A006456 -id:A006456 - cp's OEIS frontend

A282504 Expansion of 1/(1 - Sum_{k>=0} x^(2k(k+1)+1)).

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 21, 28, 37, 49, 65, 88, 119, 160, 214, 285, 381, 511, 687, 923, 1237, 1656, 2217, 2971, 3985, 5345, 7166, 9603, 12867, 17244, 23115, 30989, 41543, 55684, 74634, 100032, 134081, 179729, 240919, 322935, 432858, 580191, 777680, 1042407, 1397262, 1872911, 2510457

Offset: 0

Ilya Gutkovskiy, Feb 16 2017

nonn

Number of compositions (ordered partitions) into centered square numbers (A001844).
Conjecture: every number > 1 is the sum of at most 6 centered square numbers.
Extended conjecture: every number > 1 is the sum of at most k+2 centered k-gonal numbers.

			a(8) = 5 because we have [5, 1, 1, 1], [1, 5, 1, 1], [1, 1, 5, 1], [1, 1, 1, 5] and [1, 1, 1, 1, 1, 1, 1, 1].

Indranil Ghosh, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Centered Square Number
Index entries for sequences related to centered polygonal numbers
Index entries for sequences related to compositions

Cf. A001844, A006456, A280951.

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

Vec(1/(1 - sum(k=0, 54, x^(2*k*(k + 1) + 1))) + O(x^54)) \\ Indranil Ghosh, Mar 15 2017

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

a(n) ~ c / r^n, where r = 0.746043978237212782246711857485153004976647... is the root of the equation sqrt(r) * EllipticTheta(2, 0, r^2) = 2 and c = 0.453173429667590077751072798128748901015122665... . - Vaclav Kotesovec, Feb 17 2017

A322800 Number of compositions (ordered partitions) of n into octagonal numbers (A000567).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 37, 46, 56, 68, 83, 102, 126, 156, 195, 244, 304, 377, 466, 575, 709, 874, 1080, 1338, 1660, 2061, 2557, 3170, 3926, 4857, 6006, 7428, 9191, 11380, 14096, 17465, 21640, 26807, 33197, 41099

Offset: 0

Ilya Gutkovskiy, Dec 26 2018

nonn

Eric Weisstein's World of Mathematics, Octagonal Number
Index entries for sequences related to compositions

Cf. A000567, A006456, A023361, A181324, A279041, A279281, A322798, A322799.

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

nmax = 54; CoefficientList[Series[1/(1 - Sum[x^(k (3 k - 2)), {k, 1, nmax}]), {x, 0, nmax}], x]

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

A219331 L.g.f.: -log(1 - Sum_{n>=1} x^(n^2)) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, 1, 1, 5, 6, 7, 8, 13, 28, 36, 45, 59, 92, 134, 186, 269, 375, 538, 761, 1080, 1520, 2157, 3060, 4339, 6181, 8750, 12394, 17554, 24912, 35322, 50066, 70957, 100596, 142665, 202278, 286790, 406520, 576347, 817142, 1158528, 1642461, 2328536, 3301283, 4680417, 6635688

Offset: 1

Paul D. Hanna, Apr 12 2013

nonn

Limit a(n)/a(n+1) = 0.705346681379806989636379706393941505260078161512292870... is a real root of 1 = Sum_{n>=1} x^(n^2).

			L.g.f.: L(x) = x + x^2/2 + x^3/3 + 5*x^4/4 + 6*x^5/5 + 7*x^6/6 + 8*x^7/7 + 13*x^8/8 + 28*x^9/9 + 36*x^10/10 +...
where
exp(L(x)) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 7*x^8 + 11*x^9 + 16*x^10 + 22*x^11 + 30*x^12 +...+ A006456(n)*x^n +...
exp(-L(x)) = 1 - x - x^4 - x^9 - x^16 - x^25 - x^36 +...+ -x^(n^2) +...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A224607, A224608, A006456.

