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 A103372 #36 Feb 19 2025 11:51:54
%S A103372 1,1,1,1,1,2,2,2,2,3,4,4,4,5,7,8,8,9,12,15,16,17,21,27,31,33,38,48,58,
%T A103372 64,71,86,106,122,135,157,192,228,257,292,349,420,485,549,641,769,905,
%U A103372 1034,1190,1410,1674,1939,2224,2600,3084,3613,4163,4824,5684,6697,7776
%N A103372 a(1) = a(2) = a(3) = a(4) = a(5) = 1 and for n>5: a(n) = a(n-4) + a(n-5).
%C A103372 k=4 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) and k=3 case is A079398 (offset so as to begin 1,1,1,1).
%C 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]).
%C A103372 For this k=4 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the irreducible characteristic polynomial: x^5 - x - 1 = 0, A160155.
%C A103372 The sequence of prime values in this k=4 case is A103382; The sequence of semiprime values in this k=4 case is A103392.
%D A103372 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 A103372 Indranil Ghosh, <a href="/A103372/b103372.txt">Table of n, a(n) for n = 1..14857</a>
%H A103372 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 A103372 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 A103372 Richard Padovan, <a href="http://www.nexusjournal.com/conferences/N2002-Padovan.html">Dom Hans van der Laan and the Plastic Number</a>.
%H A103372 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 A103372 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 A103372 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1,1).
%F A103372 G.f. -x*(1+x)*(1+x^2) / ( -1+x^4+x^5 ). - _R. J. Mathar_, Aug 26 2011
%F A103372 a(n) = A124789(n-2)+A124798(n-1). - _R. J. Mathar_, Jun 30 2020
%e A103372 a(14) = 5 because a(14) = a(14-4) + a(14-5) = a(10) + a(9) = 3 + 2 = 5.
%t A103372 k = 4; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 61]
%t A103372 LinearRecurrence[{0,0,0,1,1},{1,1,1,1,1},70] (* _Harvey P. Dale_, Apr 22 2015 *)
%o A103372 (PARI) a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,1,0,0,0]^(n-1)*[1;1;1;1;1])[1,1] \\ _Charles R Greathouse IV_, Oct 03 2016
%Y A103372 Cf. A000931, A079398, A103373-A103380, A103382, A103392.
%K A103372 nonn,easy
%O A103372 1,6
%A A103372 _Jonathan Vos Post_, Feb 03 2005
%E A103372 Edited by _Ray Chandler_ and _Robert G. Wilson v_, Feb 06 2005