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A103372 a(1) = a(2) = a(3) = a(4) = a(5) = 1 and for n>5: a(n) = a(n-4) + a(n-5). 23
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 (list; graph; refs; listen; history; text; internal format)
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
J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms
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.
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
Sequence in context: A304633 A124746 A124789 * A029082 A035450 A234537
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

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Last modified May 29 12:08 EDT 2024. Contains 372940 sequences. (Running on oeis4.)