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).
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
Offset: 1
Examples
a(22) = 9 because a(22) = a(22-5) + a(22-6) = a(17) + a(16) = 5 + 4 = 9.
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
- J.-P. Allouche and T. Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [The hal link does not always work. - _N. J. A. Sloane_, Feb 19 2025]
- J.-P. Allouche and T. Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [Local copy with annotations and a correction from _N. J. A. Sloane_, Feb 19 2025]
- Richard Padovan, Dom Hans van der Laan and the Plastic Number.
- 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).
Crossrefs
Programs
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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 *) -
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
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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
Formula
G.f.: x*(1+x+x^2+x^3+x^4) / (1-x^5-x^6 ). - R. J. Mathar, Aug 26 2011
Extensions
Edited by Ray Chandler and Robert G. Wilson v, Feb 06 2005
Comments