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A098600 a(n) = Fibonacci(n-1) + Fibonacci(n+1) - (-1)^n. 9
1, 2, 2, 5, 6, 12, 17, 30, 46, 77, 122, 200, 321, 522, 842, 1365, 2206, 3572, 5777, 9350, 15126, 24477, 39602, 64080, 103681, 167762, 271442, 439205, 710646, 1149852, 1860497, 3010350, 4870846, 7881197, 12752042, 20633240, 33385281, 54018522 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Row sums of A098599.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,2,1).

FORMULA

G.f.: (1+2*x) / ((1+x)*(1-x-x^2)).

a(n) = sum{k = 0..n, binomial(k, n-k) + binomial(k-1, n-k-1)}.

a(n) = A020878(n) - 1 = A001350(n) + 1.

a(n) = Lucas(n) - (-1)^n. - Paul Barry, Dec 01 2004

a(n) = A181716(n+1).  - Richard R. Forberg, Aug 30 2014

a(n) = [x^n] ( (1 + x + sqrt(1 + 6*x + 5*x^2))/2 )^n. exp( Sum_{n >= 1} a(n)*x^n/n ) = Sum_{n >= 0} Fibonacci(n+2)*x^n. Cf. A182143. - Peter Bala, Jun 29 2015

From Colin Barker, Jun 03 2016: (Start)

a(n) = (-(-1)^n+(1/2*(1-sqrt(5)))^n+(1/2*(1+sqrt(5)))^n).

a(n) = 2*a(n-2)+a(n-3) for n>2.

(End)

E.g.f.: (2*exp(3*x/2)*cosh(sqrt(5)*x/2) - 1)*exp(-x). - Ilya Gutkovskiy, Jun 03 2016

MATHEMATICA

Table[-(-1)^n + LucasL[n], {n, 0, 39}] (* Alonso del Arte, Aug 30 2014 *)

Table[Fibonacci[n - 1] + Fibonacci[n + 1] - (-1)^n, {n, 0, 40}] (* Vincenzo Librandi, Aug 31 2014 *)

PROG

(MAGMA) [Fibonacci(n-1) + Fibonacci(n+1) - (-1)^n: n in [0..50]]: // Vincenzo Librandi, Aug 31 2014

(PARI) a(n)=fibonacci(n-1) + fibonacci(n+1) - (-1)^n; \\ Joerg Arndt, Oct 18 2014

(PARI) Vec((1+2*x)/((1+x)*(1-x-x^2)) + O(x^30)) \\ Colin Barker, Jun 03 2016

CROSSREFS

Cf. A000045, A008346, A182143.

First differences of A014217 and A062724. Cf. A000032, A099925, A181716.

Sequence in context: A238517 A099926 A181716 * A261866 A147766 A034420

Adjacent sequences:  A098597 A098598 A098599 * A098601 A098602 A098603

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Sep 17 2004

STATUS

approved

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Last modified December 2 17:17 EST 2016. Contains 278680 sequences.