A127842 - OEIS

%I #22 Sep 08 2022 08:45:29

%S 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,

%T 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,

%U 2,8,28,56,70,56,28,9,10,36,84,126,126,84,37

%N a(1)=1, a(2)=...=a(8)=0, a(n) = a(n-8)+a(n-7) for n>8.

%C 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.

%C Apart from offset same as A017857. - _Georg Fischer_, Oct 07 2018

%D S. Suter, Binet-like formulas for recurrent sequences with characteristic equation x^k=x+1, preprint, 2007. [Apparently unpublished as of May 2016]

%H Colin Barker, <a href="/A127842/b127842.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,1,1).

%F 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.

%F 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

%t LinearRecurrence[{0, 0, 0, 0, 0, 0, 1, 1}, {1, 0, 0, 0, 0, 0, 0, 0}, 100] (* _Vincenzo Librandi_, Oct 08 2018 *)

%o (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

%o (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

%o (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

%K nonn,easy

%O 1,24

%A Stephen Suter (sms5064(AT)psu.edu), Apr 02 2007