|
|
|
|
| A004018
|
|
Theta series of square lattice (or number of ways of writing n as a sum of 2 squares). Often denoted by r(n) or r_2(n).
(history;
published version)
|
|
|
|
|
#239 by Michael De Vlieger at Sat Jun 22 00:24:01 EDT 2024
|
|
|
|
|
|
|
|
#238 by Michel Marcus at Sat Jun 22 00:08:35 EDT 2024
|
|
|
|
|
|
|
|
#237 by Chai Wah Wu at Fri Jun 21 21:47:48 EDT 2024
|
|
|
|
|
|
|
|
#236 by Chai Wah Wu at Fri Jun 21 21:47:43 EDT 2024
|
|
| PROG
|
def A004018(n): return prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in factorint(n).items())<<2 if n else 1 # Chai Wah Wu, Jul 07 2022, corrected Jun 21 2024.
|
|
| STATUS
|
approved
editing
|
|
|
|
|
|
|
#235 by Hugo Pfoertner at Tue Jan 03 12:47:59 EST 2023
|
|
|
|
|
|
|
|
#234 by Hugo Pfoertner at Tue Jan 03 12:47:28 EST 2023
|
|
| PROG
|
(PARI) {a(n) = polcoeff( 1 + 4 * sum( k=1, n, x^k / (1 + x^(2*k)), x * O(x^n)), n)}; /* _Michael Somos, _, Mar 14 2003 */
|
|
| STATUS
|
approved
editing
|
|
|
|
|
|
|
#233 by Alois P. Heinz at Fri Sep 23 16:26:42 EDT 2022
|
|
|
|
|
|
|
|
#232 by Jon E. Schoenfield at Fri Sep 23 16:24:31 EDT 2022
|
|
|
|
|
|
|
|
#231 by Jon E. Schoenfield at Fri Sep 23 16:24:20 EDT 2022
|
|
| COMMENTS
|
Limit_{n-> infinity->oo} (a(n)/n - Pi*log(n)) = A062089: Sierpinski's constant. - Robert G. Wilson v, Dec 22 2016
|
|
| FORMULA
|
Expansion of theta_3(q)^2 = (Sum_{n=-infoo..+infoo} q^(n^2))^2 = Product_{m>=1} (1-q^(2*m))^2 * (1+q^(2*m-1))^4.
The constant sqrt(Pi)/GAMMAGamma(3/4)^2 produces the first 324 terms of the sequence when expanded in base exp(Pi), 450 digits of the constant are necessary. - Simon Plouffe, Mar 03 2011
|
|
| STATUS
|
approved
editing
|
|
|
|
|
|
|
#230 by Charles R Greathouse IV at Thu Sep 08 08:44:32 EDT 2022
|
|
| PROG
|
(MAGMAMagma) Basis( ModularForms( Gamma1(4), 1), 100) [1]; /* Michael Somos, Jun 10 2014 */
|
|
|
|
|
Discussion
|
| Thu Sep 08
| 08:44
| OEIS Server: https://oeis.org/edit/global/2944
|
|
|
|
|
|
