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

Author

Paul Barry, May 16 2003

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A002203, A084128, A084136.

Programs

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

Mathematica

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

PARI

a(n)=if(n<0,0,polsym(4+4*x-x^2,n)[n+1]/4+2^(n-1))

SageMath

[2^(n-2)*(2+lucas_number2(n, 2, -1)) for n in range(41)] # G. C. Greubel, Oct 13 2022

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