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A022102 Fibonacci sequence beginning 1, 12. 3
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 (list; graph; refs; listen; history; text; internal format)
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) = ((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) = 12*A000045(n) + A000045(n-1). - Paolo P. Lava, May 19 2015

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

MAPLE

with(numtheory): with(combinat): P:=proc(q) local n;

for n from 0 to q do print(12*fibonacci(n)+fibonacci(n-1));

od; end: P(30); # Paolo P. Lava, May 19 2015

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

CROSSREFS

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

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

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Last modified June 28 05:58 EDT 2017. Contains 288813 sequences.