A378259 - OEIS

%I #13 Dec 06 2024 11:09:46

%S 1,2,14,32,286,480,3532,7520,75126,138560,897876,1997408,17039340,

%T 29800896,233106072,497063872,5068814502,10555899520,67510782340,

%U 148762640480,961465207492,1640373169344,14064469534248,31207376374080,312901302489020,640840797612416,4625855789637960,9196764841428416

%N G.f. satisfies A(x) = Sum_{n>=1} A(x^(2*n))/A(x^n), with A(0) = 0 and A'(0) = 1.

%C Conjecture: a(n) == 2 (mod 4) iff n = 2^k + 1 for some k >= 0; elsewhere, a(n) is divisible by 4 for n > 1.

%C First negative term is at a(224).

%H Paul D. Hanna, <a href="/A378259/b378259.txt">Table of n, a(n) for n = 1..500</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n/4^(n-1) satisfies the following formulas.

%F (1) A(x) = Sum_{n>=1} A(x^(2*n))/A(x^n).

%F (2) A(x) = Sum_{n>=1} x^n/(1-x^n) * ([x^n] A(x^2)/A(x)).

%F (3) A(x^2)/A(x) = Sum_{n>=1} x^n * Sum_{d|n} mu(n/d) * a(d)/4^(d-1), where mu(n) = A008683(n), the Moebius function of n.

%e G.f.: A(x) = x + 2*x^2/4 + 14*x^3/4^2 + 32*x^4/4^3 + 286*x^5/4^4 + 480*x^6/4^5 + 3532*x^7/4^6 + 7520*x^8/4^7 + 75126*x^9/4^8 + 138560*x^10/4^9 + 897876*x^11/4^10 + 1997408*x^12/4^11 + ...

%e where

%e A(x) = A(x^2)/A(x) + A(x^4)/A(x^2) + A(x^6)/A(x^3) + A(x^8)/A(x^4) + A(x^10)/A(x^5) + ... + A(x^(2*n))/A(x^n) + ...

%e with

%e A(x^2)/A(x) = x - 2*x^2/4 - 2*x^3/4^2 + 30*x^5/4^4 + 96*x^6/4^5 - 564*x^7/4^6 - 672*x^8/4^7 + 17782*x^9/4^8 - 23232*x^10/4^9 - 150700*x^11/4^10 + 31328*x^12/4^11 + ...

%e SPECIFIC VALUES.

%e A(t) = 3 at t = 0.7974505378370343003451267990412475368480451135...

%e A(t) = 5/2 at t = 0.76521942129290360663050508795248283467273528299272...

%e A(t) = 2 at t = 0.72087935902801574221845019715702827377758681730558...

%e A(t) = 3/2 at t = 0.65614587043086605186186423048734410347397837010542...

%e A(t) = 1 at t = 0.55329257825045064604510648563868404064428412610006...

%e A(t) = 1/2 at t = 0.36885212407172053837959308718581567946754577165148...

%e A(4/5) = 3.04639791827460031098834746489882724258092426...

%e A(3/4) = 2.3085152427070861118573648623403955213681844300916...

%e A(2/3) = 1.5681888169382250959833931491973692522610217218632...

%e A(1/2) = 0.8198795330503962750204869589412995397389131828850...

%e A(1/3) = 0.4332475961307505208322404686096133187262861124027...

%e A(1/4) = 0.2981452060950792897168271259942284248293916912662...

%o (PARI) {a(n) = my(V=[1],A,B); for(i=1,n, V = concat(V,0); A = 4*subst(x*Ser(V),x,x/4); B = subst(A,x, x^2)/A;

%o V[#V] = 4^(#V-1)/2 * polcoef( sum(m=1,#V, subst(B,x, x^m +x*O(x^#V)) ) - A, #V) ); V[n]}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A378262, A008683.

%K nonn,new

%O 1,2

%A _Paul D. Hanna_, Dec 04 2024