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A022099 Fibonacci sequence beginning 1, 9. 6

%I

%S 1,9,10,19,29,48,77,125,202,327,529,856,1385,2241,3626,5867,9493,

%T 15360,24853,40213,65066,105279,170345,275624,445969,721593,1167562,

%U 1889155,3056717,4945872,8002589,12948461,20951050,33899511,54850561,88750072,143600633,232350705

%N Fibonacci sequence beginning 1, 9.

%C 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.

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

%C 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

%C 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

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).

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

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

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

%F 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

%F a(n) = 8*A000045(n) + A000045(n+1). - _R. J. Mathar_, Aug 10 2012

%F a(n) = 9*A000045(n) + A000045(n-1). - _Paolo P. Lava_, May 18 2015

%F a(n) = 10*A000045(n) - A000045(n-2). - _Bruno Berselli_, Feb 20 2017

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

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

%t LinearRecurrence[{1, 1}, {1, 9}, 36] (* _Robert G. Wilson v_, Apr 11 2014 *)

%o (MAGMA) a0:=1; a1:=9; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // _Bruno Berselli_, Feb 12 2013

%Y Cf. A101220, A109754.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Jun 14 1998

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Last modified November 11 18:50 EST 2019. Contains 329031 sequences. (Running on oeis4.)