A006456 -id:A006456 - cp's OEIS frontend

A354651 G.f. A(x) satisfies: 1/(1 - x) = Sum_{n>=1} (-1)^(n-1) * A(x)^(n^2).

1, 1, 1, 2, 5, 11, 25, 64, 168, 434, 1136, 3046, 8246, 22400, 61290, 169036, 468628, 1304390, 3646104, 10232796, 28814306, 81376616, 230462906, 654363034, 1862260359, 5311064061, 15176758091, 43448083792, 124593820615, 357853635931, 1029326055479, 2964817204082

Offset: 1

Paul D. Hanna, Jun 18 2022

nonn

Conjectures:
(C.1) a(4*n) = 0 (mod 2) for n >= 0.
(C.2) a(4*n+1) = a(4*n+2) = a(4*n+3) (mod 2) for n >= 0.
(C.3) a(4*n+1) = a(4*n+3) (mod 4) for n >= 0.

			G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 11*x^6 + 25*x^7 + 64*x^8 + 168*x^9 + 434*x^10 + 1136*x^11 + 3046*x^12 + 8246*x^13 + 22400*x^14 + ...
where
1/(1-x) = A(x) - A(x)^4 + A(x)^9 - A(x)^16 + A(x)^25 - A(x)^36 + A(x)^49 -+ ... + (-1)^(n-1) * A(x)^(n^2) + ...
By the Jacobi triple product
(1 - 3*x)/(1 - x) = (1 - A(x)^2)*(1 - A(x))^2 * (1 - A(x)^4) * (1 - A(x)^3)^2 * (1 - A(x)^6) * (1 - A(x)^5)^2 * (1 - A(x)^8) * (1 - A(x)^7)^2 * ...

Paul D. Hanna, Table of n, a(n) for n = 1..625
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A006456 (the series reversion of -A(-x) is the g.f. for A006456, apart from the initial term).

Cf. A355151.

{a(n) = my(A=[0,1],t); for(i=1,n, A = concat(A,0); t = sqrtint(#A)+1;
A[#A] = 1 + polcoeff( sum(n=1,t, (-1)^n * Ser(A)^(n^2)), #A-1)); H=A; A[n+1]}
for(n=1,40,print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) (1 - 3*x)/(1 - x) = Sum_{n=-oo..+oo} (-1)^n * A(x)^(n^2).
(2) (1 - 3*x)/(1 - x) = Product_{n>=1} (1 - A(x)^(2*n)) * (1 - A(x)^(2*n-1))^2, by the Jacobi triple product identity.
(3) (1 - 3*x)^2/(1 - x)^2 = 1 + 4*Sum_{n>=1} (-1)^n * A(x)^(2*n-1)/(1 + A(x)^(2*n-1)), by a q-series identity for the Jacobi theta_3 function.
(4) (1 - 3*x)^4/(1 - x)^4 = 1 + 8*Sum_{n>=1} (-1)^n * n * A(x)^n/(1 + A(x)^n), by a q-series identity for the Jacobi theta_3 function.

A363748 Number of compositions into sums of fourth powers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 22, 26, 31, 37, 44, 52, 61, 71, 82, 94, 107, 121, 136, 152, 169, 188, 210, 236, 267, 304, 348, 400, 461, 532, 614, 708, 815, 936, 1072, 1224, 1393, 1581, 1791, 2027, 2294, 2598, 2946, 3346, 3807, 4339, 4953, 5661, 6476, 7412, 8484, 9708, 11101, 12682, 14474

Offset: 0

Seiichi Manyama, Jun 19 2023

nonn

This sequence is different from A291149.

			a(18)=4 counts the compositions 1^4+1^4+1^4+2^4 = 1^4+1^4+2^4+1^4 = 1^4+2^4+1^4+1^4 = 2^4+1^4+1^4+1^4. - _R. J. Mathar_, Jun 21 2023

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

Cf. A006456, A023358, A363749.

Cf. A000583, A291149, A352529.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, ispower(j, 4)*v[i-j+1])); v;

G.f.: 1/(1 - Sum_{k>=1} x^(k^4)).

A363749 Number of compositions into sums of fifth powers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 38, 42, 47, 53, 60, 68, 77, 87, 98, 110, 123, 137, 152, 168, 185

Offset: 0

Seiichi Manyama, Jun 19 2023

nonn

This sequence is different from A291168.

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

Cf. A006456, A023358, A363748.

Cf. A000584, A291168, A352530.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, ispower(j, 5)*v[i-j+1])); v;

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

A232266 Triangle where T(n,k) = number of compositions of n^2 - k^2 into sums of squares for k=0..n, n>=0, as read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 11, 7, 3, 1, 124, 88, 30, 5, 1, 2870, 2024, 710, 124, 11, 1, 133462, 94137, 33033, 5767, 502, 22, 1, 12477207, 8800750, 3088365, 539192, 46832, 2024, 43, 1, 2344649612, 1653790807, 580347968, 101321507, 8800750, 380315, 8176, 88, 1, 885591183971, 624648802700, 219201637352, 38269865019, 3324109524, 143647802, 3088365, 33033, 175, 1

Offset: 0

Paul D. Hanna, Nov 21 2013

			Triangle begins:
1;
1, 1;
2, 1, 1;
11, 7, 3, 1;
124, 88, 30, 5, 1;
2870, 2024, 710, 124, 11, 1;
133462, 94137, 33033, 5767, 502, 22, 1;
12477207, 8800750, 3088365, 539192, 46832, 2024, 43, 1;
2344649612, 1653790807, 580347968, 101321507, 8800750, 380315, 8176, 88, 1;
885591183971, 624648802700, 219201637352, 38269865019, 3324109524, 143647802, 3088365, 33033, 175, 1; ...
where T(n,k) = coefficient of x^(n^2-k^2) in the series:
1/(1 - x - x^4 - x^9 - x^16 - x^25 - x^36 -...- x^(n^2) -...) = 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 + 43*x^13 + 62*x^14 + 88*x^15 + 124*x^16 + 175*x^17 + 249*x^18 + 354*x^19 + 502*x^20 + 710*x^21 + 1006*x^22 + 1427*x^23 + 2024*x^24 + 2870*x^25 +...

