 0, If[n == 0, p!, If[i^2 > n, 0, b[n - i^2, i - 1, p + 1]] + b[n, i - 1, p]]];
a[n_] := b[n, Sqrt[n] // Floor, 0];
a /@ Range[0, 82] (* Jean-François Alcover, Oct 29 2020, after Alois P. Heinz *)

A224366 Number of compositions of n^2 into sums of squares.

Original entry on oeis.org

1, 1, 2, 11, 124, 2870, 133462, 12477207, 2344649612, 885591183971, 672331353833716, 1025954712063362545, 3146790000180780110540, 19400015532276248131470280, 240398159948843792847457589388, 5987629866666297470033540284817068, 299759874416459708067727376075503706332

Offset: 0

Paul D. Hanna, Apr 05 2013

nonn

Equals the row sums of triangle A232266.

			Illustrate a(n) = Sum_{k=1..n} A006456(n^2-k^2):
a(1) = 1 = 1;
a(2) = 2 = 1 + 1;
a(3) = 11 = 7 + 3 + 1;
a(4) = 124 = 88 + 30 + 5 + 1;
a(5) = 2870 = 2024 + 710 + 124 + 11 + 1;
a(6) = 133462 = 94137 + 33033 + 5767 + 502 + 22 + 1;
a(7) = 12477207 = 8800750 + 3088365 + 539192 + 46832 + 2024 + 43 + 1; ...

Alois P. Heinz, Table of n, a(n) for n = 0..81

Cf. A006456, A232266.

b:= proc(n) option remember; local i; if n=0 then 1
      else 0; for i while i^2<=n do %+b(n-i^2) od fi
    end:
a:= n-> b(n^2):
seq(a(n), n=0..17);  # Alois P. Heinz, Aug 12 2017

b[0] = 1; b[n_] := b[n] = Sum[b[n-k], {k, Select[Range[n], IntegerQ[ Sqrt[#]]&]}];
a[n_] := b[n^2];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jun 09 2018 *)

{a(n)=polcoeff(1/(1-sum(k=1,n,x^(k^2))+x*O(x^(n^2))),n^2)}
for(n=0,21,print1(a(n),", "))

{A006456(n)=polcoeff(1/(1-sum(k=1,sqrtint(n+1),x^(k^2))+x*O(x^n)),n)}
{a(n)=if(n==0,1,sum(k=1,n,A006456(n^2-k^2)))}
for(n=0,21,print1(a(n),", "))

a(n) = [x^(n^2)] 1/(1 - Sum_{k>=1} x^(k^2)).
a(n) = A006456(n^2).
a(n) = Sum_{k=1..n} A006456(n^2-k^2) for n>=1 with a(0)=1.

A317665 Expansion of 1/Sum_{k>=0} x^(k^2).

Original entry on oeis.org

1, -1, 1, -1, 0, 1, -2, 3, -3, 1, 2, -6, 10, -11, 8, 0, -14, 29, -39, 38, -18, -22, 74, -123, 144, -110, 6, 161, -352, 491, -484, 251, 235, -896, 1528, -1825, 1452, -191, -1892, 4317, -6164, 6243, -3488, -2482, 10788, -18957, 23140, -19085, 3858, 22025, -52833, 77224, -80198, 47899

Offset: 0

Ilya Gutkovskiy, Aug 08 2018

sign

Convolution inverse of A010052.

			G.f. = 1 - x + x^2 - x^3 + x^5 - 2*x^6 + 3*x^7 - 3*x^8 + x^9 + 2*x^10 - 6*x^11 + ...

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

Cf. A004402, A006456, A010052, A106507, A323633, A337165.

a:=series(1/add(x^(k^2),k=0..100),x=0,54): seq(coeff(a,x,n),n=0..53); # Paolo P. Lava, Apr 02 2019
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(`if`(issqr(n-j), a(j), 0), j=0..n-1))
    end:
seq(a(n), n=0..53);  # Alois P. Heinz, Jul 26 2025

nmax = 53; CoefficientList[Series[1/Sum[x^k^2, {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 53; CoefficientList[Series[2/(1 + QPochhammer[x^2]^5/(QPochhammer[x] QPochhammer[x^4])^2), {x, 0, nmax}], x]
nmax = 53; CoefficientList[Series[2/(1 + EllipticTheta[3, 0, q]), {q, 0, nmax}], q]
a[0] = 1; a[n_] := a[n] = -Sum[Boole[IntegerQ[Sqrt[k]]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 53}]

seq(n)=Vec(1/(sum(k=0, sqrtint(n), x^(k^2)) + O(x*x^n))) \\ Andrew Howroyd, Aug 08 2018

a(n) = if(n==0, 1, -sum(k=1, n, issquare(k)*a(n-k))); \\ Seiichi Manyama, Mar 19 2022

G.f.: 2/(1 + theta_3(q)), where theta_3() is the Jacobi theta function.
a(n) = Sum_{k=0..n} (-1)^k * A337165(n,k).
a(0) = 1; a(n) = -Sum_{k=1..n} A010052(k) * a(n-k). - Seiichi Manyama, Mar 19 2022

A280542 Expansion of 1/(1 - Sum_{k>=2} x^(k^2)).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 2, 3, 1, 0, 3, 4, 3, 0, 4, 8, 6, 1, 5, 14, 10, 4, 7, 22, 20, 10, 12, 32, 39, 20, 21, 49, 70, 42, 37, 79, 116, 88, 65, 129, 193, 174, 122, 207, 326, 320, 238, 333, 551, 575, 463, 555, 914, 1029, 874, 959, 1502, 1829, 1621, 1691, 2486, 3192, 2989, 3000, 4172, 5488

Offset: 0

Ilya Gutkovskiy, Jan 05 2017

nonn

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

			a(17) = 3 because we have [9, 4, 4], [4, 9, 4] and [4, 4, 9].

