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A022391 Fibonacci sequence beginning 1, 21. 2
1, 21, 22, 43, 65, 108, 173, 281, 454, 735, 1189, 1924, 3113, 5037, 8150, 13187, 21337, 34524, 55861, 90385, 146246, 236631, 382877, 619508, 1002385, 1621893, 2624278, 4246171, 6870449, 11116620, 17987069, 29103689, 47090758, 76194447, 123285205, 199479652, 322764857, 522244509 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(21;n-1-k,k), n>=1, with a(-1)=20. These are the SW-NE diagonals in P(21;n,k), the (21,1) 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

S. Kak, The Golden Mean and the Physics of Aesthetics, arXiv:physics/0411195 [physics.hist-ph], 2004.

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)=21. a(-1):=20.

G.f.: (1+20*x)/(1-x-x^2).

a(n) = 21*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(21*fibonacci(n)+fibonacci(n-1));

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

MATHEMATICA

LinearRecurrence[{1, 1}, {1, 21}, 30] (* Jean-François Alcover, Feb 25 2018 *)

Table[Fibonacci[n + 2] + 19*Fibonacci[n], {n, 0, 50}] (* G. C. Greubel, Mar 02 2018 *)

PROG

(PARI) for(n=0, 50, print1(fibonacci(n+2) + 19*fibonacci(n), ", ")) \\ G. C. Greubel, Mar 02 2018

(MAGMA) [Fibonacci(n+2) + 19*Fibonacci(n): n in [0..50]]; // G. C. Greubel, Mar 02 2018

CROSSREFS

Sequence in context: A295747 A125737 A160782 * A041890 A041892 A041894

Adjacent sequences:  A022388 A022389 A022390 * A022392 A022393 A022394

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 13:31 EDT 2018. Contains 313863 sequences. (Running on oeis4.)