login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A000145 Number of ways of writing n as a sum of 12 squares. 7
1, 24, 264, 1760, 7944, 25872, 64416, 133056, 253704, 472760, 825264, 1297056, 1938336, 2963664, 4437312, 6091584, 8118024, 11368368, 15653352, 19822176, 24832944, 32826112, 42517728, 51425088, 61903776, 78146664, 98021616 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Apparently 8 | a(n). - Alexander R. Povolotsky, Oct 01 2011

REFERENCES

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.

Index entries for sequences related to sums of squares

FORMULA

Expansion of eta(q^2)^60 / (eta(q) * eta(q^4))^24 in powers of q.

Euler transform of period 4 sequence [24, -36, 24, -12, ...]. - Michael Somos, Sep 21 2005

G.f.: (Sum_k x^k^2)^12 = theta_3(q)^12.

a(n) = A029751(n) + 16 * A000735(n). - Michael Somos, Sep 21 2005

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 64 (t/i)^6 f(t) where q = exp(2 Pi i t).

a(n) = (24/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

EXAMPLE

G.f. = 1 + 24*x + 264*x^2 + 1760*x^3 + 7944*x^4 + 25872*x^5 + 64416*x^6 + 133056*x^7 + ...

MAPLE

(sum(x^(m^2), m=-10..10))^12; # gives g.f. for first 100 terms

coeff(((sum(x^(m^2), m=-10..10)))^12, x, 1...100) # Alexander R. Povolotsky, Oct 01 2011

t1:=(sum(x^(m^2), m=-n..n))^12; t2:=series(t1, x, n+1); t2[n+1]; # N. J. A. Sloane, Oct 01 2011

MATHEMATICA

SquaresR[12, Range[0, 30]] (* Harvey P. Dale, Sep 07 2012 *)

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^12, {q, 0, n}]; (* Michael Somos, Aug 15 2015 *)

nmax = 50; CoefficientList[Series[Product[(1-x^(2*k))^60 / ((1-x^k) * (1-x^(4*k)))^24, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 25 2016 *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^12, n))}; /* Michael Somos, Sep 21 2005 */

(MAGMA) A := Basis( ModularForms( Gamma0(4), 6), 25); A[1] + 24*A[2] + 264*A[3] + 1760*A[4]; /* Michael Somos, Aug 15 2015 */

CROSSREFS

12th column of A286815. - Seiichi Manyama, May 27 2017

Row d=12 of A122141.

Cf. A000735, A029751.

Sequence in context: A076847 A009175 A187380 * A286346 A126904 A001413

Adjacent sequences:  A000142 A000143 A000144 * A000146 A000147 A000148

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified September 24 18:42 EDT 2017. Contains 292433 sequences.