A103378 a(n) = a(n-10) + a(n-11) for n > 11, and a(n) = 1 for 1 <= n <= 11.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 12, 15, 16, 16, 16, 16, 16, 16, 16, 17, 21, 27, 31, 32, 32, 32, 32, 32, 32, 33, 38, 48, 58, 63, 64, 64, 64, 64, 64, 65, 71, 86, 106, 121, 127, 128

Offset

1,12

Comments

k=10 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 and k=9 case is A103377. 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=10 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the characteristic polynomial: x^11 - x - 1 = 0. This is the real constant 1.0682971889208412763694295883238782820936310169208334445076119466470069702... . Note that x = (1 + (1 + (1 + (1 + (1 + ...)^(1/11))^(1/11)))^(1/11))))^(1/11)))))^(1/11))))). The sequence of prime values in this k=10 case is A103388. The sequence of semiprime values in this k=10 case is A103398.

Links

Table of n, a(n) for n=1..78.

J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms

Richard Padovan, Dom Hans van der Laanand the Plastic Number.

E. S. Selmer, On the irreducibility of certain trinomials, Math. Scand., 4 (1956) 287-302.

J. Shallit, A generalization of automatic sequences, Theoretical Computer Science, 61(1988)1-16.

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.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1,1).

Formula

G.f.: x*(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9)/(1-x^10-x^11). - R. J. Mathar, Nov 22 2007

Example

a(52)=17 because a(52)=a(52-10)+a(52-11) = a(42)+a(41) = 9 + 8. The sequence has as elements 5, 17 and 257, which are all Fermat Primes.

Maple

A103378 := proc(n) option remember; if n <= 11 then 1 ; else A103378(n-10)+A103378(n-11) ; fi ; end: seq(A103378(n), n=1..78) ; # R. J. Mathar, Nov 22 2007

Mathematica

Clear[a]; k=10; Do[a[n]=1, {n, k+1}]; a[n_]:=a[n]=a[n-k]+a[n-k-1]; A103377=Array[a, 100] N[Solve[x^10 - x - 1 == 0, x], 111][[2]] (* see also A103387 for primes and A103398 for semiprimes in this sequence *)
LinearRecurrence[Join[Table[0, {9}], {1, 1}], Table[1, {11}], 80] (* Harvey P. Dale, Aug 14 2013 *)

Prog

(PARI) Vec((x^10-1)/(x-1)/(1-x^10-x^11)+O(x^80)) \\ M. F. Hasler, Sep 19 2015

Crossrefs

Cf. A000045, A000931, A079398, A103372-A103377, A103379-A103380, A103388, A103398.

Sequence in context: A156821 A025856 A350765 * A103663 A349925 A339171

Adjacent sequences: A103375 A103376 A103377 * A103379 A103380 A103381

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 15 2005

Extensions

Corrected and extended by R. J. Mathar, Nov 22 2007

Edited by M. F. Hasler, Sep 19 2015

Status

approved