OFFSET
0,1
COMMENTS
This is essentially the q-expansion of the Jacobi theta function theta_2(q). (In theta_2 one has to ignore the initial factor of 2*q^(1/4). See also A010054.) - N. J. A. Sloane, Aug 03 2014
For n > 0, a(n) is the number of partitions of n into two parts such that the larger part is equal to the square of the smaller part. - Wesley Ivan Hurt, Dec 23 2020
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Robert Price, Comments on A005369 concerning Elementary Cellular Automata, Jan 29 2016
Eric Weisstein's World of Mathematics, Jacobi Theta Functions [From Franklin T. Adams-Watters, Jun 29 2009]
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
FORMULA
Expansion of q^(-1/4) * eta(q^4)^2 / eta(q^2) in powers of q.
Euler transform of period 4 sequence [ 0, 1, 0, -1, ...].
G.f.: Product_{k>0} (1 - x^(4*k)) / (1 - x^(4*k-2)) = f(x^2, x^6) where f(, ) is Ramanujan's general theta function.
Given g.f. A(x), then B(q) = (q*A(q^4))^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 + 4*v*w^2 - u^2*w. - Michael Somos, Apr 13 2005
Given g.f. A(x), then B(q) = q*A(q^4) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u2^2*u6 - u1*u6^3 - u3^3*u2. - Michael Somos, Apr 13 2005
a(n) = b(4*n + 1) where b() = A098108() is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 if p>2. - Michael Somos, Jun 06 2005
G.f.: 1/2 x^{-1/4}theta_2(0,x), where theta_2 is a Jacobi theta function. - Franklin T. Adams-Watters, Jun 29 2009
a(n) = floor(sqrt(n+1)+1/2)-floor(sqrt(n)+1/2). - Mikael Aaltonen, Jan 02 2015
a(2*n) = A010054(n).
a(n) = A000729(n)(mod 2). - John M. Campbell, Jul 16 2016
For n > 0, a(n) = Sum_{k=1..floor(n/2)} [k^2 = n-k], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Dec 23 2020
EXAMPLE
G.f. = 1 + x^2 + x^6 + x^12 + x^20 + x^30 + x^42 + x^56 + x^72 + x^90 + ...
G.f. = q + q^9 + q^25 + q^49 + q^81 + q^121 + q^169 + q^225 + q^289 + ...
MAPLE
A005369 := proc(n)
if issqr(1+4*n) then
if type( sqrt(1+4*n)-1, 'even') then
1;
else
0;
end if;
else
0;
end if;
end proc:
seq(A005369(n), n=0..80) ; # R. J. Mathar, Feb 22 2021
MATHEMATICA
a005369[n_] := If[IntegerQ[Sqrt[4 # + 1]], 1, 0] & /@ Range[0, n]; a005369[100] (* Michael De Vlieger, Jan 02 2015 *)
a[ n_] := SquaresR[ 1, 4 n + 1] / 2; (* Michael Somos, Feb 22 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] / (2 x^(1/4)), {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)
QP = QPochhammer; s = QP[q^4]^2/QP[q^2] + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)
nmax = 200; CoefficientList[Series[Sum[x^(k*(k + 1)), {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 12 2020 *)
PROG
(PARI) {a(n) = if( n<0, 0, issquare(4*n + 1))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 / eta(x^2 + A), n))};
(Haskell)
a005369 = a010052 . (+ 1) . (* 4) -- Reinhard Zumkeller, Jul 05 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Additional comments from Michael Somos, Apr 29 2003
Erroneous formula removed by Reinhard Zumkeller, Jul 05 2014
STATUS
approved