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A022095 Fibonacci sequence beginning 1, 5.

%I A022095 #128 Aug 29 2025 11:08:08
%S A022095 1,5,6,11,17,28,45,73,118,191,309,500,809,1309,2118,3427,5545,8972,
%T A022095 14517,23489,38006,61495,99501,160996,260497,421493,681990,1103483,
%U A022095 1785473,2888956,4674429,7563385,12237814,19801199,32039013,51840212,83879225,135719437
%N A022095 Fibonacci sequence beginning 1, 5.
%C A022095 a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(5; n-1-k, k), n >= 1, with a(-1)=4. These are the sums of the SW-NE diagonals in P(5; n, k), the (5,1) Pascal triangle A093562. Observation by _Paul Barry_, Apr 29 2004. Proof via recursion relations and comparison of inputs. Also sums of the SW-NE diagonals in the (1,4)-Pascal triangle A095666.
%C A022095 Row sums of triangle A131776 starting (1, 5, 6, 11, 17, 28, ...). - _Gary W. Adamson_, Jul 14 2007
%C A022095 In general, for a Fibonacci sequence beginning with 1,b we have:
%C A022095 a(n) = (2^(-1-n)*((1 - sqrt(5))^n*(1 + sqrt(5) - 2b) + (1 + sqrt(5))^n*(-1 + sqrt(5) + 2b)))/sqrt(5). - _Herbert Kociemba_, Dec 18 2011
%C A022095 Subsequence of primes: 5, 11, 17, 73, 191, 809, 421493, 1103483, ... . - _R. J. Mathar_, Aug 09 2012
%C A022095 Pisano periods: 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 9, 60, ... (differs from A001175). - _R. J. Mathar_, Aug 10 2012
%H A022095 Muniru A Asiru, <a href="/A022095/b022095.txt">Table of n, a(n) for n = 0..960</a>
%H A022095 Élis Gardel da Costa Mesquita, Eudes Antonio Costa, Paula M. M. C. Catarino, and Francisco R. V. Alves, <a href="https://doi.org/10.14244/lajm.v4i1.39">Jacobsthal-Mulatu Numbers</a>, Latin Amer. J. Math. (2025) Vol. 4, No. 1, 23-45. See p. 24.
%H A022095 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A022095 José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, <a href="http://arxiv.org/abs/1212.1368">A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake</a>, arXiv preprint arXiv:1212.1368 [cs.DM], 2012.
%H A022095 José L. Ramírez and Gustavo N. Rubiano, <a href="http://www.mathematica-journal.com/2014/02/properties-and-generalizations-of-the-fibonacci-word-fractal/">Properties and Generalizations of the Fibonacci Word Fractal</a>, The Mathematica Journal, Vol. 16 (2014).
%H A022095 Matty van Son, <a href="https://arxiv.org/abs/1911.00746">Uniqueness conjectures for extended Markov numbers</a>, arXiv:1911.00746 [math.NT], 2019.
%H A022095 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F A022095 a(n) = a(n-1) + a(n-2), n >= 2, a(0)=1, a(1)=5.
%F A022095 G.f.: (1+4*x)/(1-x-x^2).
%F A022095 a(n) = 4*Fibonacci(n) + Fibonacci(n+1), n >= 1. - _Zerinvary Lajos_, Oct 05 2007, corrected by _R. J. Mathar_, Apr 07 2011
%F A022095 a(n-1) = ((1 + sqrt(5))^n - (1 - sqrt(5))^n)/(2^n*sqrt(5)) + 2*((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 A022095 a(n) = 4*Fibonacci(n+2) - 3*Fibonacci(n+1). - _Gary Detlefs_, Dec 21 2010
%F A022095 a(n) = (L(n-2) + 8*L(n-1) + 4*L(n) + 2*L(n+1))/5 for the Lucas numbers L(n). - _J. M. Bergot_, Oct 22 2012
%F A022095 a(n) = ((2*sqrt(5) - 1)*(((1 + sqrt(5))/2)^(n+1)) + (2*sqrt(5) + 1)*(((1 - sqrt(5))/2)^(n+1)))/(sqrt(5)). - _Bogart B. Strauss_, Jul 19 2013
%F A022095 a(n) = Lucas(n-1) + Fibonacci(n+3) = Lucas(n+2) - Fibonacci(n-3). - _Greg Dresden_ and Griffin Donaldson, Mar 03 2022
%p A022095 with(combinat): a:= n-> 4*fibonacci(n)+fibonacci(n+1): seq(a(n), n=0..32); # _Zerinvary Lajos_, Oct 05 2007
%t A022095 f[n_] := (LucasL[n - 2] + 8*LucasL[n - 1] + 4*LucasL[n] + 2*LucasL[n + 1])/5; Array[f, 38, 0] (* or *)
%t A022095 LinearRecurrence[{1, 1}, {1, 5}, 38] (* _Robert G. Wilson v_, Oct 22 2012 *)
%o A022095 (PARI) a(n)=fibonacci(n-1)+5*fibonacci(n) \\ _Charles R Greathouse IV_, Jun 05 2011
%o A022095 (Magma) a0:=1; a1:=5; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // _Bruno Berselli_, Feb 12 2013
%o A022095 (GAP) List([0..40],n->4*Fibonacci(n)+Fibonacci(n+1)); # _Muniru A Asiru_, Mar 04 2018
%o A022095 (SageMath)
%o A022095 A022095=BinaryRecurrenceSequence(1,1,1,5)
%o A022095 [A022095(n) for n in range(41)] # _G. C. Greubel_, Jun 02 2025
%Y A022095 Cf. A000032, A000045, A001175, A093562, A095666, A101220, A109754, A131776.
%Y A022095 Row n=4 of A109754 (shifted).
%K A022095 nonn,easy,changed
%O A022095 0,2
%A A022095 _N. J. A. Sloane_