A014110 -id:A014110 - cp's OEIS frontend

A045847 Matrix whose (i,j)-th entry is number of representations of j as a sum of i squares of nonnegative integers; read by diagonals.

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 4, 3, 0, 1, 0, 1, 5, 6, 1, 2, 0, 0, 1, 6, 10, 4, 3, 2, 0, 0, 1, 7, 15, 10, 5, 6, 0, 0, 0, 1, 8, 21, 20, 10, 12, 3, 0, 0, 0, 1, 9, 28, 35, 21, 21, 12, 0, 1, 1, 0, 1, 10, 36, 56, 42, 36, 30, 4, 3, 2, 0, 0, 1, 11, 45, 84, 78, 63, 61, 20, 6, 6, 2, 0, 0

Offset: 0

N. J. A. Sloane

			Rows are
1,0,0,..;
1,1,0,0,1,0..;
1,2,1,0,2,2,..;
1,3,3,1,...

Seiichi Manyama, Ascending antidiagonals n = 0..139, flattened
H. Wilf, A combinatorial determinant, arXiv:math/9809120 [math.CO], 1998.

Rows k=0-10 give: A000007, A010052, A000925, A002102, A014110, A038671, A045848, A045849, A045850, A045851, A045852.
Diagonal gives A287617.
Antidiagonal sums give A302018.

i-th row is expansion of (1+x+x^4+x^9+...)^i.

More terms from Erich Friedman

A224213 Number of nonnegative solutions to x^2 + y^2 + z^2 + u^2 <= n.

Original entry on oeis.org

1, 5, 11, 15, 20, 32, 44, 48, 54, 70, 88, 100, 108, 124, 148, 160, 165, 189, 219, 235, 253, 281, 305, 317, 329, 357, 399, 427, 439, 475, 523, 539, 545, 581, 623, 659, 688, 716, 764, 792, 810, 858, 918, 946, 970, 1030, 1078, 1102, 1110, 1154, 1226, 1274, 1304, 1352

Offset: 0

Alex Ratushnyak, Apr 01 2013

nonn

Cf. A014110 (first differences).
Cf. A224212 (number of nonnegative solutions to x^2 + y^2 <= n).
Cf. A000606 (number of nonnegative solutions to x^2 + y^2 + z^2 <= n).
Cf. A046895 (number of integer solutions to x^2 + y^2 + z^2 + u^2 <= n).

nn = 50; t = Table[0, {nn}]; Do[d = x^2 + y^2 + z^2 + u^2; If[0 < d <= nn, t[[d]]++], {x, 0, nn}, {y, 0, nn}, {z, 0, nn}, {u, 0, nn}]; Accumulate[Join[{1}, t]] (* T. D. Noe, Apr 01 2013 *)

for n in range(99):
  k = 0
  for x in range(99):
    s = x*x
    if s>n: break
    for y in range(99):
        sy = s + y*y
        if sy>n: break
        for z in range(99):
            sz = sy + z*z
            if sz>n: break
            for u in range(99):
              su = sz + u*u
              if su>n: break
              k+=1
  print(str(k), end=', ')

G.f.: (1/(1 - x))*(Sum_{k>=0} x^(k^2))^4. - Ilya Gutkovskiy, Mar 14 2017

A282288 Expansion of (Sum_{k>=0} x^(k^4))^19.

Original entry on oeis.org

1, 19, 171, 969, 3876, 11628, 27132, 50388, 75582, 92378, 92378, 75582, 50388, 27132, 11628, 3876, 988, 513, 2926, 15505, 58140, 162792, 352716, 604656, 831402, 923780, 831402, 604656, 352716, 162792, 58140, 15504, 3078, 3249, 23275, 116280, 406980, 1058148, 2116296, 3325608, 4157010, 4157010, 3325608

Offset: 0

Ilya Gutkovskiy, Feb 12 2017

Number of ways to write n as an ordered sum of 19 fourth powers (A000583).
a(n) > 0 for all n >= 0.
Every natural number is the sum of at most 19 fourth powers (Balasubramanian, 1986).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Biquadratic Number
Eric Weisstein's World of Mathematics, Waring's Problem

Cf. A000583, A002377, A002804, A014110, A173682.

nmax = 42; CoefficientList[Series[Sum[x^k^4, {k, 0, nmax}]^19, {x, 0, nmax}], x]

G.f.: (Sum_{k>=0} x^(k^4))^19.

