A261329 Euler transform of Pell numbers.

1, 1, 3, 8, 23, 62, 175, 477, 1319, 3602, 9851, 26779, 72726, 196724, 531157, 1430144, 3842911, 10303055, 27570786, 73637306, 196333303, 522584286, 1388786089, 3685169795, 9764703347, 25838430572, 68282175170, 180221449469, 475102410065, 1251038486529

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015.

Eric Weisstein's World of Mathematics, Pell Number

Wikipedia, Pell number

Formula

G.f.: Product_{k>=1} 1/(1-x^k)^(A000129(k)).

a(n) ~ (1+sqrt(2))^n * exp(-1/8 + 2^(1/4)*sqrt(n) + s) / (2^(11/8) * sqrt(Pi) * n^(3/4)), where s = Sum_{k>=2} 1/(((sqrt(2)+1)^k - (sqrt(2)-1)^k - 2)*k) = 0.17615706029370539578355193664752741450665073523628663099586621933373...

G.f.: exp(Sum_{k>=1} x^k/(k*(1 - 2*x^k - x^(2*k)))). - Ilya Gutkovskiy, May 30 2018

Mathematica

nmax=40; Pell[0]=0; Pell[1]=1; Pell[n_]:=Pell[n] = 2*Pell[n-1] + Pell[n-2]; CoefficientList[Series[Product[1/(1-x^k)^Pell[k], {k, 1, nmax}], {x, 0, nmax}], x]

Prog

(SageMath) # uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(2, 1)
b = EulerTransform(a)
print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020

Crossrefs

Cf. A000129, A261330, A261331, A261332, A166861, A261031.

Sequence in context: A318861 A318819 A146998 * A375468 A248870 A103819

Adjacent sequences: A261326 A261327 A261328 * A261330 A261331 A261332

Keyword

nonn

Author

Vaclav Kotesovec, Aug 15 2015

Status

approved