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A022393 Fibonacci sequence beginning 1, 23. 0
1, 23, 24, 47, 71, 118, 189, 307, 496, 803, 1299, 2102, 3401, 5503, 8904, 14407, 23311, 37718, 61029, 98747, 159776, 258523, 418299, 676822, 1095121, 1771943, 2867064, 4639007, 7506071, 12145078 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n-1) = sum_{k=0..ceiling((n-1)/2)} P(23;n-1-k,k), n>=1, with a(-1)=22. These are the SW-NE diagonals in P(23;n,k), the (23,1) 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..29.

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)=23. a(-1):=22.

G.f.: (1+22*x)/(1-x-x^2).

(n) = 23*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(23*fibonacci(n)+fibonacci(n-1));

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

MATHEMATICA

a={}; b=1; c=23; AppendTo[a, b]; AppendTo[a, c]; Do[b=b+c; AppendTo[a, b]; c=b+c; AppendTo[a, c], {n, 4!}]; a (* Vladimir Joseph Stephan Orlovsky, Sep 18 2008 *)

a[1]=1; a[2]=23; a[n_]:=a[n]=a[n - 1]+a[n - 2] [From José María Grau Ribas, Feb 15 2010]

LinearRecurrence[{1, 1}, {1, 23}, 30] (* Harvey P. Dale, Sep 30 2011 *)

CROSSREFS

Sequence in context: A007638 A031332 A122470 * A042068 A042066 A042070

Adjacent sequences:  A022390 A022391 A022392 * A022394 A022395 A022396

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 23 23:53 EDT 2017. Contains 286937 sequences.