

A103380


k=12 case of family of sequences beyond Fibonacci and Padovan: a(n) = a(n12) + a(n13).


12



1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 21, 27, 31, 32, 32, 32, 32, 32, 32, 32, 32, 33, 38, 48, 58, 63, 64, 64, 64, 64, 64, 64, 64, 65
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OFFSET

1,14


COMMENTS

k=12 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, k=7 case is A103375, k=8 case is A103376, k=9 case is A103377, k=10 case is A103378 and k=11 case is A103379.
The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1)= 1 and for n>(k+1) a(n) = a(nk) + a(n(k+1)).
For this k=11 case, the ratio of successive terms a(n)/a(n1) approaches the unique positive root of the characteristic polynomial: x^13  x  1 = 0. This is the real constant 1.0570505752... . Note that x = (1 + (1 + (1 + (1 + (1 + ...)^(1/13))^(1/13)))^(1/13))))^(1/13)))))^(1/13))))).
The sequence of prime values in this k=12 case is A103390.
The sequence of semiprime values in this k=12 case is A103400.


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

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,1,1).


FORMULA

For n>13: a(n) = a(n12) + a(n13). a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = a(10) = a(11) = a(12) a(13) = 1.
G.f.: x*(1x^12) / ((1x)*(1x^12x^13)).  Colin Barker, Mar 27 2013


MAPLE

A103380 := proc(n) option remember ; if n <= 13 then 1; else procname(n12)+procname(n13) ; fi; end: for n from 1 to 120 do printf("%d, ", A103380(n)) ; od: # R. J. Mathar, Aug 30 2008


MATHEMATICA

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, 90] (* Harvey P. Dale, Jul 16 2012 *)


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



