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

%I #17 Jul 26 2022 01:35:53

%S 1,2,4,9,19,39,80,163,330,666,1341,2695,5409,10846,21733,43526,87140,

%T 174409,349007,698291,1396988,2794571,5590014,11181306,22364485,

%U 44731715,89467453,178940802,357890245,715793154,1431604868,2863236937

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

%C Row sums of number triangle A099567.

%H G. C. Greubel, <a href="/A099568/b099568.txt">Table of n, a(n) for n = 0..1000</a>

%H Denis Neiter and Amsha Proag, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Proag/proag3.html">Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,1,-2).

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

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

%t LinearRecurrence[{3,-2,1,-2}, {1,2,4,9}, 40] (* _G. C. Greubel_, Jul 26 2022 *)

%o (PARI) Vec((1-x)/((1-2*x)*(1-x-x^3)) + O(x^40)) \\ _Michel Marcus_, Oct 18 2016

%o (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

%o (SageMath)

%o @CachedFunction

%o def a(n): # a = A099568

%o if (n<4): return round(9^(n/3))

%o else: return 3*a(n-1) -2*a(n-2) + a(n-3) - 2*a(n-4)

%o [a(n) for n in (0..40)] # _G. C. Greubel_, Jul 26 2022

%Y Cf. A099567.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Oct 22 2004