

A103373


a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1 and for n>6: a(n) = a(n5) + a(n6).


19



1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 7, 8, 8, 8, 9, 12, 15, 16, 16, 17, 21, 27, 31, 32, 33, 38, 48, 58, 63, 65, 71, 86, 106, 121, 128, 136, 157, 192, 227, 249, 264, 293, 349, 419, 476, 513, 557, 642, 768, 895, 989, 1070, 1199, 1410, 1663, 1884, 2059, 2269
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,7


COMMENTS

k=5 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) and k=4 case is A103372.
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=5 case, the ratio of successive terms a(n)/a(n1) approaches the unique positive root of the characteristic polynomial: x^6  x  1 = 0. This is the real constant 1.1347241384015194926054460545064728402796672263828014859251495516682....
The sequence of prime values in this k=5 case is A103383; the sequence of semiprime values in this k=5 case is A103393.


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) / (1x^5x^6 ).  R. J. Mathar, Aug 26 2011


EXAMPLE

a(22) = 9 because a(22) = a(225) + a(226) = a(17) + a(16) = 5 + 4 = 9.


MATHEMATICA

k = 5; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n  k] + a[n  k  1]; Array[a, 65]
RecurrenceTable[{a[n] == a[n  5] + a[n  6], a[1] == a[2] == a[3] == a[4] == a[5] == a[6] == 1}, a, {n, 65}] (* or *)
Rest@ CoefficientList[Series[x (1 + x + x^2 + x^3 + x^4)/(1 + x^5 + x^6), {x, 0, 65}], x] (* Michael De Vlieger, Oct 03 2016 *)
LinearRecurrence[{0, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1}, 70] (* Harvey P. Dale, Jul 20 2019 *)


PROG

(PARI) a(n)=([0, 1, 0, 0, 0, 0; 0, 0, 1, 0, 0, 0; 0, 0, 0, 1, 0, 0; 0, 0, 0, 0, 1, 0; 0, 0, 0, 0, 0, 1; 1, 1, 0, 0, 0, 0]^(n1)*[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+x^2+x^3+x^4)/(1x^5x^6 )) \\ G. C. Greubel, May 01 2017


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



