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A127840 a(1)=1, a(2)=...=a(6)=0, a(n) = a(n-6)+a(n-5) for n>6.

%I A127840 #14 Apr 24 2025 10:46:46
%S A127840 1,0,0,0,0,0,1,0,0,0,0,1,1,0,0,0,1,2,1,0,0,1,3,3,1,0,1,4,6,4,1,1,5,10,
%T A127840 10,5,2,6,15,20,15,7,8,21,35,35,22,15,29,56,70,57,37,44,85,126,127,94,
%U A127840 81,129,211,253,221,175,210,340,464,474,396,385,550
%N A127840 a(1)=1, a(2)=...=a(6)=0, a(n) = a(n-6)+a(n-5) for n>6.
%C A127840 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.
%D A127840 S. Suter, Binet-like formulas for recurrent sequences with characteristic equation x^k=x+1, preprint, 2007. [Apparently unpublished as of May 2016]
%H A127840 Colin Barker, <a href="/A127840/b127840.txt">Table of n, a(n) for n = 1..1000</a>
%H A127840 Sadjia Abbad and Hacène Belbachir, <a href="https://math.colgate.edu/~integers/z38/z38.pdf">The r-Fibonacci polynomial and its companion sequences linked with some classical sequences</a>, Integers (2025), Vol. 25, Art. No. A38. See p. 17.
%H A127840 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,1,1).
%F A127840 Binet-like formula: a(n) = Sum_{i=1..6} (r_i^n)/(5(r_i)^2+6(r_i)) where r_i is a root of x^6=x+1.
%F A127840 a(n) = A017837(n-6). - _R. J. Mathar_, Sep 20 2012
%F A127840 G.f.: x*(1-x)*(1+x+x^2+x^3+x^4) / (1-x^5-x^6). - _Colin Barker_, May 30 2016
%o A127840 (PARI) Vec(x*(1-x)*(1+x+x^2+x^3+x^4)/(1-x^5-x^6) + O(x^100)) \\ _Colin Barker_, May 30 2016
%K A127840 nonn,easy
%O A127840 1,18
%A A127840 Stephen Suter (sms5064(AT)psu.edu), Apr 02 2007