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

0,2

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).

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

Mathematica

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

Crossrefs

Cf. A001045, A114202.

Sequence in context: A193617 A233139 A308246 * A339837 A100132 A176720

Adjacent sequences: A114200 A114201 A114202 * A114204 A114205 A114206

Keyword

easy,nonn

Author

Paul Barry, Nov 16 2005

Status

approved