{a(n)=n*polcoeff(-log(1-sum(r=1,sqrtint(n+1),x^(r^2)+x*O(x^n))),n)}
for(n=1,50,print1(a(n),", "))

Logarithmic derivative of A006456, where A006456(n) is the number of compositions of n into sums of squares.

A240944 Number of compositions of n into square parts k^2 where there are k sorts of part k^2.

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 7, 9, 15, 28, 45, 66, 99, 164, 269, 422, 651, 1028, 1654, 2637, 4149, 6522, 10350, 16467, 26091, 41205, 65174, 103339, 163833, 259361, 410376, 649827, 1029543, 1630725, 2581848, 4087797, 6473832, 10253370, 16237375, 25711316, 40714953, 64478427, 102111230, 161701086, 256062990, 405499697, 642156651

Offset: 0

Joerg Arndt, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A006456 (compositions into squares).

a:= proc(n) option remember; `if`(n=0, 1, `if`(n<0, 0,
      add(k*a(n-k^2), k=1..isqrt(n))))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 04 2014

a[n_] := a[n] = If[n == 0, 1, If[n<0, 0, Sum[k*a[n-k^2], {k, Sqrt[n]}]]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Aug 29 2021, after Alois P. Heinz *)

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

G.f.: 1/(1 - sum(k>=1, k * x^(k^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)).

A303667 Expansion of 2/((1 - x)*(3 - theta_3(x))), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 13, 18, 25, 36, 52, 74, 104, 147, 209, 297, 421, 596, 845, 1199, 1701, 2411, 3417, 4844, 6868, 9738, 13806, 19573, 27749, 39342, 55778, 79079, 112112, 158944, 225342, 319479, 452941, 642152, 910404, 1290719, 1829911, 2594344, 3678108, 5214606, 7392970, 10481335

Offset: 0

Ilya Gutkovskiy, Apr 28 2018

nonn

Partial sums of A006456.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
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, A006456, A010052, A302833, A303668.

b:= proc(n) option remember;
      `if`(n=0, 1, add(b(n-i^2), i=1..isqrt(n)))
    end:
a:= proc(n) option remember;
      `if`(n<0, 0, b(n)+a(n-1))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 28 2018

nmax = 45; CoefficientList[Series[2/((1 - x) (3 - EllipticTheta[3, 0, x])), {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[1/((1 - x) (1 - Sum[x^k^2, {k, 1, nmax}])), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Boole[IntegerQ[k^(1/2)]] a[n - k], {k, 1, n}]; Accumulate[Table[a[n], {n, 0, 45}]]

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

A339419 Number of compositions (ordered partitions) of n into an odd number of squares.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 3, 1, 5, 5, 7, 14, 10, 27, 27, 44, 69, 73, 144, 158, 260, 366, 466, 775, 940, 1490, 2031, 2803, 4264, 5551, 8460, 11525, 16399, 23864, 32435, 47981, 66005, 94701, 135072, 187999, 272678, 379095, 543626, 769490, 1083788, 1553661, 2177681, 3113333

Offset: 0

Ilya Gutkovskiy, Dec 03 2020

nonn

			a(9) = 5 because we have [9], [4, 4, 1], [4, 1, 4], [1, 4, 4] and [1, 1, 1, 1, 1, 1, 1, 1, 1].

Cf. A000290, A006456, A166444, A317665, A339365, A339417, A339418.

b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+2*g-1, g+1 od; r fi
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 03 2020

nmax = 47; CoefficientList[Series[1/(3 - EllipticTheta[3, 0, x]) - 1/(1 + EllipticTheta[3, 0, x]), {x, 0, nmax}], x]

G.f.: 1 / (3 - theta_3(x)) - 1 / (1 + theta_3(x)), where theta_3() is the Jacobi theta function.
a(n) = (A006456(n) - A317665(n)) / 2.
a(n) = -Sum_{k=0..n-1} A006456(k) * A317665(n-k).

