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Fibonacci sequence beginning 1, 11.
5

%I #55 Jun 08 2024 00:01:46

%S 1,11,12,23,35,58,93,151,244,395,639,1034,1673,2707,4380,7087,11467,

%T 18554,30021,48575,78596,127171,205767,332938,538705,871643,1410348,

%U 2281991,3692339,5974330,9666669,15640999,25307668,40948667,66256335,107205002,173461337,280666339

%N Fibonacci sequence beginning 1, 11.

%C a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(11;n-1-k,k) with n >= 1, a(-1)=10. These are the SW-NE diagonals in P(11;n,k), the (11,1) 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, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60, ... (is this 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)=11. a(-1)=10.

%F G.f.: (1+10*x)/(1-x-x^2).

%F a(n-1) = ((1+sqrt(5))^n - (1-sqrt(5))^n)/(2^n*sqrt(5)) + 5*((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) = 10*A000045(n) + A000045(n+1). - _R. J. Mathar_, Apr 07 2011

%F a(n) = 12*A000045(n) - A000045(n-2). - _Bruno Berselli_, Feb 20 2017

%F a(n) = A000045(n+4) + A000032(n-4) for n > 0. - _Bruno Berselli_, Sep 27 2017

%t LinearRecurrence[{1,1},{1,11},40] (* _Harvey P. Dale_, Aug 16 2015 *)

%o (Magma) a0:=1; a1:=11; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // _Bruno Berselli_, Feb 12 2013

%o (PARI) a(n) = 10*fibonacci(n)+fibonacci(n+1) \\ _Charles R Greathouse IV_, Jun 11 2015

%Y Cf. A000032, A000045.

%Y a(n) = A109754(10, n+1) = A101220(10, 0, n+1).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_