A005369 a(n) = 1 if n is of the form m(m+1), else 0.
1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0
Examples
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 + ...
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
- Index entries for sequences related to cellular automata
- Index to Elementary Cellular Automata
- Index entries for characteristic functions
Programs
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Haskell
a005369 = a010052 . (+ 1) . (* 4) -- Reinhard Zumkeller, Jul 05 2014
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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
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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 *)
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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))}; -
Python
from sympy.ntheory.primetest import is_square def A005369(n): return int(is_square((n<<2)|1)) # Chai Wah Wu, Jun 07 2025
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.
From Michael Somos, Apr 13 2005: (Start)
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.
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. (End)
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
Extensions
Additional comments from Michael Somos, Apr 29 2003
Erroneous formula removed by Reinhard Zumkeller, Jul 05 2014
Comments