

A103376


a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = 1 and for n>9: a(n) = a(n8) + a(n9).


15



1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 16, 16, 17, 21, 27, 31, 32, 32, 32, 32, 33, 38, 48, 58, 63, 64, 64, 64, 65, 71, 86, 106, 121, 127, 128, 128, 129, 136, 157, 192, 227, 248, 255, 256, 257, 265, 293
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OFFSET

1,10


COMMENTS

k=8 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, k=6 case is A103374 and k=7 case is A103375.
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=8 case, the ratio of successive terms a(n)/a(n1) approaches the unique positive root of the characteristic polynomial: x^9  x  1 = 0. This is the real constant (to 50 digits accuracy): 1.0850702454914508283368958640973142340506536310308. Note that x = (1 + (1 + (1 + (1 + (1 + ...)^(1/9))^(1/9)))^(1/9))))^(1/9)))))^(1/9))))).
The sequence of prime values in this k=8 case is A103386; The sequence of semiprime values in this k=8 case is A103396.


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)/(1x^8x^9).  R. J. Mathar, Dec 14 2009
a(1)=1, a(2)=1, a(3)=1, a(4)=1, a(5)=1, a(6)=1, a(7)=1, a(8)=1, a(9)=1, a(n)=a(n8)+a(n9).  Harvey P. Dale, May 07 2015


EXAMPLE

a(93) = 1200 because a(93) = a(938) + a(939) = a(85) + a(84) = 642 + 558.


MATHEMATICA

k = 8; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n  k] + a[n  k  1]; Array[a, 76]
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1}, 80] (* Harvey P. Dale, May 07 2015 *)


PROG

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


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



