A114203 Row sums of a Pascal-Jacobsthal triangle.

1, 2, 4, 8, 18, 44, 110, 272, 662, 1596, 3838, 9240, 22286, 53812, 129974, 313888, 757878, 1829644, 4416910, 10662952, 25742302, 62147556, 150038438, 362226480, 874493446, 2111213372, 5096916094, 12305037368, 29706982638, 71719002644, 173145004310, 418009044032

Offset: 0

Author

Paul Barry, Nov 16 2005

Keywords

• easy
• nonn

Comments

Binomial transform of double Jacobsthal sequence 1,1,1,1,3,3,5,5,11,11,... Row sums of A114202.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2613
• Sergio Falcón, Binomial Transform of the Generalized k-Fibonacci Numbers, Communications in Mathematics and Applications (2019) Vol. 10, No. 3, 643-651.
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2,2).

Crossrefs

Cf. A001045, A114202.

Programs

Mathematica

LinearRecurrence[{4,-5,2,2},{1,2,4,8},40] (* Harvey P. Dale, Jun 05 2012 *)

Formula

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

a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) + 2*a(n-4).

a(n) = Sum_{k=0..n} Sum_{i=0..n-k} C(n-k, i)*C(k, i)*J(i).

a(n) = Sum_{k=0..n} C(n, k)*J(floor((k+2)/2)), J(n) = A001045(n).

E.g.f.: exp(x)*(cos(x) + 2*cosh(sqrt(2)*x) + sin(x) + sqrt(2)*sinh(sqrt(2)*x))/3. - Stefano Spezia, May 29 2024