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A022109 Fibonacci sequence beginning 1, 19. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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)=19. a(-1):=18.

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

a(n) = 19*A000045(n) + A000045(n-1). [Paolo P. Lava, May 19 2015]

MAPLE

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

for n from 0 to q do print(19*fibonacci(n)+fibonacci(n-1));

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

MATHEMATICA

a={}; b=1; c=19; AppendTo[a, b]; AppendTo[a, c]; Do[b=b+c; AppendTo[a, b]; c=b+c; AppendTo[a, c], {n, 1, 12, 1}]; a (* Vladimir Joseph Stephan Orlovsky, Jul 23 2008 *)

PROG

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

CROSSREFS

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

Sequence in context: A241849 A054304 A151979 * A041730 A041732 A041728

Adjacent sequences:  A022106 A022107 A022108 * A022110 A022111 A022112

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 23 01:31 EDT 2017. Contains 285313 sequences.