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A227375
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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.
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8
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1, 1, 1, 2, 3, 5, 9, 14, 24, 41, 69, 118, 200, 340, 579, 985, 1677, 2854, 4858, 8270, 14078, 23966, 40798, 69453, 118235, 201280, 342655, 583328, 993046, 1690543, 2877949, 4899369, 8340598, 14198887, 24171937, 41149884, 70052848, 119256753, 203020631, 345618810, 588375486, 1001640259
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OFFSET
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0,4
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COMMENTS
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Radius of convergence r is a root of 1 - r - r^2 - r^3 + r^5 + r^6 + r^7 = 0,
where r = Limit a(n)/a(n+1) = 0.587411973105598587998520092901249815195963...
Compare to sequence A227376, generated by 1/(1-x-x^2-x^3+x^5+x^6+x^7).
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, 0, 0, -2, -2, -1, 0, 1, 1, 1).
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FORMULA
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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
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EXAMPLE
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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 +...
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MATHEMATICA
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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 *)
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 *)
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PROG
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(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)
for(n=0, 50, print1(a(n), ", "))
(PARI) /* From R. J. Mathar's g.f. formula: */
{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)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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