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Fibonacci sequence beginning 1, 12.
4

%I #53 Jun 08 2024 00:01:51

%S 1,12,13,25,38,63,101,164,265,429,694,1123,1817,2940,4757,7697,12454,

%T 20151,32605,52756,85361,138117,223478,361595,585073,946668,1531741,

%U 2478409,4010150,6488559,10498709

%N Fibonacci sequence beginning 1, 12.

%C a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(12;n-1-k,k) with n>=1, a(-1)=11. These are the SW-NE diagonals in P(12;n,k), the (12,1) 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)=12. a(-1):=11.

%F G.f.: (1+11*x)/(1-x-x^2).

%F a(n) = A109754(11, n+1) = A101220(11, 0, n+1).

%F a(n) = ((1+sqrt(5))^n - (1-sqrt(5))^n)/(2^n*sqrt(5)) + (11/2)*((1+sqrt(5))^(n-1)-(1-sqrt(5))^(n-1))/(2^(n-2)*sqrt(5)). Offset 1. a(3)=13. - Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009

%F a(n) = 11*A000045(n) + A000045(n+1). - _R. J. Mathar_, Aug 10 2012

%F a(n) = 13*A000045(n) - A000045(n-2). - _Bruno Berselli_, Feb 20 2017

%F a(n) = F(n+5) + F(n-1) - F(n-5) for F(n) the Fibonacci number A000045(n). - _Greg Dresden_ and _Aamen Muharram_, Jun 09 2022

%t LinearRecurrence[{1,1},{1,12},40] (* _Harvey P. Dale_, Jan 23 2012 *)

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

%o (PARI) a(n) = if(n==0, 1, if(n==1, 12, a(n-1)+a(n-2))) \\ _Felix Fröhlich_, Jun 09 2022

%o (PARI) Vec((1+11*x)/(1-x-x^2) + O(x^20)) \\ _Felix Fröhlich_, Jun 09 2022

%Y Cf. A000045, A001175, A093645, A101220, A109754.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_