A378262 G.f. satisfies A(x) = Sum_{n>=1} 2^(n-1) * A(x^(2*n))/A(x^n), with A(0) = 0 and A'(0) = 1.

1, 1, 3, 5, 14, 25, 60, 117, 257, 504, 1053, 2067, 4197, 8248, 16491, 32533, 64919, 128893, 257923, 515770, 1036024, 2080743, 4185781, 8411269, 16893534, 33867059, 67793691, 135460708, 270330258, 538950081, 1074174949, 2141296967, 4271640535, 8530158021, 17054867115, 34138204669, 68398842318

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..520

Formula

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

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

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

Example

G.f.: A(x) = x + x^2 + 3*x^3 + 5*x^4 + 14*x^5 + 25*x^6 + 60*x^7 + 117*x^8 + 257*x^9 + 504*x^10 + 1053*x^11 + 2067*x^12 + 4197*x^13 + 8248*x^14 + 16491*x^15 + ...
where
A(x) = A(x^2)/A(x) + 2*A(x^4)/A(x^2) + 2^2*A(x^6)/A(x^3) + 2^3*A(x^8)/A(x^4) + 2^4*A(x^10)/A(x^5) + ... + 2^(n-1)*A(x^(2*n))/A(x^n) + ...
with
A(x^2)/A(x) = x - x^2 - x^3 - x^4 - 2*x^5 - x^6 - 4*x^7 - x^8 + 5*x^9 + 12*x^10 + 29*x^11 + 65*x^12 + 101*x^13 + 128*x^14 + 131*x^15 - 97*x^16 + ...
From the coefficients of x^n in A(x^2)/A(x) we may form A(x) like so
A(x) = x/(1-2*x) - x^2/(1-2*x^2) - x^3/(1-2*x^3) - x^4/(1-2*x^4) - 2*x^5/(1-2*x^5) - x^6/(1-2*x^6) - 4*x^7/(1-2*x^7) - x^8/(1-2*x^8) + 5*x^9/(1-2*x^9) + 12*x^10/(1-2*x^10) + 29*x^11/(1-2*x^11) + 65*x^12/(1-2*x^12) + ... + ([x^n] A(x^2)/A(x)) * x^n/(1 - 2*x^n) + ...
SPECIFIC VALUES.
A(t) = 3 at t = 0.43733814488282063301680910555646948170999223090757...
A(t) = 2 at t = 0.41397004526177482844685461509046496062468756075210...
A(t) = 3/2 at t = 0.39330222443727410417944329172595352498212069922925...
A(t) = 1 at t = 0.35749402156274824941243704576004713164226979305829...
A(t) = 2/3 at t = 0.31333794936725576481707196951511233228426643397267...
A(t) = 1/2 at t = 0.27748211306635183931075005365310539042990531754751...
A(2/5) = 1.63758629524692945778740862999490236385081146642444...
A(1/3) = 0.79365065960676358258532147504412273666409442743886...
A(1/4) = 0.40608804950229998821473272051417196450715525962124...
A(1/5) = 0.27936643003430423804549184204540603443863382773108...

Prog

(PARI) {a(n) = my(V=[1], A, B); for(i=1, n, V = concat(V, 0); A = x*Ser(V); B = subst(A, x, x^2)/A;
V[#V] = (1/2) * polcoef( sum(m=1, #V, 2^(m-1) * subst(B, x, x^m +x*O(x^#V)) ) - A, #V) ); V[n]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A378259.

Sequence in context: A372436 A295359 A104208 * A364314 A026777 A007136

Adjacent sequences: A378259 A378260 A378261 * A378263 A378264 A378265

Keyword

nonn,new

Author

Paul D. Hanna, Dec 05 2024

Status

approved