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

Jean-Paul Allouche and Tom Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [The hal link does not always work. - N. J. A. Sloane, Feb 19 2025]

Jean-Paul Allouche and Tom Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [Local copy with annotations and a correction from N. J. A. Sloane, Feb 19 2025]

Hùng Việt Chu and Zachary Louis Vasseur, Linear Recurrences of Generalized Schreier Sets Revisited, J. Int. Seq. 29 (2026), Article 26.2.2. See p. 4 (Table 3).

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

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

Jeffrey Shallit, A generalization of automatic sequences, Theor. Comp. Sci. 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