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A212804 Expansion of (1-x)/(1-x-x^2). 11
1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, 2971215073, 4807526976 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

A variant of the Fibonacci number A000045.

Number of compositions of n into parts >= 2. - Joerg Arndt, Aug 13 2012

LINKS

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

J. J. Madden, A generating function for the distribution of runs in binary words, arXiv:1707.04351 [math.CO], 2017, Theorem 1.1, r=1,k=0.

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

FORMULA

G.f.: (1-x)/(1-x-x^2) = (1-x)*G(0)/(x*sqrt(5)) where G(k)= 1 -((-1)^k)*2^k/(a^k - b*x*a^k*2^k/(b*x*2^k - 2*((-1)^k)*c^k/G(k+1))) and a=3+sqrt(5), b=1+sqrt(5), c=3-sqrt(5); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jun 04 2012

G.f.: 1/(1-(Sum_{k>=2} x^k)). - Joerg Arndt, Aug 13 2012

a(n) = Fibonacci(n+1) - Fibonacci(n). - Arkadiusz Wesolowski, Oct 29 2012

G.f.: 1 - x*Q(0) where Q(k) = 1 - (1+x)/(1 - x/(x - 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 06 2013

G.f.: 3*x^3/(3*x - Q(0)) - x^2 + 1, where Q(k) = 1 - 1/(4^k - x*16^k/(x*4^k - 1/(1 + 1/(2*4^k - 4*x*16^k/(2*x*4^k +1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 21 2013

G.f.: G(0)*(1-x)/(2-x), where G(k)= 1 + 1/(1 - (x*(5*k-1))/((x*(5*k+4)) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 15 2013

G.f.: 1 + Q(0)*x^2/2, where Q(k) = 1 + 1/(1 - x*(2*k+1 + x)/( x*(2*k+2 + x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 29 2013

a(n) = Sum_{k=0..n} C(k,n-k) - C(k,n-k-1). - Peter Luschny, Oct 01 2014

a(n) = (2^(-1-n)*((1-sqrt(5))^n*(1+sqrt(5))+(-1+sqrt(5))*(1+sqrt(5))^n))/sqrt(5). - Colin Barker, Sep 25 2016

MATHEMATICA

Table[Fibonacci[n-1], {n, 0, 40}] (* Vladimir Reshetnikov, Sep 24 2016 *)

PROG

(MAGMA) [Fibonacci(n + 1) - Fibonacci(n): n in [0..50]]; // Vincenzo Librandi, Dec 09 2012

CROSSREFS

Cf. A000045, A105809 (alternating row sums).

Sequence in context: A236191 A000045 A020695 * A132916 A274163 A177194

Adjacent sequences:  A212801 A212802 A212803 * A212805 A212806 A212807

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, May 27 2012, following a suggestion from R. K. Guy.

STATUS

approved

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Last modified February 18 20:32 EST 2018. Contains 299330 sequences. (Running on oeis4.)