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A127842
a(1)=1, a(2)=...=a(8)=0, a(n) = a(n-8)+a(n-7) for n>8.
1
1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 1, 5, 10, 10, 5, 1, 0, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 2, 8, 28, 56, 70, 56, 28, 9, 10, 36, 84, 126, 126, 84, 37
OFFSET
1,24
COMMENTS
Part of the phi_k family of sequences defined by a(1)=1,a(2)=...=a(k)=0, a(n)=a(n-k)+a(n-k+1) for n>k. phi_2 is a shift of the Fibonacci sequence and phi_3 is a shift of the Padovan sequence.
Apart from offset same as A017857. - Georg Fischer, Oct 07 2018
REFERENCES
S. Suter, Binet-like formulas for recurrent sequences with characteristic equation x^k=x+1, preprint, 2007. [Apparently unpublished as of May 2016]
FORMULA
Binet-like formula: a(n) = Sum_{i=1..8} (r_i^n)/(7(r_i)^2+8(r_i)) where r_i is a root of x^8=x+1.
G.f.: x*(1-x)*(1+x+x^2+x^3+x^4+x^5+x^6) / (1-x^7-x^8). - Colin Barker, May 30 2016
MATHEMATICA
LinearRecurrence[{0, 0, 0, 0, 0, 0, 1, 1}, {1, 0, 0, 0, 0, 0, 0, 0}, 100] (* Vincenzo Librandi, Oct 08 2018 *)
PROG
(PARI) Vec(x*(1-x)*(1+x+x^2+x^3+x^4+x^5+x^6)/(1-x^7-x^8) + O(x^100)) \\ Colin Barker, May 30 2016
(GAP) a:=[1, 0, 0, 0, 0, 0, 0, 0];; for n in [9..80] do a[n]:=a[n-7]+a[n-8]; od; a; # Muniru A Asiru, Oct 07 2018
(Magma) I:=[1, 0, 0, 0, 0, 0, 0, 0]; [n le 8 select I[n] else Self(n-7)+Self(n-8): n in [1..100]]; // Vincenzo Librandi, Oct 08 2018
CROSSREFS
Sequence in context: A349903 A308118 A017857 * A127512 A263787 A307710
KEYWORD
nonn,easy
AUTHOR
Stephen Suter (sms5064(AT)psu.edu), Apr 02 2007
STATUS
approved