A022391 Fibonacci sequence beginning 1, 21.

1, 21, 22, 43, 65, 108, 173, 281, 454, 735, 1189, 1924, 3113, 5037, 8150, 13187, 21337, 34524, 55861, 90385, 146246, 236631, 382877, 619508, 1002385, 1621893, 2624278, 4246171, 6870449, 11116620, 17987069, 29103689, 47090758, 76194447, 123285205, 199479652, 322764857, 522244509

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(21;n-1-k,k), n>=1, with a(-1)=20. These are the SW-NE diagonals in P(21;n,k), the (21,1) 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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• S. Kak, The Golden Mean and the Physics of Aesthetics, arXiv:physics/0411195 [physics.hist-ph], 2004.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (1, 1).

Programs

Magma

[Fibonacci(n+2) + 19*Fibonacci(n): n in [0..50]]; // G. C. Greubel, Mar 02 2018

Mathematica

LinearRecurrence[{1, 1}, {1, 21}, 30] (* Jean-François Alcover, Feb 25 2018 *)
Table[Fibonacci[n + 2] + 19*Fibonacci[n], {n, 0, 50}] (* G. C. Greubel, Mar 02 2018 *)

PARI

for(n=0, 50, print1(fibonacci(n+2) + 19*fibonacci(n), ", ")) \\ G. C. Greubel, Mar 02 2018

Formula

a(n) = a(n-1) + a(n-2), n>=2, a(0)=1, a(1)=21. a(-1):=20.

G.f.: (1+20*x)/(1-x-x^2).

Extensions

Terms a(30) onward added by G. C. Greubel, Mar 02 2018