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%I #30 Jan 01 2024 11:37:45
%S 0,1,3,14,58,247,1045,4428,18756,79453,336567,1425722,6039454,
%T 25583539,108373609,459077976,1944685512,8237820025,34895965611,
%U 147821682470,626182695490,2652552464431,11236392553213,47598122677284
%N a(n+3) = 3*a(n+2) + 5*a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 3.
%C A001076(n) = a(n) + a(n+1). Program "Superseeker" finds: A033887(n+1) = a(n+2) - a(n); Elements of even index in the sequence: A049661(n) = (F(6n+1)-1)/4; A015448(n+2) = a(n+2) + 2*a(n+1) + a(n)
%H G. C. Greubel, <a href="/A110526/b110526.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, 5, 1).
%F G.f.: -x/((1+x)*(x^2+4*x-1)).
%F a(n) = (-1)^n/2 * Sum_{k=0..n} (-1)^k*Fibonacci(3*k). - _Gary Detlefs_, Jan 03 2013
%F a(n) = (Fibonacci(3*n+1)-(-1)^n)/4, where Fibonacci(n) = A000045(n). - _Vladimir Reshetnikov_, Oct 28 2015
%p seriestolist(series(-x/((1+x)*(x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: 1jbaseseq[(- 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj')(+ .5'i + .5i' + .5'jj' + .5'kk')]
%t Table[(Fibonacci[3n+1]-(-1)^n)/4, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 28 2015 *)
%o (PARI) concat(0, Vec(x/((1+x)*(1-x^2-4*x)) + O(x^100))) \\ _Altug Alkan_, Oct 28 2015
%Y Cf. A110526, A110527, A033887, A001076, A049661, A033887, A000045.
%K easy,nonn
%O 0,3
%A _Creighton Dement_, Jul 24 2005
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