A166952 -id:A166952

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A066535

Number of ways of writing n as a sum of n squares.

+10
25

1, 2, 4, 8, 24, 112, 544, 2368, 9328, 34802, 129064, 491768, 1938336, 7801744, 31553344, 127083328, 509145568, 2035437440, 8148505828, 32728127192, 131880275664, 532597541344, 2153312518240, 8710505815360, 35250721087168, 142743029326162, 578472382307304

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

John Holley-Reid and Jeremy Rouse, The number of representations of n as a growing number of squares, arXiv:1910.01001 [math.NT], 2019.

FORMULA

a(n) equals the coefficient of x^n in the n-th power of Jacobi theta_3(x) where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). - Paul D. Hanna, Oct 25 2009

a(n) ~ c * d^n / sqrt(n), where d = 4.13273137623493996302796465... (= 1/radius of convergence A166952), c = 0.2820942036723951157919967... . - Vaclav Kotesovec, Sep 12 2014

EXAMPLE

There are a(3) = 8 solutions (x,y,z) of 3 = x^2 + y^2 + z^2: (1,1,1), (-1,-1,-1), 3 permutations of (1,1,-1) and 3 permutations of (1,-1,-1).

MAPLE

b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,

b(n, t-1) +2*add(b(n-j^2, t-1), j=1..isqrt(n))))

end:

a:= n-> b(n$2):

seq(a(n), n=0..30); # Alois P. Heinz, Jul 16 2014

MATHEMATICA

Join[{1}, Table[SquaresR[n, n], {n, 24}]]

PROG

(PARI) {a(n)=local(THETA3=1+2*sum(k=1, sqrtint(n), x^(k^2))+x*O(x^n)); polcoeff(THETA3^n, n)} /* Paul D. Hanna, Oct 25 2009 */

CROSSREFS

Cf. A004018, A005875, A000118, A066536.

Cf. A122141, A166952. - Paul D. Hanna, Oct 25 2009

a(n^2) gives A361431.

KEYWORD

nonn

AUTHOR

Peter Bertok (peter(AT)bertok.com), Jan 07 2002

EXTENSIONS

Edited by Dean Hickerson, Jan 12 2002

a(0) added by Paul D. Hanna, Oct 25 2009

Edited by R. J. Mathar, Oct 29 2009

STATUS

approved

A302860

a(n) = [x^n] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.

+10
6

1, 3, 9, 27, 89, 333, 1341, 5449, 21697, 84663, 327829, 1275739, 5020457, 19964623, 79883141, 320317827, 1284656385, 5152761033, 20686311261, 83182322509, 335110196569, 1352277390001, 5463873556381, 22097867887045, 89441286136465, 362277846495883, 1468465431530457

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

a(n) = number of integer lattice points inside the n-dimensional hypersphere of radius sqrt(n).

LINKS

Table of n, a(n) for n=0..26.

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Index entries for sequences related to sums of squares

FORMULA

a(n) = A122510(n,n).

a(n) ~ c / (sqrt(n) * r^n), where r = 0.241970723224463308846762732757915397312... (= radius of convergence A166952) and c = 0.716940866073606328... - Vaclav Kotesovec, Apr 14 2018

MATHEMATICA

Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n/(1 - x), {x, 0, n}], {n, 0, 26}]

Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, -n, n}]^n, {x, 0, n}], {n, 0, 26}]

CROSSREFS

Main diagonal of A122510.

Cf. A000122, A001650, A046895, A057655, A066535, A117609, A302861, A066536, A166952.

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Apr 14 2018

STATUS

approved

A066536

Number of ways of writing n as a sum of n+1 squares.

+10
4

1, 4, 12, 32, 90, 312, 1288, 5504, 22608, 88660, 339064, 1297056, 5043376, 19975256, 80027280, 321692928, 1291650786, 5177295432, 20748447108, 83279292960, 335056780464, 1351064867328, 5456890474248, 22063059606912

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

a(n) equals the coefficient of x^n in the (n+1)-th power of Jacobi theta_3(x) where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). - Paul D. Hanna, Oct 26 2009

a(n) is divisible by n+1: a(n)/(n+1) = A166952(n) for n>=0. - Paul D. Hanna, Oct 26 2009

a(n) ~ c * d^n / sqrt(n), where d = 4.13273137623493996302796465... (= 1/radius of convergence A166952), c = 0.70710538549959357505200... . - Vaclav Kotesovec, Sep 10 2014

EXAMPLE

There are a(2)=12 solutions (x,y,z) of 2=x^2+y^2+z^2: 3 permutations of (1,1,0), 3 permutations of (-1,-1,0) and 6 permutations of (1, -1,0).

MATHEMATICA

Join[{1}, Table[SquaresR[n+1, n], {n, 24}]]

(* Calculation of constants {d, c}: *) {1/r, Sqrt[s/(2*Pi*r^2*Derivative[0, 0, 2][EllipticTheta][3, 0, r*s])]} /. FindRoot[{s == EllipticTheta[3, 0, r*s], r*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s] == 1}, {r, 1/4}, {s, 5/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Nov 16 2023 *)

PROG

(PARI) {a(n)=local(THETA3=1+2*sum(k=1, sqrtint(n), x^(k^2))+x*O(x^n)); polcoeff(THETA3^(n+1), n)} /* Paul D. Hanna, Oct 26 2009*/

CROSSREFS

Cf. A004018, A005875, A000118, A066535.

