OFFSET
0,2
COMMENTS
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.
Row sums of triangle A131776 starting (1, 5, 6, 11, 17, 28, ...). - Gary W. Adamson, Jul 14 2007
In general, for a Fibonacci sequence beginning with 1,b we have:
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
Subsequence of primes: 5, 11, 17, 73, 191, 809, 421493, 1103483, ... . - R. J. Mathar, Aug 09 2012
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
LINKS
Muniru A Asiru, Table of n, a(n) for n = 0..960
Bruce M. Boman, How Does Multicellular Life Happen? Modeling Fibonacci Patterns in Biological Tissues Unveils Underlying Mechanisms, Fibonacci Quarterly (September 2025). See pp. 5 (Table 1), 8 (Fig. 3), 9 (Table 2), 10 (Fig. 4).
Eudes Antonio Costa, Élis Gardel Da Costa Mesquita, and Paula M. M. C. Catarino, On the connections between Fibonacci and Mulatu Numbers, Intermaths 6(1) (2025) 17-36. See p. 34.
Élis Gardel da Costa Mesquita, Eudes Antonio Costa, Paula M. M. C. Catarino, and Francisco R. V. Alves, Jacobsthal-Mulatu Numbers, Latin Amer. J. Math. (2025) Vol. 4, No. 1, 23-45. See p. 24.
Tanya Khovanova, Recursive Sequences.
Élis Gardel Da Costa Mesquita, Paula Maria Machado Cruz Catarino, and Eudes Antonio Costa, Exploring Generalized 2^k-Fibonacci Sequence: A New Family of the Fibonacci Sequence, Comm. Adv. Math. Sci. 8(4) (2025) 225-246. See pp. 225-226.
José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368 [cs.DM], 2012.
José L. Ramírez and Gustavo N. Rubiano, Properties and Generalizations of the Fibonacci Word Fractal, Mathematica J. 16 (2014).
Matty van Son, Uniqueness conjectures for extended Markov numbers, arXiv:1911.00746 [math.NT], 2019.
Index entries for linear recurrences with constant coefficients, signature (1,1).
FORMULA
a(n) = a(n-1) + a(n-2), n >= 2, a(0)=1, a(1)=5.
G.f.: (1+4*x)/(1-x-x^2).
a(n) = 4*Fibonacci(n) + Fibonacci(n+1), n >= 1. - Zerinvary Lajos, Oct 05 2007, corrected by R. J. Mathar, Apr 07 2011
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
a(n) = 4*Fibonacci(n+2) - 3*Fibonacci(n+1). - Gary Detlefs, Dec 21 2010
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
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
a(n) = Lucas(n-1) + Fibonacci(n+3) = Lucas(n+2) - Fibonacci(n-3). - Greg Dresden and Griffin Donaldson, Mar 03 2022
E.g.f.: exp(x/2)*(5*cosh(sqrt(5)*x/2) + 9*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Dec 12 2025
MAPLE
with(combinat): a:= n-> 4*fibonacci(n)+fibonacci(n+1): seq(a(n), n=0..32); # Zerinvary Lajos, Oct 05 2007
MATHEMATICA
f[n_] := (LucasL[n - 2] + 8*LucasL[n - 1] + 4*LucasL[n] + 2*LucasL[n + 1])/5; Array[f, 38, 0] (* or *)
LinearRecurrence[{1, 1}, {1, 5}, 38] (* Robert G. Wilson v, Oct 22 2012 *)
PROG
(PARI) a(n)=fibonacci(n-1)+5*fibonacci(n) \\ Charles R Greathouse IV, Jun 05 2011
(Magma) a0:=1; a1:=5; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // Bruno Berselli, Feb 12 2013
(GAP) List([0..40], n->4*Fibonacci(n)+Fibonacci(n+1)); # Muniru A Asiru, Mar 04 2018
(SageMath)
A022095=BinaryRecurrenceSequence(1, 1, 1, 5)
[A022095(n) for n in range(41)] # G. C. Greubel, Jun 02 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved