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A022098 Fibonacci sequence beginning 1, 8. 9
1, 8, 9, 17, 26, 43, 69, 112, 181, 293, 474, 767, 1241, 2008, 3249, 5257, 8506, 13763, 22269, 36032, 58301, 94333, 152634, 246967, 399601, 646568, 1046169, 1692737, 2738906, 4431643, 7170549, 11602192, 18772741, 30374933, 49147674, 79522607, 128670281 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(8; n-1-k, k) with n >= 1, a(-1) = 7. These are the SW-NE diagonals in P(8; n, k), the (8, 1) Pascal triangle A093565. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

Pisano period lengths: 1, 3, 8, 6, 4, 24, 16, 12, 24, 12, 10, 24, 28, 48, 8, 24, 36, 24, 18, 12, ... (is this the same as A106291?). - R. J. Mathar, Aug 10 2012

Also the sum of five consecutive Lucas numbers starting with L(-3). - Alonso del Arte, Sep 26 2013

LINKS

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

Tanya Khovanova, Recursive Sequences

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

FORMULA

a(n) = a(n-1) + a(n-2) for n>1, a(0)=1, a(1)=8, and a(-1):=7.

G.f.: (1 + 7*x)/(1 - x - x^2).

a(n) = ((1 + sqrt(5))^n - (1 - sqrt(5))^n)/(2^n*sqrt(5)) + 3.5*((1 + sqrt(5))^(n-1) - (1 - sqrt(5))^(n-1))/(2^(n-2)*sqrt(5)) for n>0. - Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009

a(n) = 2^(-1-n)*((1 - sqrt(5))^n*(-15 + sqrt(5)) + (1 + sqrt(5))^n*(15 + sqrt(5)))/sqrt(5). - Herbert Kociemba, Dec 18 2011

a(n) = 7*A000045(n) + A000045(n+1). - R. J. Mathar, Aug 10 2012

a(n) = 8*A000045(n) + A000045(n-1). - Paolo P. Lava, May 18 2015

a(n) = 9*A000045(n) - A000045(n-2). - Bruno Berselli, Feb 20 2017

a(n) = Lucas(n+3) + Lucas(n-3) - 3*Lucas(n) for n>1. - Bruno Berselli, Dec 29 2016

MATHEMATICA

LinearRecurrence[{1, 1}, {1, 8}, 40] (* Alonso del Arte, Sep 26 2013 *)

CoefficientList[Series[(1 + 7 x)/(1 - x - x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 27 2013 *)

Table[LucasL[n + 3] + LucasL[n - 3] - 3 LucasL[n], {n, 2, 40}] (* Bruno Berselli, Dec 30 2016 *)

PROG

(MAGMA) a0:=1; a1:=8; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // Bruno Berselli, Feb 12 2013

(PARI) a(n)=([0, 1; 1, 1]^n*[1; 8])[1, 1] \\ Charles R Greathouse IV, Oct 07 2016

CROSSREFS

Cf. A000032, A000045.

a(n) = A109754(7, n+1) = A101220(7, 0, n+1).

Sequence in context: A175053 A274406 A261454 * A129659 A041130 A041307

Adjacent sequences:  A022095 A022096 A022097 * A022099 A022100 A022101

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified June 27 21:39 EDT 2017. Contains 288804 sequences.