This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A103375 #30 Feb 19 2025 11:55:45
%S A103375 1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,4,4,4,4,4,5,7,8,8,8,8,8,9,12,15,16,
%T A103375 16,16,16,17,21,27,31,32,32,32,33,38,48,58,63,64,64,65,71,86,106,121,
%U A103375 127,128,129,136,157,192,227,248,255,257,265,293,349,419,475,503,512
%N A103375 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = 1 and for n>8: a(n) = a(n-7) + a(n-8).
%C A103375 k=7 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, k=5 case is A103373 and k=6 case is A103374.
%C A103375 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 A103375 For this k=7 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^8 - x - 1 = 0. This is the real constant 1.09698155779855981790827896716753708959253010821278671381232885124855898059....
%C A103375 The sequence of prime values in this k=7 case is A103385; the sequence of semiprime values in this k=7 case is A103395.
%D A103375 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 A103375 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 A103375 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 A103375 Richard Padovan, <a href="http://www.nexusjournal.com/conferences/N2002-Padovan.html">Dom Hans van der Laan and the Plastic Number</a>.
%H A103375 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 A103375 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 A103375 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,1,1).
%F A103375 G.f.: -x*(1+x+x^2+x^3+x^4+x^5+x^6)/(-1+x^7+x^8). - _R. J. Mathar_, Dec 14 2009
%e A103375 a(30) = 12 because a(30) = a(30-7) + a(30-8) = a(24) + a(23) = 7 + 5 = 12.
%t A103375 k = 7; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 73]
%t A103375 LinearRecurrence[{0,0,0,0,0,0,1,1},{1,1,1,1,1,1,1,1},80]
%o A103375 (PARI) a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; 1,1,0,0,0,0,0,0]^(n-1)*[1;1;1;1;1;1;1;1])[1,1] \\ _Charles R Greathouse IV_, Oct 03 2016
%Y A103375 Cf. A000045, A000931, A079398, A103372-A103374, A103376-A103380, A103385, A103395.
%K A103375 nonn,easy
%O A103375 1,9
%A A103375 _Jonathan Vos Post_, Feb 03 2005
%E A103375 Edited by _Ray Chandler_ and _Robert G. Wilson v_, Feb 06 2005
%E A103375 Corrected (one more 8 inserted) by _R. J. Mathar_, Dec 14 2009