A255175 - OEIS

%I #23 Aug 05 2022 07:45:44

%S 1,0,-1,-2,-1,0,1,2,3,2,1,0,-1,-2,-3,-4,-3,-2,-1,0,1,2,3,4,5,4,3,2,1,

%T 0,-1,-2,-3,-4,-5,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,6,5,4,3,2,1,0,-1,

%U -2,-3,-4,-5,-6,-7,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3

%N Expansion of phi(-x) / (1 - x)^2 in powers of x where phi() is a Ramanujan theta function.

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

%H G. C. Greubel, <a href="/A255175/b255175.txt">Table of n, a(n) for n = 0..1000</a>

%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 G.f.: Product_{k>0} (1 - x^(2*k)) * (1 - x^(2*k+1))^2.

%F A053615(n) = abs(A196199(n)) = abs(a(n-1)).

%F Euler transform of -A134451.

%F a(n) = Sum_{i=0..n}( (-1)^(floor(sqrt(i))) ). - _John M. Campbell_, Dec 22 2016

%e G.f. = 1 - x^2 - 2*x^3 - x^4 + x^6 + 2*x^7 + 3*x^8 + 2*x^9 + x^10 - x^12 + ...

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] / (1 - x)^2, {x, 0, n}];

%t a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^(Mod[k, 2] + 1), {k, 2, n}], {x, 0, n}];

%t a[ n_] := If[ n < 0, 0, With[{m = Floor[ Sqrt[ n + 1]]}, (-1)^m (n + 1 - m - m^2)]];

%t Table[Sum[(-1)^(Floor[Sqrt[i]]), {i,0,n}], {n,0,50}] (* _G. C. Greubel_, Dec 22 2016 *)

%o (PARI) {a(n) = my(m); if( n<0, 0, m = sqrtint(n + 1); (-1)^m * (n + 1 - m - m^2))};

%o (PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=2, n, (1 - x^k)^(k%2+1), 1 + x * O(x^n)), n))};

%o (Python)

%o from math import isqrt

%o def A255175(n): return ((1+(t:=isqrt(n)))*t-n-1)*(1 if t&1 else -1) # _Chai Wah Wu_, Aug 04 2022

%Y Cf. A053615, A134451, A196199, A329116 (essentially the same), A339265 (first differences).

%K sign

%O 0,4

%A _Michael Somos_, Feb 16 2015