A347711 Number of compositions (ordered partitions) of n into at most 4 squares.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 1, 4, 8, 3, 5, 6, 6, 12, 2, 5, 16, 7, 8, 10, 15, 12, 3, 15, 14, 16, 12, 8, 30, 16, 1, 18, 23, 18, 17, 18, 21, 28, 8, 11, 42, 19, 15, 32, 30, 24, 5, 23, 39, 30, 20, 20, 48, 36, 6, 34, 44, 21, 36, 24, 36, 52, 2, 34, 60, 31, 23, 36, 66, 36

Offset: 0

Ilya Gutkovskiy, Sep 10 2021

nonn

Index entries for sequences related to sums of squares

Cf. A000290, A002654, A006456, A063730, A337165, A347710, A347712, A347713.

a(n) = Sum_{k=0..4} A337165(n,k). - Alois P. Heinz, Sep 10 2021

A347806 Expansion of (theta_3(x) - 1)^3 / (4 * (3 - theta_3(x))).

Original entry on oeis.org

1, 1, 1, 4, 5, 6, 10, 14, 22, 30, 41, 62, 88, 123, 173, 248, 354, 500, 710, 1006, 1427, 2024, 2867, 4066, 5767, 8176, 11591, 16436, 23301, 33032, 46832, 66396, 94137, 133461, 189209, 268252, 380315, 539190, 764431, 1083764, 1536498, 2178364, 3088363, 4378502, 6207581

Offset: 3

Ilya Gutkovskiy, Sep 14 2021

nonn

Number of compositions (ordered partitions) of n into 3 or more squares.

Index entries for sequences related to sums of squares

Cf. A000290, A006456, A337165, A347805, 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, 3):
seq(a(n), n=3..47);  # Alois P. Heinz, Sep 14 2021

nmax = 47; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^3/(4 (3 - EllipticTheta[3, 0, x])), {x, 0, nmax}], x] // Drop[#, 3] &

a(n) = Sum_{k=3..n} A337165(n,k). - Alois P. Heinz, Sep 14 2021

A347807 Expansion of (theta_3(x) - 1)^4 / (8 * (3 - theta_3(x))).

Original entry on oeis.org

1, 1, 1, 5, 6, 7, 14, 19, 29, 41, 56, 88, 123, 170, 245, 351, 500, 704, 1003, 1427, 2021, 2867, 4060, 5763, 8176, 11585, 16430, 23301, 33032, 46826, 66393, 94131, 133458, 189209, 268243, 380315, 539190, 764422, 1083758, 1536495, 2178361, 3088357, 4378496, 6207581

Offset: 4

Ilya Gutkovskiy, Sep 14 2021

nonn

Number of compositions (ordered partitions) of n into 4 or more squares.

Index entries for sequences related to sums of squares

Cf. A000290, A006456, A337165, A347805, A347806, 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, 4):
seq(a(n), n=4..47);  # Alois P. Heinz, Sep 14 2021

nmax = 47; CoefficientList[Series[(EllipticTheta[3, 0, x] - 1)^4/(8 (3 - EllipticTheta[3, 0, x])), {x, 0, nmax}], x] // Drop[#, 4] &

a(n) = Sum_{k=4..n} A337165(n,k). - Alois P. Heinz, Sep 14 2021

cp's OEIS Frontend

A282504 Expansion of 1/(1 - Sum_{k>=0} x^(2*k*(k+1)+1)).

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A322800 Number of compositions (ordered partitions) of n into octagonal numbers (A000567).

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A219331 L.g.f.: -log(1 - Sum_{n>=1} x^(n^2)) = Sum_{n>=1} a(n)*x^n/n.

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A240944 Number of compositions of n into square parts k^2 where there are k sorts of part k^2.

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

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A303667 Expansion of 2/((1 - x)*(3 - theta_3(x))), where theta_3() is the Jacobi theta function.

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A339419 Number of compositions (ordered partitions) of n into an odd number of squares.

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A347711 Number of compositions (ordered partitions) of n into at most 4 squares.

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A282504 Expansion of 1/(1 - Sum_{k>=0} x^(2k(k+1)+1)).