A347803 Expansion of ( Sum_{k>=0} k^2 * q^(k^2) )^4.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 16, 0, 0, 96, 0, 36, 256, 0, 432, 256, 0, 1728, 64, 486, 2304, 768, 3888, 0, 3072, 7776, 1728, 7112, 0, 13824, 12864, 0, 27648, 6336, 15552, 9261, 18688, 62208, 21744, 24576, 0, 72576, 51456, 24300, 117504, 38400, 101088, 9216, 93184, 155520, 86400, 142382, 62208, 352512, 67344, 0, 202752, 286176

Offset: 0

Seiichi Manyama, Sep 14 2021

nonn

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

Cf. A000414, A014110, A037216, A347801, A347802.

a(n) = sum(i=1, n, sum(j=1, n, sum(k=1, n, sum(m=1, n, (i^2+j^2+k^2+m^2==n)*(i*j*k*m)^2))));

my(N=66, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=0, sqrtint(N), k^2*x^k^2)^4))

a(n) is sum of i^2 * j^2 * k^2 *m^2 for positive integers i,j,k,m such that i^2+j^2+k^2+m^2=n.

A123337 Number of ordered ways to write n as the sum of 5 squares less than 5^2.

Original entry on oeis.org

1, 5, 10, 10, 10, 21, 30, 20, 15, 35, 50, 40, 30, 45, 70, 60, 30, 55, 100, 80, 56, 90, 110, 80, 60, 85, 120, 100, 60, 90, 130, 80, 35, 90, 120, 80, 65, 85, 90, 60, 35, 60, 90, 50, 30, 61, 60, 20, 10, 50, 40, 30, 25, 20, 30, 0, 10, 20, 20, 10, 0, 20, 0, 0, 5, 5, 10, 0, 5, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Jonathan Vos Post, Oct 11 2006

Through n = 24, a(n) = number of ordered ways to write n as the sum of 5 squares. For n > 24, we must exclude sums which include 5^2, 6^2 and the like. The values of n such that a(n) = 0 are 55, 60, 62, 63, 67, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79 and all n > 80. Without the restriction on the size of squares, all natural numbers can be written as the sum of 4 squares, as Lagrange proved in 1750.

			a(0) = 1 because the unique such sum is 0 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2.
a(1) = 5 because there are 5 permutations of 1 = 1^2 + 0^2 + 0^2 + 0^2 + 0^2, such as 1 = 0^2 + 1^2 + 0^2 + 0^2 + 0^2.
a(2) = 10 because there are 10 permutations of 2 = 1^2 + 1^2 + 0^2 + 0^2 + 0^2, such as 2 = 1^2 + 0^2 + 1^2 + 0^2 + 0^2.
a(5) = 21 because of the unique sum 5 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 and also 20 permutations of 5 = 2^2 + 1^2 + 0^2 + 0^2 + 0^2.
a(16) = 30 because there are 5 permutations of 16 = 4^2 + 0^2 + 0^2 + 0^2 + 0^2 and 5 permutations of 16 = 0^2 + 2^2 + 2^2 + 2^2 + 2^2 and 20 permutations of 16 = 3^2 + 2^2 + 1^2 + 1^2 + 1^2.

Cf. A000118, A014110.

a[n_] := Total[ Length /@ Permutations /@ IntegerPartitions[n, {5}, Range[0, 4]^2]]; a /@ Range[0, 80] (* Giovanni Resta, Jun 13 2016 *)

23 terms corrected by Giovanni Resta, Jun 13 2016

A059160 Number of ordered ways of writing n as a sum of 5 generalized pentagonal numbers (A001318).

