A227375 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.
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
Offset: 0
Keywords
Examples
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 +...
Links
- Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, 0, 0, -2, -2, -1, 0, 1, 1, 1).
Programs
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Mathematica
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 *) -
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),", "))
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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)} for(n=0,50,print1(a(n),", ")) \\ Paul D. Hanna, Jul 18 2013
Formula
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
Comments