login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A103378 a(n) = a(n-10) + a(n-11) for n > 11, and a(n) = 1 for 1 <= n <= 11. 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 (list; graph; refs; listen; history; text; internal format)
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
J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms
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.
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
Sequence in context: A156821 A025856 A350765 * A103663 A349925 A339171
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 21 14:30 EDT 2024. Contains 371874 sequences. (Running on oeis4.)