A080995 - OEIS

%I #98 Jan 13 2024 03:33:15

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

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

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

%N Characteristic function of generalized pentagonal numbers A001318.

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

%C Repeatedly [1,[0,]^2k,1,[0,]^k], k>=0; characteristic function of generalized pentagonal numbers: a(A001318(n))=1, a(A090864(n))=0. - _Reinhard Zumkeller_, Apr 22 2006

%C Starting with offset 1 with 1's signed (++--++,...), i.e., (1, 1, 0, 0, -1, 0, -1, 0, ...); is the INVERTi transform of A000041 starting (1, 2, 3, 5, 7, 11, ...). - _Gary W. Adamson_, May 17 2013

%C Number 9 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - _Michael Somos_, May 04 2016

%D Percy A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p. 81, Article 331.

%H Seiichi Manyama, <a href="/A080995/b080995.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1001 from T. D. Noe)

%H S. Cooper and M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00079-7">Results of Hurwitz type for three squares.</a> Discrete Math. 274 (2004), no. 1-3, 9-24. See P(q).

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.

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

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

%H Don Zagier, <a href="http://dx.doi.org/10.1007/978-3-540-74119-0">Elliptic modular forms and their applications</a>, in "The 1-2-3 of modular forms", Springer-Verlag, 2008.

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.

%F Expansion of phi(-x^3) / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions. - _Michael Somos_, Sep 14 2007

%F Expansion of psi(x) - x * psi(x^9) in powers of x^3 where psi() is a Ramanujan theta function. - _Michael Somos_, Sep 14 2007

%F Expansion of f(x, x^2) in powers of x where f() is Ramanujan's two-variable theta function.

%F Expansion of q^(-1/24) * eta(q^2) * eta(q^3)^2 / (eta(q) * eta(q^6)) in powers of q.

%F a(n) = b(24*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p>3. - _Michael Somos_, Jun 06 2005

%F Euler transform of period 6 sequence [ 1, 0, -1, 0, 1, -1, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 2^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089810.

%F G.f.: Product_{k>0} (1 - x^(3*k)) / (1 - x^k + x^(2*k)). - _Michael Somos_, Jan 26 2008

%F G.f.: Sum x^(n*(3n+1)/2), n=-inf..inf [the exponents are the pentagonal numbers, A000326].

%F a(n) = |A010815(n)| = A089806(2*n) = A033683(24*n + 1).

%F For n > 0, a(n) = b(n) - b(n-1) + c(n) - c(n-1), where b(n) = floor(sqrt(2n/3+1/36)+1/6) (= A180447(n)) and c(n) = floor(sqrt(2n/3+1/36)-1/6) (= A085141(n)). - _Mikael Aaltonen_, Mar 08 2015

%F a(n) = (-1)^n * A133985(n). - _Michael Somos_, Jul 12 2015

%F a(n) = A000009(n) (mod 2). - _John M. Campbell_, Jun 29 2016

%F Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 2*sqrt(2/3) = 1.632993... . - _Amiram Eldar_, Jan 13 2024

%e G.f. = 1 + x + x^2 + x^5 + x^7 + x^12 + x^15 + x^22 + x^26 + x^35 + x^40 + x^51 + ...

%e G.f. = q + q^25 + q^49 + q^121 + q^169 + q^289 + q^361 + q^529 + q^625 + ...

%t a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (Series[ EllipticTheta[ 3, Log[y] / (2 I), x^(3/2)], {x, 0, n + Floor@Sqrt[n]}] // Normal // TrigToExp) /. {y -> x^(1/2)}, {x, 0, n}]]; (* _Michael Somos_, Nov 18 2011 *)

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^3] / QPochhammer[ x, x^2], {x, 0, n}]; (* _Michael Somos_, Jun 08 2013 *)

%t a[ n_] := If[ n < 0, 0, Boole[ IntegerQ[ Sqrt[ 24 n + 1]]]]; (* _Michael Somos_, Jun 08 2013 *)

%o (PARI) {a(n) = if( n<0, 0, abs( polcoeff( eta(x + x * O(x^n)), n)))};

%o (PARI) {a(n) = issquare( 24*n + 1)}; /* _Michael Somos_, Apr 13 2005 */

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A) * eta(x^6 + A)), n))};

%o (Haskell)

%o a080995 = a033683 . (+ 1) . (* 24)  -- _Reinhard Zumkeller_, Nov 14 2015

%Y Cf. A001318 (support), A010815 (absolute values), A033683, A089806.

%Y Cf. A000326, A180447, A085141, A133985.

%Y Cf. A000009, A000041, A000122, A000700, A010054, A090864, A121373.

%K nonn,easy

%O 0,1

%A _Michael Somos_, Feb 27 2003

%E Minor edits by _N. J. A. Sloane_, Feb 03 2012