

A103375


a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = 1 and for n>8: a(n) = a(n7) + a(n8).


15



1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 16, 17, 21, 27, 31, 32, 32, 32, 33, 38, 48, 58, 63, 64, 64, 65, 71, 86, 106, 121, 127, 128, 129, 136, 157, 192, 227, 248, 255, 257, 265, 293, 349, 419, 475, 503, 512
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OFFSET

1,9


COMMENTS

k=7 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), k=3 case is A079398 (offset so as to begin 1,1,1,1), k=4 case is A103372, k=5 case is A103373 and k=6 case is A103374.
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=7 case, the ratio of successive terms a(n)/a(n1) approaches the unique positive root of the characteristic polynomial: x^8  x  1 = 0. This is the real constant 1.09698155779855981790827896716753708959253010821278671381232885124855898059....
The sequence of prime values in this k=7 case is A103385; the sequence of semiprime values in this k=7 case is A103395.


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+x^2+x^3+x^4+x^5+x^6)/(1+x^7+x^8).  R. J. Mathar, Dec 14 2009


EXAMPLE

a(30) = 12 because a(30) = a(307) + a(308) = a(24) + a(23) = 7 + 5 = 12.


MATHEMATICA

k = 7; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n  k] + a[n  k  1]; Array[a, 73]
LinearRecurrence[{0, 0, 0, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1}, 80]


PROG

(PARI) a(n)=([0, 1, 0, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 0, 1; 1, 1, 0, 0, 0, 0, 0, 0]^(n1)*[1; 1; 1; 1; 1; 1; 1; 1])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS

Corrected (one more 8 inserted) by R. J. Mathar, Dec 14 2009


STATUS

approved



