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Fibonacci sequence beginning 1, 13.
6

%I #50 Feb 18 2024 01:59:12

%S 1,13,14,27,41,68,109,177,286,463,749,1212,1961,3173,5134,8307,13441,

%T 21748,35189,56937,92126,149063,241189,390252,631441,1021693,1653134,

%U 2674827,4327961,7002788,11330749,18333537,29664286,47997823,77662109,125659932,203322041,328981973

%N Fibonacci sequence beginning 1, 13.

%C a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(13;n-1-k,k) for n>=1, a(-1)=12. These are the SW-NE diagonals in P(13;n,k), the (13,1) Pascal triangle. Cf. A093645 for the (10,1) Pascal triangle. 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, 4, 24, 16, 12, 24, 12, 10, 24, 28, 48, 8, 24, 36, 24, 18, 12, ... (is this A106291?). - _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) for n>=2, a(0)=1, a(1)=13, and a(-1):=12.

%F G.f.: (1 + 12*x)/(1 - x - x^2).

%F a(n) = ((1 + sqrt(5))^n-(1 - sqrt(5))^n)/(2^n*sqrt(5))+ 6*((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

%F a(n) = 12*A000045(n) + A000045(n+1). - _R. J. Mathar_, Aug 10 2012

%F a(n) = 14*A000045(n) - A000045(n-2). - _Bruno Berselli_, Feb 20 2017

%F a(n) = Lucas(n+5) - 5*Lucas(n). - _Bruno Berselli_, Dec 30 2016

%t LinearRecurrence[{1, 1}, {1, 13}, 40] (* or *) Table[LucasL[n + 5] - 5 LucasL[n], {n, 0, 40}] (* _Bruno Berselli_, Dec 30 2016 *)

%o (Magma) a0:=1; a1:=13; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // _Bruno Berselli_, Feb 12 2013

%Y a(n) = A109754(12, n+1) = A101220(12, 0, n+1).

%Y Cf. A000032, A000045.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_