Original entry on oeis.org

1, 5, 15, 30, 45, 56, 65, 85, 115, 150, 171, 175, 185, 205, 260, 300, 325, 340, 350, 415, 440, 485, 500, 505, 580, 581, 650, 645, 675, 815, 815, 910, 845, 865, 985, 951, 1130, 1030, 1060, 1155, 1150, 1370, 1265, 1410, 1495, 1420, 1545, 1460, 1600, 1675, 1690

Offset: 0

Judson Neer, Feb 14 2001

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A001318, A008443, A014110, A080995.

a[n_] := SeriesCoefficient[(QPochhammer[-q, q^3]* QPochhammer[-q^2, q^3]*QPochhammer[q^3, q^3])^5, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jun 12 2017 *)

G.f.: f(x, x^2)^5 where f(,) is Ramanujan's two-variable theta function. - Michael Somos, Jun 08 2012

A123999 Number of ordered ways of writing n as a sum of 4 squares of nonnegative numbers less than 4.

Original entry on oeis.org

1, 4, 6, 4, 5, 12, 12, 4, 6, 16, 18, 12, 8, 16, 24, 12, 1, 12, 18, 12, 6, 4, 12, 12, 0, 0, 6, 4, 4, 0, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Jonathan Vos Post, Oct 31 2006

Through n = 15, a(n) = number of ordered ways to write n as the sum of 4 squares. For n > 15, we must exclude sums which include 4^2, 5^2, 6^2 and the like. The values of n such that a(n) = 0 are 16, 24, 25, 29, 30, 32, 33, 34, 35 and all n > 36. Without the restriction on the size of squares, all natural numbers can be written as the sum of 4 squares, as Lagrange proved in 1750. This sequence is to 4 as A123337 Number of ordered ways to write n as the sum of 5 squares less than 5, is to 5.

			a(0) = 1 because of the unique sum 0 = 0^2 + 0^2 + 0^2 + 0^2.
a(1) = 4 because of the 4 permutations 1 = 0^2 + 0^2 + 0^2 + 1^2 = 0^2 + 0^2 + 1^2 + 0^2 = 0^2 + 1^2 + 0^2 + 0^2 = 1^2 + 0^2 + 0^2 + 0^2.
a(4) = 5 because of 4 = 1^2 + 1^2 + 1^2 + 1^2 plus the 4 permutations of 4 = 0^2 + 0^2 + 0^2 + 2^2.
a(16) = 1 because 16 = 2^2 + 2^2 + 2^2 + 2^2.

Cf. A000118, A014110, A123337.

a[n_] := Total[ Length /@ Permutations /@ IntegerPartitions[n, {4}, Range[0, 3]^2]]; a /@ Range[0, 72] (* Giovanni Resta, Jun 13 2016 *)

a(n) = Card{(a,b,c,d) such that 0<=a,b,c,d<4 and a^2 + b^2 + c^2 + d^2 = n}.

Corrected typo in third example Dave Zobel (dzobel(AT)alumni.caltech.edu), Mar 07 2009

a(16) and related example corrected by Giovanni Resta, Jun 13 2016

A317645 Expansion of (1 + theta_3(q))^3*(1 + theta_3(q^2))/16, where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 3, 4, 4, 6, 7, 6, 6, 7, 9, 12, 10, 10, 15, 10, 6, 12, 15, 16, 18, 16, 16, 18, 12, 12, 18, 24, 22, 24, 25, 10, 18, 19, 18, 30, 26, 24, 33, 30, 12, 24, 27, 30, 36, 28, 31, 24, 24, 22, 33, 32, 30, 42, 43, 36, 24, 34, 24, 48, 46, 24, 51, 34, 30, 36, 30, 34, 54, 48, 42, 48, 30, 37, 45, 54, 38

Offset: 0

Ilya Gutkovskiy, Aug 02 2018

nonn

Number of nonnegative integer solutions to the equation x^2 + y^2 + z^2 + 2*w^2 = n.

			G.f. = 1 + 3*q + 4*q^2 + 4*q^3 + 6*q^4 + 7*q^5 + 6*q^6 + 6*q^7 + 7*q^8 + ...

