A103372 a(1) = a(2) = a(3) = a(4) = a(5) = 1 and for n>5: a(n) = a(n-4) + a(n-5).

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 7, 8, 8, 9, 12, 15, 16, 17, 21, 27, 31, 33, 38, 48, 58, 64, 71, 86, 106, 122, 135, 157, 192, 228, 257, 292, 349, 420, 485, 549, 641, 769, 905, 1034, 1190, 1410, 1674, 1939, 2224, 2600, 3084, 3613, 4163, 4824, 5684, 6697, 7776

Offset

1,6

Comments

k=4 case of the family of sequences whose k=1 case is the Fibonacci sequence A000045, k=2 case is the Padovan sequence A000931 (offset so as to begin 1,1,1) and k=3 case is A079398 (offset so as to begin 1,1,1,1).

The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1) and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]).

For this k=4 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the irreducible characteristic polynomial: x^5 - x - 1 = 0, A160155.

The sequence of prime values in this k=4 case is A103382; The sequence of semiprime values in this k=4 case is A103392.

References

Zanten, A. J. van, The golden ratio in the arts of painting, building and mathematics, Nieuw Archief voor Wiskunde, 4 (17) (1999) 229-245.

Links

Indranil Ghosh, Table of n, a(n) for n = 1..14857

J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms

Richard Padovan, Dom Hans van der Laan and the Plastic Number.

E. S. Selmer, On the irreducibility of certain trinomials, Math. Scand., 4 (1956) 287-302.

J. Shallit, A generalization of automatic sequences, Theoretical Computer Science, 61 (1988), 1-16.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,1).

Formula

G.f. -x*(1+x)*(1+x^2) / ( -1+x^4+x^5 ). - R. J. Mathar, Aug 26 2011

a(n) = A124789(n-2)+A124798(n-1). - R. J. Mathar, Jun 30 2020

Example

a(14) = 5 because a(14) = a(14-4) + a(14-5) = a(10) + a(9) = 3 + 2 = 5.

Mathematica

k = 4; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 61]
LinearRecurrence[{0, 0, 0, 1, 1}, {1, 1, 1, 1, 1}, 70] (* Harvey P. Dale, Apr 22 2015 *)

Prog

(PARI) a(n)=([0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1; 1, 1, 0, 0, 0]^(n-1)*[1; 1; 1; 1; 1])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016

Crossrefs

Cf. A000931, A079398, A103373-A103380, A103382, A103392.

Sequence in context: A304633 A124746 A124789 * A029082 A035450 A234537

Adjacent sequences: A103369 A103370 A103371 * A103373 A103374 A103375

Keyword

nonn,easy

Author

Jonathan Vos Post, Feb 03 2005

Extensions

Edited by Ray Chandler and Robert G. Wilson v, Feb 06 2005

Status

approved