OFFSET
0,2
COMMENTS
a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(6;n-1-k,k), n>=1, with a(-1)=5. These are the sums of the SW-NE diagonals in P(6;n,k), the (6,1) Pascal triangle A093563. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs. Also sums of SW-NE diagonals in (1,5)-Pascal triangle A096940.
Subsequence of primes: 7, 13, 53, 139, 953, 44771, 189653, 1494692464747, ... - R. J. Mathar, Aug 09 2012
a(n) is the sum of seven consecutive Fibonacci numbers. a(n) = F(n-4) + F(n-3) + F(n-2) + F(n-1) + F(n) + F(n+1) + F(n+2), where F(n)=A000045(n), extended so that F(-1)=1, F(-2)=-1, F(-3)=2, and F(-4)=-3. - Graeme McRae, Apr 24 2014
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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), 9 (Table 2) and 10 (Fig. 4).
Jia Huang, Hecke algebras of simply-laced type with independent parameters, arXiv:1902.11139 [math.RT], 2019.
Tanya Khovanova, Recursive Sequences
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], 2012-2014.
José L. Ramírez and Gustavo N. Rubiano, Properties and Generalizations of the Fibonacci Word Fractal, Mathematica J. 16 (2014).
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)=6.
G.f.: (1+5*x)/(1-x-x^2).
a(n) = A109754(5, n+1).
a(n) = 5*Fibonacci(n+2) - 4*Fibonacci(n+1). - Gary Detlefs, Dec 21 2010
a(n) = (2^(-1-n)*((1 - sqrt(5))^n*(-11 + sqrt(5)) + (1 + sqrt(5))^n*(11 + sqrt(5))))/sqrt(5). - Herbert Kociemba, Dec 18 2011
a(n) = Fibonacci(n+3) - Fibonacci(n-4). - Greg Dresden and Sam Neale, Mar 08 2022
MATHEMATICA
CoefficientList[Series[(1+5 x)/(1-x-x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 25 2014 *)
LinearRecurrence[{1, 1}, {1, 6}, 40] (* Harvey P. Dale, Aug 07 2023 *)
PROG
(Magma) a0:=1; a1:=6; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // Bruno Berselli, Feb 12 2013
(PARI) a(n)=([0, 1; 1, 1]^n*[1; 6])[1, 1] \\ Charles R Greathouse IV, Jan 29 2016
(SageMath)
A022096=BinaryRecurrenceSequence(1, 1, 1, 6)
print([A022096(n) for n in range(41)]) # G. C. Greubel, Jun 02 2025
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
Spelling correction by Jason G. Wurtzel, Aug 22 2010
STATUS
approved
