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A103374 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = 1 and for n>7: a(n) = a(n-6) + a(n-7).

%I A103374 #39 Feb 19 2025 11:53:36
%S A103374 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,
%T A103374 17,21,27,31,32,32,33,38,48,58,63,64,65,71,86,106,121,127,129,136,157,
%U A103374 192,227,248,256,265,293,349,419,475,504,521,558,642,768,894,979,1025,1079
%N A103374 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = 1 and for n>7: a(n) = a(n-6) + a(n-7).
%C A103374 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.
%C A103374 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 A103374 For this k=6 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^7 - x - 1 = 0. This is the real constant 1.1127756842787... (see A230160).
%C A103374 The sequence of prime values in this k=6 case is A103384; the sequence of semiprime values in this k=6 case is A103394.
%D A103374 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 A103374 G. C. Greubel, <a href="/A103374/b103374.txt">Table of n, a(n) for n = 1..1000</a>
%H A103374 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 A103374 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 A103374 Richard Padovan, <a href="http://www.nexusjournal.com/conferences/N2002-Padovan.html">Dom Hans van der Laan and the Plastic Number</a>.
%H A103374 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 A103374 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 A103374 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,1,1).
%F A103374 G.f.: x*(1+x)*(1+x+x^2)*(x^2-x+1) / ( 1-x^6-x^7 ). - _R. J. Mathar_, Aug 26 2011
%e A103374 a(32) = 17 because a(32) = a(32-6) + a(32-7) = a(26) + a(25) = 9 + 8 = 17.
%t A103374 k = 6; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 70]
%t A103374 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 *)
%t A103374 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 *)
%t A103374 LinearRecurrence[{0,0,0,0,0,1,1},{1,1,1,1,1,1,1},80] (* _Harvey P. Dale_, Sep 02 2024 *)
%o A103374 (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]^(n-1)*[1;1;1;1;1;1;1])[1,1] \\ _Charles R Greathouse IV_, Oct 03 2016
%o A103374 (PARI) x='x+O('x^50); Vec(x*(1+x)*(1+x+x^2)*(x^2-x+1)/(1-x^6-x^7)) \\ _G. C. Greubel_, May 01 2017
%Y A103374 Cf. A000045, A000931, A079398, A103384, A103394, A230160.
%Y A103374 Cf. A103372, A103373, A103375, A103376, A103377, A103378, A103379, A103380
%K A103374 nonn,easy
%O A103374 1,8
%A A103374 _Jonathan Vos Post_, Feb 03 2005
%E A103374 Edited by _Ray Chandler_ and _Robert G. Wilson v_, Feb 06 2005