A099568 Expansion of (1-x)/((1-2*x)*(1-x-x^3)).

1, 2, 4, 9, 19, 39, 80, 163, 330, 666, 1341, 2695, 5409, 10846, 21733, 43526, 87140, 174409, 349007, 698291, 1396988, 2794571, 5590014, 11181306, 22364485, 44731715, 89467453, 178940802, 357890245, 715793154, 1431604868, 2863236937

Offset

0,2

Comments

Row sums of number triangle A099567.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.

Index entries for linear recurrences with constant coefficients, signature (3,-2,1,-2).

Formula

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

a(n) = Sum_{k=0..n} Sum_{j=0..floor(n/3)} binomial(n-2*j, k+j).

Mathematica

LinearRecurrence[{3, -2, 1, -2}, {1, 2, 4, 9}, 40] (* G. C. Greubel, Jul 26 2022 *)

Prog

(PARI) Vec((1-x)/((1-2*x)*(1-x-x^3)) + O(x^40)) \\ Michel Marcus, Oct 18 2016
(Magma) [n le 4 select Round(9^((n-1)/3)) else 3*Self(n-1) -2*Self(n-2) +Self(n-3) -2*Self(n-4): n in [1..41]]; // G. C. Greubel, Jul 26 2022
(SageMath)
@CachedFunction
def a(n): # a = A099568
    if (n<4): return round(9^(n/3))
    else: return 3*a(n-1) -2*a(n-2) + a(n-3) - 2*a(n-4)
[a(n) for n in (0..40)] # G. C. Greubel, Jul 26 2022

Crossrefs

Cf. A099567.

Sequence in context: A267157 A054135 A171858 * A018001 A018099 A011955

Adjacent sequences: A099565 A099566 A099567 * A099569 A099570 A099571

Keyword

easy,nonn

Author

Paul Barry, Oct 22 2004

Status

approved