Cf. A000290, A006456, A078134.

nmax = 75; CoefficientList[Series[1/(1 - Sum[x^k^2, {k, 2, nmax}]), {x, 0, nmax}], x]

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

A322798 Number of compositions (ordered partitions) of n into hexagonal numbers (A000384).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 22, 29, 37, 47, 60, 77, 101, 133, 174, 226, 292, 376, 486, 632, 823, 1072, 1394, 1808, 2342, 3036, 3939, 5116, 6648, 8636, 11211, 14548, 18875, 24493, 31795, 41283, 53604, 69594, 90338, 117251, 152184, 197540

Offset: 0

Ilya Gutkovskiy, Dec 26 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..8828
Eric Weisstein's World of Mathematics, Hexagonal Number
Index entries for sequences related to compositions

Cf. A000384, A006456, A023361, A181324, A278949, A279279, A322799, A322800.

h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(2*t-1)>n, t-1, t))(1+h(n-1)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i*(2*i-1)), i=1..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018

nmax = 50; CoefficientList[Series[1/(1 - Sum[x^(k (2 k - 1)), {k, 1, nmax}]), {x, 0, nmax}], x]

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

A322799 Number of compositions (ordered partitions) of n into heptagonal numbers (A000566).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 22, 29, 37, 46, 57, 71, 89, 112, 143, 183, 233, 295, 372, 468, 588, 741, 937, 1188, 1506, 1908, 2414, 3049, 3848, 4857, 6136, 7757, 9812, 12414, 15702, 19852, 25089, 31703, 40061, 50631, 64004, 80923, 102318

Offset: 0

Ilya Gutkovskiy, Dec 26 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..9828
Eric Weisstein's World of Mathematics, Heptagonal Number
Index entries for sequences related to compositions

Cf. A000566, A006456, A023361, A181324, A279012, A279280, A322798, A322800.

h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(5*t-3)/2>n, t-1, t))(1+h(n-1)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i*(5*i-3)/2), i=1..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018

nmax = 53; CoefficientList[Series[1/(1 - Sum[x^(k (5 k - 3)/2), {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=1} x^(k*(5*k-3)/2)).

A347805 Expansion of (theta_3(x) - 1)^2 / (2 * (3 - theta_3(x))).

Original entry on oeis.org

1, 1, 1, 3, 4, 5, 7, 10, 16, 22, 30, 43, 62, 88, 123, 175, 249, 354, 502, 710, 1006, 1427, 2024, 2869, 4068, 5767, 8176, 11593, 16436, 23301, 33033, 46832, 66398, 94137, 133461, 189211, 268252, 380315, 539192, 764433, 1083764, 1536498, 2178364, 3088365, 4378502, 6207581, 8800750

Offset: 2

Ilya Gutkovskiy, Sep 14 2021

nonn

Number of compositions (ordered partitions) of n into two or more squares.

Index entries for sequences related to sums of squares

Cf. A000290, A006456, A010052, A033985, A219610, A337165, A347806, A347807, A347808, A347809.

b:= proc(n, t) option remember; `if`(n=0, `if`(t=0, 1, 0), add((
     s->`if`(s>n, 0, b(n-s, max(0, t-1))))(j^2), j=1..isqrt(n)))
    end:
a:= n-> b(n, 2):
seq(a(n), n=2..48);  # Alois P. Heinz, Sep 14 2021

nmax = 48; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^2/(2 (3 - EllipticTheta[3, 0, x])), {x, 0, nmax}], x] // Drop[#, 2] &

From Alois P. Heinz, Sep 14 2021: (Start)
a(n) = A006456(n) - A010052(n).
a(n) = Sum_{k=2..n} A337165(n,k). (End)

A322340 Number of compositions (ordered partitions) of n into square pyramidal numbers (A000330).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 27, 36, 48, 64, 85, 114, 153, 205, 274, 365, 487, 651, 871, 1165, 1557, 2080, 2780, 3716, 4967, 6639, 8873, 11860, 15853, 21189, 28320, 37850, 50589, 67618, 90379, 120799, 161456, 215797, 288430, 385512, 515269, 688699

Offset: 0

Ilya Gutkovskiy, Dec 26 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..7939
Eric Weisstein's World of Mathematics, Square Pyramidal Number
Index entries for sequences related to compositions

Cf. A000330, A006456, A279220, A282582, A298246.

h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(t+1)*(2*t+1)/6>n, t-1, t))(1+h(n-1)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i*(i+1)*(2*i+1)/6), i=1..h(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 28 2018

nmax = 49; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1) (2 k + 1)/6), {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=1} x^(k*(k+1)*(2*k+1)/6)).

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)).

cp's OEIS Frontend

A337165 Number T(n,k) of compositions of n into k nonzero squares; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A331844 Number of compositions (ordered partitions) of n into distinct squares.

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A224366 Number of compositions of n^2 into sums of squares.

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A317665 Expansion of 1/Sum_{k>=0} x^(k^2).

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A280542 Expansion of 1/(1 - Sum_{k>=2} x^(k^2)).

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A322798 Number of compositions (ordered partitions) of n into hexagonal numbers (A000384).

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A322799 Number of compositions (ordered partitions) of n into heptagonal numbers (A000566).

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