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%I #26 Jul 08 2023 15:19:43
%S 1,1,1,2,3,5,9,14,24,41,69,118,200,340,579,985,1677,2854,4858,8270,
%T 14078,23966,40798,69453,118235,201280,342655,583328,993046,1690543,
%U 2877949,4899369,8340598,14198887,24171937,41149884,70052848,119256753,203020631,345618810,588375486,1001640259
%N G.f.: 1/(1 - x*(1-x^6)/(1 - x^2*(1-x^7)/(1 - x^3*(1-x^8)/(1 - x^4*(1-x^9)/(1 - x^5*(1-x^10)/(1 - ...)))))), a continued fraction.
%C Radius of convergence r is a root of 1 - r - r^2 - r^3 + r^5 + r^6 + r^7 = 0,
%C where r = Limit a(n)/a(n+1) = 0.587411973105598587998520092901249815195963...
%C Compare to sequence A227376, generated by 1/(1-x-x^2-x^3+x^5+x^6+x^7).
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 1, 0, 0, -2, -2, -1, 0, 1, 1, 1).
%F Conjecture: G.f. -(x^4+x-1)*(x^5+x^4+x^3-x-1) / ( (x-1)*(x^4+x^3+x^2+x+1)*(x^7+x^6+x^5-x^3-x^2-x+1) ). - _R. J. Mathar_, Jul 17 2013
%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 9*x^6 + 14*x^7 + 24*x^8 +...
%t nMax = 42; col[m_ /; 0 <= m <= nMax] := 1/(1 + ContinuedFractionK[-x^k (1 - x^(m + k)), 1, {k, 1, Ceiling[nMax/2]}]) + O[x]^(2 nMax) // CoefficientList[#, x]&; A227375 = col[5][[1 ;; nMax]] (* _Jean-François Alcover_, Nov 03 2016 *)
%t LinearRecurrence[{1,1,1,0,0,-2,-2,-1,0,1,1,1},{1,1,1,2,3,5,9,14,24,41,69,118},50] (* _Harvey P. Dale_, Jul 08 2023 *)
%o (PARI) a(n)=local(CF); CF=1+x; for(k=0, n, CF=1/(1 - x^(n-k+1)*(1 - x^(n-k+6))*CF+x*O(x^n))); polcoeff(CF, n)
%o for(n=0,50,print1(a(n),", "))
%o (PARI) /* From R. J. Mathar's g.f. formula: */
%o {a(n)=polcoeff((1-x-x^4)*(1+x-x^3-x^4-x^5)/((1-x^5)*(1-x-x^2-x^3+x^5+x^6+x^7) +x*O(x^n)),n)}
%o for(n=0,50,print1(a(n),", ")) \\ _Paul D. Hanna_, Jul 18 2013
%Y Cf. A173173, A227374, A227360, A227376, A228644, A228645.
%Y Column m=5 of A185646.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jul 09 2013
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