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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(n-8) + a(n-9). 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 (list; graph; refs; listen; history; text; internal format)
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(n-k) + a(n-[k+1]).
For this k=8 case, the ratio of successive terms a(n)/a(n-1) 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) 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)/(1-x^8-x^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(n-8)+a(n-9). - Harvey P. Dale, May 07 2015
EXAMPLE
a(93) = 1200 because a(93) = a(93-8) + a(93-9) = 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]^(n-1)*[1; 1; 1; 1; 1; 1; 1; 1; 1])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016
CROSSREFS
Sequence in context: A029241 A226749 A277090 * A189819 A145992 A045818
KEYWORD
nonn,easy
AUTHOR
Jonathan Vos Post, Feb 05 2005
EXTENSIONS
Edited by Ray Chandler, Feb 10 2005
STATUS
approved

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Last modified March 4 07:29 EST 2024. Contains 370522 sequences. (Running on oeis4.)