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A022099 Fibonacci sequence beginning 1, 9. 5
1, 9, 10, 19, 29, 48, 77, 125, 202, 327, 529, 856, 1385, 2241, 3626, 5867, 9493, 15360, 24853, 40213, 65066, 105279, 170345, 275624, 445969, 721593, 1167562, 1889155, 3056717, 4945872, 8002589, 12948461, 20951050, 33899511, 54850561, 88750072, 143600633, 232350705 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(9;n-1-k,k) with n>=1, a(-1)=8. These are the SW-NE diagonals in P(9;n,k), the (9,1) Pascal triangle A093644. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

In general, for b Fibonacci sequence beginning with 1, h, we have:

b(n) = (2^(-1-n)*((1 - sqrt(5))^n*(1 + sqrt(5) - 2*h) + (1 + sqrt(5))^n*(-1 + sqrt(5) + 2*h)))/sqrt(5). - Herbert Kociemba, Dec 18 2011

Pisano period lengths:  1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60, ... (perhaps the same as A001175). - R. J. Mathar, Aug 10 2012

LINKS

Table of n, a(n) for n=0..37.

Tanya Khovanova, Recursive Sequences

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

FORMULA

a(n) = a(n-1) + a(n-2), n>=2, a(0)=1, a(1)=9. a(-1):=8.

G.f.: (1+8*x)/(1-x-x^2).

a(n) = A109754(8, n+1) = A101220(8, 0, n+1).

a(n+1) = ((1 + sqrt(5))^n - (1 - sqrt(5))^n)/(2^n*sqrt(5))+ 4*((1 + sqrt(5))^(n-1) - (1 - sqrt(5))^(n-1))/(2^(n-2)*sqrt(5)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009

a(n) =  8*A000045(n) + A000045(n+1). - R. J. Mathar, Aug 10 2012

a(n) =  9*A000045(n) + A000045(n-1). - Paolo P. Lava, May 18 2015

a(n) = 10*A000045(n) - A000045(n-2). - Bruno Berselli, Feb 20 2017

MAPLE

with(combinat):  P:=proc(q) local n; for n from 0 to q do

print(9*fibonacci(n)+fibonacci(n-1)); od; end: P(10^2); # Paolo P. Lava, May 18 2015

MATHEMATICA

LinearRecurrence[{1, 1}, {1, 9}, 36] (* Robert G. Wilson v, Apr 11 2014 *)

PROG

(MAGMA) a0:=1; a1:=9; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // Bruno Berselli, Feb 12 2013

CROSSREFS

Cf. A101220, A109754.

Sequence in context: A015898 A231504 A050551 * A042113 A041166 A042613

Adjacent sequences:  A022096 A022097 A022098 * A022100 A022101 A022102

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Jun 14 1998

STATUS

approved

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Last modified August 19 23:35 EDT 2017. Contains 290821 sequences.