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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).

%I A103373 #37 Feb 19 2025 11:52:55
%S A103373 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,
%T A103373 32,33,38,48,58,63,65,71,86,106,121,128,136,157,192,227,249,264,293,
%U A103373 349,419,476,513,557,642,768,895,989,1070,1199,1410,1663,1884,2059,2269
%N 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).
%C A103373 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.
%C 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(n-k) + a(n-[k+1]).
%C A103373 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....
%C A103373 The sequence of prime values in this k=5 case is A103383; the sequence of semiprime values in this k=5 case is A103393.
%D A103373 Zanten, A. J. van, "The golden ratio in the arts of painting, building and mathematics", Nieuw Archief voor Wiskunde, 4 (17) (1999) 229-245.
%H A103373 G. C. Greubel, <a href="/A103373/b103373.txt">Table of n, a(n) for n = 1..1000</a>
%H A103373 J.-P. Allouche and T. Johnson, <a href="https://hal.science/hal-02986050v1">Narayana's cows and delayed morphisms</a>, 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]
%H A103373 J.-P. Allouche and T. Johnson,  <a href="/A000930/a000930.pdf">Narayana's cows and delayed morphisms</a>, 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]
%H A103373 Richard Padovan, <a href="http://www.nexusjournal.com/conferences/N2002-Padovan.html">Dom Hans van der Laan and the Plastic Number</a>.
%H A103373 E. S. Selmer, <a href="http://www.mscand.dk/article/view/10478/8499">On the irreducibility of certain trinomials</a>, Math. Scand., 4 (1956) 287-302.
%H A103373 J. Shallit, <a href="http://dx.doi.org/10.1016/0304-3975(88)90103-X">A generalization of automatic sequences</a>, Theoretical Computer Science, 61 (1988) 1-16.
%H A103373 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,1,1).
%F A103373 G.f.: x*(1+x+x^2+x^3+x^4) / (1-x^5-x^6 ). - _R. J. Mathar_, Aug 26 2011
%e A103373 a(22) = 9 because a(22) = a(22-5) + a(22-6) = a(17) + a(16) = 5 + 4 = 9.
%t A103373 k = 5; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 65]
%t A103373 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 *)
%t A103373 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 *)
%t A103373 LinearRecurrence[{0,0,0,0,1,1},{1,1,1,1,1,1},70] (* _Harvey P. Dale_, Jul 20 2019 *)
%o A103373 (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
%o A103373 (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
%Y A103373 Cf. A000045, A000931, A079398, A103383, A103393.
%Y A103373 Cf. A103372, A103374, A103375, A103376, A103377, A103378, A103379, A103380
%K A103373 nonn,easy
%O A103373 1,7
%A A103373 _Jonathan Vos Post_, Feb 03 2005
%E A103373 Edited by _Ray Chandler_ and _Robert G. Wilson v_, Feb 06 2005