A022096 Fibonacci sequence beginning 1, 6.

1, 6, 7, 13, 20, 33, 53, 86, 139, 225, 364, 589, 953, 1542, 2495, 4037, 6532, 10569, 17101, 27670, 44771, 72441, 117212, 189653, 306865, 496518, 803383, 1299901, 2103284, 3403185, 5506469, 8909654, 14416123, 23325777, 37741900, 61067677, 98809577, 159877254

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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
• 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.
• J. L. Ramírez and G. N. Rubiano, Properties and Generalizations of the Fibonacci Word Fractal, The Mathematica Journal, Vol. 16 (2014).
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

Cf. A000045, A093563, A096940, A101220, A109754.

Programs

Magma

a0:=1; a1:=6; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // Bruno Berselli, Feb 12 2013

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 *)

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

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

Extensions

Spelling correction by Jason G. Wurtzel, Aug 22 2010