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A103373 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1 and for n>6: a(n) = a(n-5) + a(n-6). 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(n-k) + a(n-[k+1]).

For this k=5 case, the ratio of successive terms a(n)/a(n-1) 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) 229-245.

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) 287-302.

J. Shallit, A generalization of automatic sequences, Theoretical Computer Science, 61 (1988) 1-16.

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

FORMULA

G.f.: x*(1+x+x^2+x^3+x^4) / (1-x^5-x^6 ). - R. J. Mathar, Aug 26 2011

EXAMPLE

a(22) = 9 because a(22) = a(22-5) + a(22-6) = 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]^(n-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+x^2+x^3+x^4)/(1-x^5-x^6 )) \\ G. C. Greubel, May 01 2017

CROSSREFS

Cf. A000045, A000931, A079398, A103372 to A103381, A103383, A103393.

Sequence in context: A025780 A199121 A109697 * A038539 A275891 A109368

Adjacent sequences:  A103370 A103371 A103372 * A103374 A103375 A103376

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

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Last modified January 29 08:12 EST 2020. Contains 331337 sequences. (Running on oeis4.)