A022102 Fibonacci sequence beginning 1, 12.

1, 12, 13, 25, 38, 63, 101, 164, 265, 429, 694, 1123, 1817, 2940, 4757, 7697, 12454, 20151, 32605, 52756, 85361, 138117, 223478, 361595, 585073, 946668, 1531741, 2478409, 4010150, 6488559, 10498709

Offset

0,2

Comments

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.

In general, for b Fibonacci sequence beginning with 1, h, we have:

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

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

Links

Table of n, a(n) for n=0..30.

Tanya Khovanova, Recursive Sequences

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)=12. a(-1):=11.

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

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

a(n) = ((1+sqrt5)^n-(1-sqrt5)^n)/(2^n*sqrt5)+ 5.5*((1+sqrt5)^(n-1)-(1-sqrt5)^(n-1))/(2^(n-2)*sqrt5). Offset 1. a(3)=13. - Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009

a(n) = 11*A000045(n) + A000045(n+1). - R. J. Mathar, Aug 10 2012

a(n) = 13*A000045(n) - A000045(n-2). - Bruno Berselli, Feb 20 2017

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

Mathematica

LinearRecurrence[{1, 1}, {1, 12}, 40] (* Harvey P. Dale, Jan 23 2012 *)

Prog

(Magma) a0:=1; a1:=12; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013
(PARI) a(n) = if(n==0, 1, if(n==1, 12, a(n-1)+a(n-2))) \\ Felix Fröhlich, Jun 09 2022
(PARI) Vec((1+11*x)/(1-x-x^2) + O(x^20)) \\ Felix Fröhlich, Jun 09 2022

Crossrefs

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

Sequence in context: A108710 A108709 A138821 * A041292 A041679 A041294

Adjacent sequences: A022099 A022100 A022101 * A022103 A022104 A022105

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved