A022101 Fibonacci sequence beginning 1, 11.

1, 11, 12, 23, 35, 58, 93, 151, 244, 395, 639, 1034, 1673, 2707, 4380, 7087, 11467, 18554, 30021, 48575, 78596, 127171, 205767, 332938, 538705, 871643, 1410348, 2281991, 3692339, 5974330, 9666669, 15640999, 25307668, 40948667, 66256335, 107205002, 173461337, 280666339

Offset

0,2

Comments

a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(11;n-1-k,k) with n >= 1, a(-1)=10. These are the SW-NE diagonals in P(11;n,k), the (11,1) 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..37.

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)=11. a(-1)=10.

G.f.: (1+10*x)/(1-x-x^2).

a(n-1) = ((1+sqrt(5))^n - (1-sqrt(5))^n)/(2^n*sqrt(5)) + 5*((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) = 10*A000045(n) + A000045(n+1). - R. J. Mathar, Apr 07 2011

a(n) = 12*A000045(n) - A000045(n-2). - Bruno Berselli, Feb 20 2017

a(n) = A000045(n+4) + A000032(n-4) for n > 0. - Bruno Berselli, Sep 27 2017

Mathematica

LinearRecurrence[{1, 1}, {1, 11}, 40] (* Harvey P. Dale, Aug 16 2015 *)

Prog

(Magma) a0:=1; a1:=11; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013
(PARI) a(n) = 10*fibonacci(n)+fibonacci(n+1) \\ Charles R Greathouse IV, Jun 11 2015

Crossrefs

Cf. A000032, A000045.

a(n) = A109754(10, n+1) = A101220(10, 0, n+1).

Sequence in context: A105945 A139114 A367556 * A041246 A042633 A197221

Adjacent sequences: A022098 A022099 A022100 * A022102 A022103 A022104

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved