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A267541 Expansion of (2 + 4*x + x^2 + x^3 + 2*x^4 + x^5)/(1 - x - x^5 + x^6).

%I A267541 #39 Sep 08 2022 08:46:15
%S A267541 2,6,7,8,10,13,17,18,19,21,24,28,29,30,32,35,39,40,41,43,46,50,51,52,
%T A267541 54,57,61,62,63,65,68,72,73,74,76,79,83,84,85,87,90,94,95,96,98,101,
%U A267541 105,106,107,109,112,116,117,118,120,123,127,128,129,131,134,138,139,140
%N A267541 Expansion of (2 + 4*x + x^2 + x^3 + 2*x^4 + x^5)/(1 - x - x^5 + x^6).
%C A267541 Also, numbers that are congruent to {2, 6, 7, 8, 10} mod 11.
%C A267541 (m^k+1)/11 is a nonnegative integer when
%C A267541 . m is a member of this sequence and k is an odd multiple of 5 (A017329),
%C A267541 . m is a member of A017509 and k is odd but not multiple of 5 (A045572).
%C A267541 If k is even, (m^k+1)/11 is never an integer.
%C A267541 The product of two terms does not belong to the sequence.
%H A267541 Bruno Berselli, <a href="/A267541/b267541.txt">Table of n, a(n) for n = 0..1000</a>
%H A267541 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).
%F A267541 G.f.: (2 + 4*x + x^2 + x^3 + 2*x^4 + x^5)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
%F A267541 a(n) = a(n-1) + a(n-5) - a(n-6).
%F A267541 a(-n) = -A267755(n-1).
%e A267541 From the linear recurrence:
%e A267541 (-A267755) ..., -12, -9, -5, -4, -3, -1, 2, 6, 7, 8, 10, 13, ... (A267541)
%p A267541 gf := (2+4*x+x^2+x^3+2*x^4+x^5)/((1-x)^2*(1+x+x^2+x^3+ x^4)): deg := 64: series(gf,x,deg): seq(coeff(%,x,n), n=0..deg-1); # _Peter Luschny_, Jan 19 2016
%t A267541 CoefficientList[Series[(2 + 4 x + x^2 + x^3 + 2 x^4 + x^5)/(1 - x - x^5 + x^6), {x, 0, 70}], x]
%t A267541 LinearRecurrence[{1, 0, 0, 0, 1, -1}, {2, 6, 7, 8, 10, 13}, 70]
%t A267541 Select[Range[150], MemberQ[{2, 6, 7, 8, 10}, Mod[#, 11]]&]
%o A267541 (PARI) Vec((2+4*x+x^2+x^3+2*x^4+x^5)/(1-x-x^5+x^6)+O(x^70))
%o A267541 (Magma) m:=70; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((2+4*x+x^2+x^3+2*x^4+x^5)/(1-x-x^5+x^6)));
%o A267541 (Sage)
%o A267541 gf = (2+4*x+x^2+x^3+2*x^4+x^5)/((1-x)^2*(1+x+x^2+x^3+ x^4))
%o A267541 print(taylor(gf, x, 0, 63).list()) # _Peter Luschny_, Jan 19 2016
%Y A267541 Cf. A002266, A017329, A017509, A267755.
%Y A267541 Cf. A088225: numbers congruent to {2,6,7,8} mod 11.
%K A267541 nonn,easy
%O A267541 0,1
%A A267541 _Bruno Berselli_, Jan 16 2016