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A022393 Fibonacci sequence beginning 1, 23. 1
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, 19651149, 31796227, 51447376, 83243603, 134690979, 217934582, 352625561, 570560143 (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

G. C. Greubel, Table of n, a(n) for n = 0..1000

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).

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

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

Table[Fibonacci[n + 2] + 21*Fibonacci[n], {n, 0, 50}] (* G. C. Greubel, Mar 02 2018 *)

PROG

(PARI) for(n=0, 50, print1(fibonacci(n+2) + 21*fibonacci(n), ", ")) \\ G. C. Greubel, Mar 02 2018

(MAGMA) [Fibonacci(n+2) + 21*Fibonacci(n): n in [0..50]]; // G. C. Greubel, Mar 02 2018

(GAP) List([0..40], n->Fibonacci(n+2)+21*Fibonacci(n)); # Muniru A Asiru, Mar 03 2018

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

EXTENSIONS

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

STATUS

approved

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Last modified August 19 13:43 EDT 2018. Contains 313863 sequences. (Running on oeis4.)