

A103374


a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = 1 and for n>7: a(n) = a(n6) + a(n7).


16



1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 17, 21, 27, 31, 32, 32, 33, 38, 48, 58, 63, 64, 65, 71, 86, 106, 121, 127, 129, 136, 157, 192, 227, 248, 256, 265, 293, 349, 419, 475, 504, 521, 558, 642, 768, 894, 979, 1025, 1079
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OFFSET

1,8


COMMENTS

k=6 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 and k=5 case is A103373.
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=6 case, the ratio of successive terms a(n)/a(n1) approaches the unique positive root of the characteristic polynomial: x^7  x  1 = 0. This is the real constant 1.1127756842787054706297040205710929356068592718552836814857016280071663325....
The sequence of prime values in this k=6 case is A103384; the sequence of semiprime values in this k=6 case is A103394.


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

G. C. Greubel, Table of n, a(n) for n = 1..1000
Richard Padovan, Dom Hans van der Laan and the Plastic Number.
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) 287302.
J. Shallit, A generalization of automatic sequences, Theoretical Computer Science, 61 (1988) 116.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1,1).


FORMULA

G.f.: x*(1+x)*(1+x+x^2)*(x^2x+1) / ( 1x^6x^7 ).  R. J. Mathar, Aug 26 2011


EXAMPLE

a(32) = 17 because a(32) = a(326) + a(327) = a(26) + a(25) = 9 + 8 = 17.


MATHEMATICA

k = 6; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n  k] + a[n  k  1]; Array[a, 70]
RecurrenceTable[{a[n] == a[n  6] + a[n  7], a[1] == a[2] == a[3] == a[4] == a[5] == a[6] == a[7] == 1}, a, {n, 70}] (* or *)
Rest@ CoefficientList[Series[x (1 + x) (1 + x + x^2) (x^2  x + 1)/(1 + x^6 + x^7), {x, 0, 70}], x] (* Michael De Vlieger, Oct 03 2016 *)


PROG

(PARI) a(n)=([0, 1, 0, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0, 0; 0, 0, 0, 1, 0, 0, 0; 0, 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 0, 1; 1, 1, 0, 0, 0, 0, 0]^(n1)*[1; 1; 1; 1; 1; 1; 1])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016
(PARI) x='x+O('x^50); Vec(x*(1+x)*(1+x+x^2)*(x^2x+1)/(1x^6x^7)) \\ G. C. Greubel, May 01 2017


CROSSREFS

Cf. A000045, A000931, A079398, A103372A103381, A103384, A103394.
Sequence in context: A317245 A109701 A124751 * A208251 A241087 A137722
Adjacent sequences: A103371 A103372 A103373 * A103375 A103376 A103377


KEYWORD

nonn,easy


AUTHOR

Jonathan Vos Post, Feb 03 2005


EXTENSIONS

Edited by Ray Chandler and Robert G. Wilson v, Feb 06 2005


STATUS

approved