Cf. A122141, A166952. - Paul D. Hanna, Oct 26 2009

KEYWORD

nonn

AUTHOR

Peter Bertok (peter(AT)bertok.com), Jan 07 2002

EXTENSIONS

Edited by Dean Hickerson, Jan 12, 2002

a(0) added by Paul D. Hanna, Oct 26 2009

Edited by R. J. Mathar, Oct 29 2009

STATUS

approved

A212326

G.f. satisfies: A(x) = theta_3( x*A(x) )^2, where theta_3(x) is Jacobi's theta_3 function.

+10
0

1, 4, 20, 112, 676, 4312, 28704, 197600, 1397060, 10090676, 74152456, 552666448, 4167528000, 31736182776, 243698432960, 1884809367456, 14668777816708, 114789815231560, 902661488046900, 7129068237647408, 56524456978032904, 449752267499647104

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..21.

FORMULA

G.f. A(x) satisfies:

(1) A(x) = (1 + 2*Sum_{n>=1} (x*A(x))^(n^2) )^2.

(2) A(x) = 1 + 4*Sum_{n>=1} (x*A(x))^n / (1 + (x*A(x))^(2*n)).

(3) A(x) = Product_{n>=1} (1 - (-x)^n*A(x)^n)^2 / (1 + (-x)^n*A(x)^n)^2.

(4) A( x/theta_3(x)^2 ) = theta_3(x)^2.

(5) A(x) = (1/x)*Series_Reversion(x/theta_3(x)^2), where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

a(n) = [x^n] theta_3(x)^(2*n+2) / (n+1).

a(n) ~ c * d^n / n^(3/2), where d = 8.54148362320612002563896433934021488424489314523756456892173912667254... and c = 1.2437677914754786190190604348779334425700766084860016245397106832001... - Vaclav Kotesovec, Nov 16 2023

EXAMPLE

G.f.: A(x) = 1 + 4*x + 20*x^2 + 112*x^3 + 676*x^4 + 4312*x^5 + 28704*x^6 +...

Given g.f. A(x), let q = x*A(x), then by a q-series identity:

A(x) = 1 + 4*q/(1+q^2) + 4*q^2/(1+q^4) + 4*q^3/(1+q^6) + 4*q^4/(1+q^8) +...

A(x) = (1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + 2*q^25 +...)^2.

...

Illustrate a(n) = [x^n] theta_3(x)^(2*n+2) / (n+1) by the following table of coefficients in powers theta_3(x)^(2*n+2) for n>=0:

n=0: [(1), 4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0,...];

n=1: [1, (8), 24, 32, 24, 48, 96, 64, 24, 104, 144, 96, 96, 112,...];

n=2: [1, 12, (60), 160, 252, 312, 544, 960, 1020, 876, 1560, 2400,...];

n=3: [1, 16, 112, (448), 1136, 2016, 3136, 5504, 9328, 12112,...];

n=4: [1, 20, 180, 960, (3380), 8424, 16320, 28800, 52020, 88660,...];

n=5: [1, 24, 264, 1760, 7944, (25872), 64416, 133056, 253704,...];

n=6: [1, 28, 364, 2912, 16044, 64792, (200928), 503360, ...];

n=7: [1, 32, 480, 4480, 29152, 140736, 525952, (1580800), ...]; ...

where the coefficients in parenthesis form the initial terms of this sequence:

A = [1/1, 8/2, 60/3, 448/4, 3380/5, 25872/6, 200928/7, 1580800/8, ...].

MATHEMATICA

CoefficientList[1/x * InverseSeries[Series[x/EllipticTheta[3, 0, x]^2, {x, 0, 25}], x], x] (* Vaclav Kotesovec, Nov 16 2023 *)

(* Calculation of constants {d, c}: *) {1/r, s/Sqrt[Pi*(1 + 4 * r^2 * s^(3/2) * Derivative[0, 0, 2][EllipticTheta][3, 0, r*s])]} /. FindRoot[{s == EllipticTheta[3, 0, r*s]^2, 2*r*Sqrt[s]*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s] == 1}, {r, 1/8}, {s, 3/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Nov 16 2023 *)

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+4*sum(m=1, n, (x*A)^m/(1+(x*A+x*O(x^n))^(2*m)))); polcoeff(A, n)}

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1+2*sum(m=1, sqrtint(n+1), (x*A+x*O(x^n))^(m^2)))^2); polcoeff(A, n)}

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n, (1-(-x)^m*A^m)/(1+(-x)^m*A^m +x*O(x^n)))^2); polcoeff(A, n)}

for(n=0, 30, print1(a(n), ", "))

CROSSREFS

Cf. A166952.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 14 2012

STATUS

approved

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