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Characteristic function of squares or five times squares.
3

%I #33 Jan 22 2024 00:14:15

%S 1,1,0,0,1,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,

%T 0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0

%N Characteristic function of squares or five times squares.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%C A195198 is a similar sequence except with three instead of five. - _Michael Somos_, Oct 22 2017

%H G. C. Greubel, <a href="/A214284/b214284.txt">Table of n, a(n) for n = 0..1000</a>

%H S. Cooper and M. Hirschhorn, <a href="http://dx.doi.org/10.1216/rmjm/1008959672">On some infinite product identities</a>, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 134 Theorem 4.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of f(q, q^9) * f(-q^8, -q^12) / f(-q^4, -q^16) in powers of q where f(, ) is Ramanujan's general theta function.

%F Expansion of f(q^3, q^7) * f(-q^2, -q^3) / f(-q, -q^4) in powers of q where f(, ) is Ramanujan's general theta function.

%F Euler transform of period 20 sequence [1, -1, 0, 1, 0, 0, 0, -1, 1, -1, 1, -1, 0, 0, 0, 1, 0, -1, 1, -1, ...].

%F a(n) is multiplicative with a(0) = a(5^e) = 1, a(p^e) = 1 if e is even, 0 otherwise.

%F G.f.: (theta_3(q) + theta_3(q^5)) / 2 = 1 + (Sum_{k>0} x^(k^2) + x^(5*k^2)).

%F Dirichlet g.f.: zeta(2*s) * (1 + 5^-s).

%F a(4*n + 2) = a(4*n + 3) = 0. a(4*n + 1) = A127693(n). a(5*n) = a(n).

%F Sum_{k=0..n} a(k) ~ c * sqrt(n), where c = 1+1/sqrt(5) = 1.447213... (A344212). - _Amiram Eldar_, Sep 14 2023

%e G.f. = 1 + x + x^4 + x^5 + x^9 + x^16 + x^20 + x^25 + x^36 + x^45 + x^49 + ...

%t a[ n_] := SeriesCoefficient[ Series[ (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^5]) / 2, {q, 0, n}], {q, 0, n}];

%t a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors @ n] || OddQ [ Length @ Divisors[5 n]]]];

%o (PARI) {a(n) = issquare(n) || issquare(5*n)};

%o (PARI) {a(n) = if( n<1, n==0, direuler( p=2, n, if( p==5, 1 + X, 1) / (1 - X^2))[n])};

%Y Cf. A127693, A195198, A344212.

%Y Cf. A000700, A000122, A010054, A121373.

%K nonn,mult,easy

%O 0,1

%A _Michael Somos_, Jul 09 2012