A084137 Binomial transform of A084136.

1, 2, 8, 32, 144, 672, 3200, 15360, 73984, 356864, 1722368, 8314880, 40144896, 193830912, 935886848, 4518838272, 21818834944, 105350561792, 508677324800, 2456111022080, 11859152338944, 57261051346944, 276480810549248

Offset

0,2

Comments

Exponential self-convolution of companion Pell numbers (A002203), divided by 4. - Vladimir Reshetnikov, Oct 07 2016

Links

Michael De Vlieger, Table of n, a(n) for n = 0..1463

Sergio Falcon, Half self-convolution of the k-Fibonacci sequence, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.

Index entries for linear recurrences with constant coefficients, signature (6,-4,-8).

Formula

G.f.: (1-4*x)/((1-2*x)*(1-4*x-4*x^2)).

E.g.f.: exp(2*x)*cosh(sqrt(2)*x)^2 = (exp(x)*cosh(sqrt(2)*x))^2.

a(n) = ((2+sqrt(8))^n + (2-sqrt(8))^n + 2^(n+1))/4.

a(n) = (A084128(n) + 2^n)/2.

a(n) = 2^(n-2)*(2 + A002203(n)). - Vladimir Reshetnikov, Oct 07 2016

a(n) = 6*a(n-1) - 4*a(n-2) - 8*a(n-3). - G. C. Greubel, Oct 13 2022

Mathematica

Table[2^(n-2)*(2+LucasL[n, 2]), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 07 2016 *)

Prog

(PARI) a(n)=if(n<0, 0, polsym(4+4*x-x^2, n)[n+1]/4+2^(n-1))
(Magma)
A002203:= func< n | Round((1+Sqrt(2))^n + (1-Sqrt(2))^n) >;
[2^(n-2)*(2+A002203(n)): n in [0..40]]; // G. C. Greubel, Oct 13 2022
(SageMath) [2^(n-2)*(2+lucas_number2(n, 2, -1)) for n in range(41)] # G. C. Greubel, Oct 13 2022

Crossrefs

Cf. A002203, A084128, A084136.

Sequence in context: A150861 A150862 A150863 * A129400 A003304 A150864

Adjacent sequences: A084134 A084135 A084136 * A084138 A084139 A084140

Keyword

easy,nonn

Author

Paul Barry, May 16 2003

Status

approved