

A103378


a(n) = a(n10) + a(n11) 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
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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(nk) + a(n[k+1]). For this k=10 case, the ratio of successive terms a(n)/a(n1) 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

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)/(1x^10x^11).  R. J. Mathar, Nov 22 2007


EXAMPLE

a(52)=17 because a(52)=a(5210)+a(5211) = a(42)+a(41) = 9 + 8. The sequence has as elements 5, 17 and 257, which are all Fermat Primes.


MAPLE



MATHEMATICA

Clear[a]; k=10; Do[a[n]=1, {n, k+1}]; a[n_]:=a[n]=a[nk]+a[nk1]; 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^101)/(x1)/(1x^10x^11)+O(x^80)) \\ M. F. Hasler, Sep 19 2015


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



