

A103372


a(1) = a(2) = a(3) = a(4) = a(5) = 1 and for n>5: a(n) = a(n4) + a(n5).


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
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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(nk) + a(n[k+1]).
For this k=4 case, the ratio of successive terms a(n)/a(n1) 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) 229245.


LINKS



FORMULA

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


EXAMPLE

a(14) = 5 because a(14) = a(144) + a(145) = 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]^(n1)*[1; 1; 1; 1; 1])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