Cf. A006456, A224366.

{T(n,k)=polcoeff(1/(1-sum(m=1,n+1,x^(m^2))+x*O(x^(n^2-k^2))),n^2-k^2)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

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

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.

A282583 Number of compositions (ordered partitions) of n into quarter-squares (A002620).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 19, 33, 60, 107, 193, 345, 621, 1113, 1999, 3586, 6439, 11554, 20741, 37223, 66814, 119916, 215237, 386310, 693375, 1244494, 2233686, 4009113, 7195757, 12915268, 23180946, 41606232, 74676840, 134033474, 240569601, 431785583, 774989076, 1390986741, 2496608365, 4481029864, 8042762869

Offset: 0

Ilya Gutkovskiy, Feb 19 2017

nonn

			a(4) = 6 because we have [4], [2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2] and [1, 1, 1, 1].

Indranil Ghosh, Table of n, a(n) for n = 0..200
Index entries for sequences related to compositions

Cf. A002620, A006456, A197081, A197122.

nmax = 40; CoefficientList[Series[1/(1 - Sum[x^Floor[k^2/4], {k, 2, nmax}]), {x, 0, nmax}], x]

Vec(1/(1 - sum(k=2, 40, x^floor(k^2/4)) + O(x^41))) \\ Indranil Ghosh, Mar 15 2017

G.f.: 1/(1 - Sum_{k>=2} x^(floor(k^2/4))).

A300715 Number of compositions (ordered partitions) of n into squares that do not divide n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 4, 3, 0, 0, 7, 6, 0, 0, 14, 10, 4, 0, 22, 20, 10, 0, 32, 39, 20, 0, 49, 70, 42, 0, 12, 116, 88, 0, 128, 156, 174, 11, 207, 3, 320, 0, 333, 551, 575, 0, 555, 914, 0, 0, 959, 1502, 1829, 44, 1691, 2486, 3192, 0, 3000, 4172, 4005

Offset: 0

Ilya Gutkovskiy, Mar 11 2018

nonn

			a(21) = 4 because we have [9, 4, 4, 4], [4, 9, 4, 4], [4, 4, 9, 4] and [4, 4, 4, 9].

Cf. A000290, A006456, A294105, A294265, A294266, A300702, A300703, A300704, A300706.

a:= proc(m) option remember; local b; b:= proc(n) option
      remember; `if`(n=0, 1, add((s->`if`(s>n or irem(m, s)
       =0, 0, b(n-s)))(j^2), j=2..isqrt(n))) end; b(m)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 11 2018

Table[SeriesCoefficient[1/(1 - Sum[Boole[Mod[n, k] != 0 && IntegerQ[k^(1/2)]] x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 75}]

A301503 Number of compositions (ordered partitions) of n into square parts (A000290) such that no two adjacent parts are equal (Carlitz compositions).

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 1, 0, 0, 2, 4, 2, 0, 2, 7, 8, 4, 3, 7, 14, 16, 11, 9, 18, 32, 35, 30, 32, 49, 74, 87, 83, 84, 120, 178, 209, 205, 219, 305, 434, 515, 523, 572, 785, 1080, 1255, 1303, 1488, 2002, 2644, 3058, 3284, 3849, 5077, 6518, 7525, 8319, 9927, 12803, 16051, 18623, 21081

Offset: 0

Ilya Gutkovskiy, Mar 22 2018

nonn

			a(10) = 4 because we have [9, 1], [4, 1, 4, 1], [1, 9] and [1, 4, 1, 4].

Cf. A000290, A003242, A006456, A301502.

nmax = 61; CoefficientList[Series[1/(1 - Sum[x^k^2/(1 + x^k^2), {k, 1, nmax}]), {x, 0, nmax}], x]

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

A332006 Number of compositions (ordered partitions) of n into distinct centered square numbers.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 6, 0, 0, 0, 0, 0, 0, 2, 6, 0, 1, 2, 6, 24, 0, 2, 6, 0, 0, 0, 0, 0, 0, 2, 6, 0, 0, 0, 6, 24, 1, 2, 0, 0, 0, 4, 12, 0, 0, 0, 6, 24, 0, 2, 6, 0, 0, 0, 12, 48, 0, 0, 0, 24, 121, 4, 6

Offset: 0

Ilya Gutkovskiy, Feb 04 2020

nonn

			a(19) = 6 because we have [13, 5, 1], [13, 1, 5], [5, 13, 1], [5, 1, 13], [1, 13, 5] and [1, 5, 13].

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

Cf. A001844, A006456, A280951, A281082, A282504, A331844, A331984, A332005.

cp's OEIS Frontend

A354651 G.f. A(x) satisfies: 1/(1 - x) = Sum_{n>=1} (-1)^(n-1) * A(x)^(n^2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A363748 Number of compositions into sums of fourth powers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A363749 Number of compositions into sums of fifth powers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A232266 Triangle where T(n,k) = number of compositions of n^2 - k^2 into sums of squares for k=0..n, n>=0, as read by rows.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A282583 Number of compositions (ordered partitions) of n into quarter-squares (A002620).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A300715 Number of compositions (ordered partitions) of n into squares that do not divide n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A301503 Number of compositions (ordered partitions) of n into square parts (A000290) such that no two adjacent parts are equal (Carlitz compositions).

Views

Author

Keywords

Examples

Links