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A099568
Expansion of (1-x)/((1-2*x)*(1-x-x^3)).
2
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
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.
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
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Oct 22 2004
STATUS
approved