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A214284 Characteristic function of squares or five times squares.

%I A214284 #34 Feb 16 2025 08:33:18
%S A214284 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 A214284 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 A214284 0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0
%N A214284 Characteristic function of squares or five times squares.
%C A214284 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A214284 A195198 is a similar sequence except with three instead of five. - _Michael Somos_, Oct 22 2017
%H A214284 G. C. Greubel, <a href="/A214284/b214284.txt">Table of n, a(n) for n = 0..1000</a>
%H A214284 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 A214284 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A214284 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A214284 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 A214284 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 A214284 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 A214284 a(n) is multiplicative with a(0) = a(5^e) = 1, a(p^e) = 1 if e is even, 0 otherwise.
%F A214284 G.f.: (theta_3(q) + theta_3(q^5)) / 2 = 1 + (Sum_{k>0} x^(k^2) + x^(5*k^2)).
%F A214284 Dirichlet g.f.: zeta(2*s) * (1 + 5^-s).
%F A214284 a(4*n + 2) = a(4*n + 3) = 0. a(4*n + 1) = A127693(n). a(5*n) = a(n).
%F A214284 Sum_{k=0..n} a(k) ~ c * sqrt(n), where c = 1+1/sqrt(5) = 1.447213... (A344212). - _Amiram Eldar_, Sep 14 2023
%e A214284 G.f. = 1 + x + x^4 + x^5 + x^9 + x^16 + x^20 + x^25 + x^36 + x^45 + x^49 + ...
%t A214284 a[ n_] := SeriesCoefficient[ Series[ (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^5]) / 2, {q, 0, n}], {q, 0, n}];
%t A214284 a[ n_] := If[ n < 0, 0, Boole[ OddQ [ Length @ Divisors @ n] || OddQ [ Length @ Divisors[5 n]]]];
%o A214284 (PARI) {a(n) = issquare(n) || issquare(5*n)};
%o A214284 (PARI) {a(n) = if( n<1, n==0, direuler( p=2, n, if( p==5, 1 + X, 1) / (1 - X^2))[n])};
%Y A214284 Cf. A127693, A195198, A344212.
%Y A214284 Cf. A000700, A000122, A010054, A121373.
%K A214284 nonn,mult,easy
%O A214284 0,1
%A A214284 _Michael Somos_, Jul 09 2012