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

%I #23 Aug 14 2024 23:14:04

%S 1,1,3,2,6,2,11,-1,21,-12,44,-44,101,-131,247,-362,626,-970,1615,

%T -2565,4201,-6744,10968,-17688,28681,-46343,75051,-121366,196446,

%U -317782,514259,-832009,1346301,-2178276,3524612,-5702852,9227501,-14930315,24157855,-39088130,63246026,-102334114

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

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

%F a(n) = (-1)^n*Fibonacci(n-1) + n. - _Vladeta Jovovic_, Jul 18 2004

%F a(n) = A001924(-3-n) = 2*a(n-2) - a(n-3) + 1. - _Michael Somos_, Dec 31 2012

%F If 0 is prepended then BINOMIAL transform is A079282 with 0 prepended. - _Michael Somos_, Dec 31 2012

%F a(n) = (-1)^n * Sum_{k=0..n} binomial(k-2,n-k). - _Seiichi Manyama_, Aug 14 2024

%e 1 + x + 3*x^2 + 2*x^3 + 6*x^4 + 2*x^5 + 11*x^6 - x^7 + 21*x^8 - 12*x^9 + 44*x^10 + ...

%t Table[(-1)^n*Fibonacci[n - 1] + n, {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 28 2011 *)

%o (PARI) {a(n) = fibonacci(1-n) + n} /* _Michael Somos_, Dec 31 2012 */

%Y Cf. A000045.

%K sign

%O 0,3

%A _N. J. A. Sloane_, Nov 17 2002