A022109 Fibonacci sequence beginning 1, 19.

1, 19, 20, 39, 59, 98, 157, 255, 412, 667, 1079, 1746, 2825, 4571, 7396, 11967, 19363, 31330, 50693, 82023, 132716, 214739, 347455, 562194, 909649, 1471843, 2381492, 3853335, 6234827, 10088162, 16322989, 26411151, 42734140, 69145291, 111879431, 181024722

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n-1) = Sum(P(19;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=18. These are the SW-NE diagonals in P(19;n,k), the (19,1) 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

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

a(n) = A109754(18, n+1) = A101220(18, 0, n+1).

Programs

Magma

a0:=1; a1:=19; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013

Mathematica

LinearRecurrence[{1, 1}, {1, 19}, 35] (* Paolo Xausa, Feb 22 2024 *)

Formula

a(n) = a(n-1)+a(n-2), n >= 2, a(0) = 1, a(1) = 19.

G.f.: (1+18*x)/(1-x-x^2).