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A022108 Fibonacci sequence beginning 1, 18. 2
1, 18, 19, 37, 56, 93, 149, 242, 391, 633, 1024, 1657, 2681, 4338, 7019, 11357, 18376, 29733, 48109, 77842, 125951, 203793, 329744, 533537, 863281, 1396818, 2260099, 3656917, 5917016, 9573933, 15490949 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n-1)=sum(P(18;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=17. These are the SW-NE diagonals in P(18;n,k), the (18,1) 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)=18. a(-1):=17.

G.f.: (1+17*x)/(1-x-x^2).

a(n) = 18*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(18*fibonacci(n)+fibonacci(n-1));

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

MATHEMATICA

a={}; b=1; c=18; 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 *)

LinearRecurrence[{1, 1}, {1, 18}, 40] (* Harvey P. Dale, Apr 15 2018 *)

PROG

(MAGMA) a0:=1; a1:=18; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013

CROSSREFS

a(n) = A109754(17, n+1) = A101220(17, 0, n+1).

Sequence in context: A301601 A056022 A118510 * A041654 A041656 A042507

Adjacent sequences:  A022105 A022106 A022107 * A022109 A022110 A022111

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 10 19:01 EST 2018. Contains 318049 sequences. (Running on oeis4.)