A103377 - OEIS

%I #40 Sep 01 2024 00:14:20

%S 1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,5,7,8,8,8,8,

%T 8,8,8,9,12,15,16,16,16,16,16,16,17,21,27,31,32,32,32,32,32,33,38,48,

%U 58,63,64,64,64,64,65,71,86,106,121,127,128,128,128,129,136,157,192,227

%N a(1)=a(2)=...=a(10)=1, a(n)=a(n-9)+a(n-10).

%C k=9 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, k=6 case is A103374, k=7 case is A103375, k=8 case is A103376, k=10 case is A103378 and k=11 case is A103379. The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1)= 1 and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]). For this k=9 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^10 - x - 1 = 0. This is the real constant (to 50 digits accuracy): 1.0757660660868371580595995241652758206925302476392 = A230163. Note that x = (1 + x)^(1/10) = (1 + (1 + (1 + ...)^(1/10))^(1/10))^(1/10). The sequence of prime values in this k=9 case is A103387; The sequence of semiprime values in this k=9 case is A103397.

%C In analogy to the Fibonacci sequence, one might prefer to start this sequence with offset 0. - _M. F. Hasler_, Sep 19 2015

%D A. J. van Zanten, The golden ratio in the arts of painting, building and mathematics, Nieuw Archief voor Wiskunde, vol 17 no 2 (1999) 229-245.

%H J.-P. Allouche and T. Johnson, <a href="http://www.math.jussieu.fr/~allouche/johnson2.pdf">Narayana's Cows and Delayed Morphisms</a>

%H J.-P. Allouche and T. Johnson, <a href="http://recherche.ircam.fr/equipes/repmus/jim96/actes/Allouche.ps">Narayana's Cows and Delayed Morphisms</a>

%H Richard Padovan, <a href="https://www.nexusjournal.com/the-nexus-conferences/nexus-2002/148-n2002-padovan.html">Dom Hans Van Der Laan And The Plastic Number</a>.

%H E. S. Selmer, <a href="https://doi.org/10.7146/math.scand.a-10478">On the irreducibility of certain trinomials</a>, Math. Scand., 4 (1956) 291-306

%H 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 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,1,1).

%F a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = a(10) = 1 and for n>10: a(n) = a(n-9) + a(n-10).

%F O.g.f.: -x*(x^2+x+1)*(x^6+x^3+1)/(-1+x^9+x^10). - _R. J. Mathar_, May 02 2008

%e a(83) = 257 because a(83) = a(83-9) + a(83-10). a(74) + a(73) = 129 + 128. This sequence has as elements 5, 17 and 257, which are all Fermat Primes.

%t LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1, 1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, 90] (* _Charles R Greathouse IV_, Jan 11 2013 *)

%o (PARI) Vec((1+x+x^2)*(1+x^3+x^6)/(1-x^9-x^10)+O(x^99)) \\ _Charles R Greathouse IV_, Jan 11 2013

%Y Cf. A000045, A000931, A079398, A103372-A103376, A103378-A103380, A103387, A103397.

%K easy,nonn

%O 1,11

%A _Jonathan Vos Post_, Feb 15 2005

%E Edited by _R. J. Mathar_, May 02 2008

%E Edited by _M. F. Hasler_, Sep 19 2015