A377249 G.f. A(x) satisfies 1 - x = Sum_{n=-oo..+oo} (x^(2*n) - A(x))^n.

1, 3, 5, 11, 20, 38, 67, 119, 211, 398, 830, 1940, 4902, 12784, 33165, 84136, 207240, 495964, 1157767, 2654461, 6029627, 13704225, 31463620, 73498385, 175220708, 425631952, 1048102141, 2599306042, 6453178098, 15967452038, 39281184601, 96019973309, 233425343306, 565413231173

Offset

1,2

Comments

a(n+1)/a(n) tends to 2.55118... - Vaclav Kotesovec, Nov 14 2024

Links

Paul D. Hanna, Table of n, a(n) for n = 1..502

Formula

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

(1) 1 - x = Sum_{n=-oo..+oo} (x^(2*n) - A(x))^n.

(2) 1 - x = Sum_{n=-oo..+oo} x^(2*n^2)/(1 - x^(2*n)*A(x))^n.

(3) 1 - x = Sum_{n=-oo..+oo} x^(2*n) * (x^(2*n) + A(x))^(n-1).

(4) 1 - x = Sum_{n=-oo..+oo} x^(2*n^2)/(1 + x^(2*n)*A(x))^(n+1).

(5) 0 = Sum_{n=-oo..+oo} (-1)^n * (x^(2*n) - A(x))^(n-1).

Example

G.f.: A(x) = x + 3*x^2 + 5*x^3 + 11*x^4 + 20*x^5 + 38*x^6 + 67*x^7 + 119*x^8 + 211*x^9 + 398*x^10 + 830*x^11 + 1940*x^12 + ...
RELATED SERIES.
Let A = A(x), then 1 - x = P + Q where
P = 1 + (x^2 - A) + (x^4 - A)^2 + (x^6 - A)^3 + (x^8 - A)^4 + (x^10 - A)^5 + (x^12 - A)^6 + ... + (x^(2*n) - A)^n + ...
Q = x^2/(1 - x^2*A) + x^8/(1 - x^4*A)^2 + x^18/(1 - x^6*A)^3 + x^32/(1 - x^8*A)^4 + x^50/(1 - x^10*A)^5 + ... + x^(2*n^2)/(1 - x^(2*n)*A)^n + ...
Explicitly,
P = 1 - x - x^2 - x^5 - 3*x^6 - 5*x^7 - 13*x^8 - 26*x^9 - 57*x^10 - 120*x^11 - 259*x^12 - 561*x^13 - 1238*x^14 - 2780*x^15 + ...
Q = x^2 + x^5 + 3*x^6 + 5*x^7 + 13*x^8 + 26*x^9 + 57*x^10 + 120*x^11 + 259*x^12 + 561*x^13 + 1238*x^14 + 2780*x^15 + ...
SPECIFIC VALUES.
A(t) = 1 at t = 0.31160833954190659544044203165981407225865730702613...
  notice that Sum_{n=-oo..+oo} (t^(2*n) - 1)^n = 3/2 - t, which deviates from 1 - t; this is due to the term (t^(2*n) - 1)^n taking on the indeterminate form 0^0 at n = 0.
A(t) = 3/4 at t = 0.27771666015004017042369933635782020387857285503045...
  where 1-t = Sum_{n=-oo..+oo} (t^(2*n) - 3/4)^n.
A(t) = 2/3 at t = 0.26361225583793268826267306259869479301133547781221...
  where 1-t = Sum_{n=-oo..+oo} (t^(2*n) - 2/3)^n.
A(t) = 1/2 at t = 0.22935306806508598431890965331979164445097661571539...
  where 1-t = Sum_{n=-oo..+oo} (t^(2*n) - 1/2)^n.
A(t) = 1/3 at t = 0.18321842757370009004270801439324522647367156844515...
  where 1-t = Sum_{n=-oo..+oo} (t^(2*n) - 1/3)^n.
A(t) = 1/4 at t = 0.15319590401223075722696025027321147967336641684269...
  where 1-t = Sum_{n=-oo..+oo} (t^(2*n) - 1/4)^n.
A(1/4) = 0.59500579234891734482663421150554161897595924474890...
  where 3/4 = Sum_{n=-oo..+oo} (1/4^(2*n) - A(1/4))^n.
A(1/5) = 0.38777623383308495901886663021576078479818829432566...
  where 4/5 = Sum_{n=-oo..+oo} (1/5^(2*n) - A(1/5))^n.

Prog

(PARI) {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax = Ser(A);
A[#A] = polcoef(-(1-x) + sum(n=-#A, #A, (x^(2*n) - Ax)^n ), #A-1) ); A[n+1]}
for(n=1, 40, print1(a(n), ", "))

Crossrefs

Sequence in context: A328660 A058932 A118037 * A094588 A299027 A339006

Adjacent sequences: A377246 A377247 A377248 * A377250 A377251 A377252

Keyword

nonn

Author

Paul D. Hanna, Nov 14 2024

Status

approved