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A014110, A156384, A236928.

nmax = 75; CoefficientList[Series[(1 + EllipticTheta[3, 0, q])^3 (1 + EllipticTheta[3, 0, q^2])/16, {q, 0, nmax}], q]
nmax = 75; CoefficientList[Series[(1 + QPochhammer[-q, -q]/QPochhammer[q, -q])^3 (1 + QPochhammer[-q^2, -q^2]/QPochhammer[q^2, -q^2])/16, {q, 0, nmax}], q]

A374014 Expansion of (Sum_{k>=0} x^(k^4))^16.

Original entry on oeis.org

1, 16, 120, 560, 1820, 4368, 8008, 11440, 12870, 11440, 8008, 4368, 1820, 560, 120, 16, 17, 240, 1680, 7280, 21840, 48048, 80080, 102960, 102960, 80080, 48048, 21840, 7280, 1680, 240, 16, 120, 1680, 10920, 43680, 120120, 240240, 360360, 411840, 360360, 240240, 120120, 43680, 10920, 1680, 120, 0

Offset: 0

Seiichi Manyama, Jun 25 2024

nonn

Number of ways to write n as an ordered sum of 16 fourth powers (A000583).

Seiichi Manyama, Table of n, a(n) for n = 0..20000
Eric Weisstein's World of Mathematics, Waring's Problem

Cf. A000583, A002377, A014110, A099591.

Cf. A282288, A374016.

my(N=50, x='x+O('x^N)); Vec(sum(k=0, sqrtnint(N, 4), x^k^4)^16)

a(A099591(n)) = 0.

a(n) > 0 for n > 13792.

A317646 Expansion of (1 + theta_3(q))^2*(1 + theta_3(q^2))^2/16, where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 5, 4, 5, 8, 8, 8, 11, 8, 6, 8, 5, 10, 14, 12, 16, 12, 11, 8, 11, 14, 14, 20, 18, 12, 14, 12, 5, 20, 19, 20, 30, 16, 17, 16, 16, 18, 24, 20, 25, 28, 14, 16, 11, 22, 25, 28, 34, 20, 30, 24, 18, 28, 26, 28, 42, 24, 20, 32, 5, 28, 36, 28, 41, 32, 32, 20, 30, 30, 28, 44

Offset: 0

Ilya Gutkovskiy, Aug 02 2018

nonn

Number of nonnegative integer solutions to the equation x^2 + y^2 + 2*z^2 + 2*w^2 = n.

			G.f. = 1 + 2*q + 3*q^2 + 4*q^3 + 5*q^4 + 4*q^5 + 5*q^6 + 4*q^7 + 5*q^8 + ...

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A014110, A097057, A317645.

nmax = 75; CoefficientList[Series[(1 + EllipticTheta[3, 0, q])^2 (1 + EllipticTheta[3, 0, q^2])^2/16, {q, 0, nmax}], q]
nmax = 75; CoefficientList[Series[(1 + QPochhammer[-q, -q]/QPochhammer[q, -q])^2 (1 + QPochhammer[-q^2, -q^2]/QPochhammer[q^2, -q^2])^2/16, {q, 0, nmax}], q]

cp's OEIS Frontend

A045847 Matrix whose (i,j)-th entry is number of representations of j as a sum of i squares of nonnegative integers; read by diagonals.

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A224213 Number of nonnegative solutions to x^2 + y^2 + z^2 + u^2 <= n.

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A282288 Expansion of (Sum_{k>=0} x^(k^4))^19.

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A347803 Expansion of ( Sum_{k>=0} k^2 * q^(k^2) )^4.

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A123337 Number of ordered ways to write n as the sum of 5 squares less than 5^2.

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A059160 Number of ordered ways of writing n as a sum of 5 generalized pentagonal numbers (A001318).

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A123999 Number of ordered ways of writing n as a sum of 4 squares of nonnegative numbers less than 4.

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A317645 Expansion of (1 + theta_3(q))^3*(1 + theta_3(q^2))/16, where theta_3() is the Jacobi theta function.

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A374014 Expansion of (Sum_{k>=0} x^(k^4))^16.