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A323941
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Total number of isomers (nonisomorphic systems) of unbranched tri-4-catafusenes as a function of the number of hexagons (see Cyvin et al. (1996) for precise definition).
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4
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1, 3, 16, 62, 275, 1121, 4584, 18012, 69573, 262495, 974704, 3562714, 12859127, 45881213, 162093320, 567579192, 1971791241, 6801382203, 23309839120, 79421200630, 269160513115, 907726206233, 3047449980392, 10188384020372, 33930772031565, 112595241877911, 372383348102640, 1227721195083922
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OFFSET
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3,2
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COMMENTS
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Let I(r, k) be the total number of isomers (nonisomorphic systems) of unbranched k-4-catafusenes, which are generated from catafusenes by converting k of its r hexagons to tetragons. According to Cyvin et al. (1996), for r >= k, we have I(r, k) = 1/4 *(binomial(r, k) + (r - 2)! * (r^2 + (4 * k - 1) * r + 4 * k * (k - 2)) * 3^(r - k - 2)/(k! * (r - k)!) + (2 + (-1)^k - (-1)^r) * (binomial(floor(r/2), floor(k/2)) + 2 * binomial(floor(r/2) - 1, floor(k/2) - 1)) * 3^(floor(r/2) - floor(k/2) - 1)). See Eq. (48) on p. 503 in the paper.
Letting k = 0 - 10, we get the eleven columns of Table 2 on p. 501 of Cyvin et al. (1996). (We need r >= max(k, 2) because the number of hexagons r should be greater than or equal to the number of converted polygons k.)
(End)
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (16,-102,304,-247,-1056,3372,-3168,-2223,8208,-8262,3888,-729).
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FORMULA
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a(n) = I(r = n, k = 3) in the formula above in the comments (for n >= 3). - Petros Hadjicostas, May 26 2019
G.f.: -x^3*(-1 +13*x -70*x^2 +192*x^3 -250*x^4 +22*x^5 +402*x^6 -672*x^7 +663*x^8 -387*x^9 +72*x^10) / ( (-1+3*x^2)^2 *(3*x-1)^4 *(x-1)^4 ). - R. J. Mathar, Jul 25 2019
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MAPLE
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CyvinI := proc(r, k)
if r >= k then
1/4 *(binomial(r, k) + (r - 2)! * (r^2 + (4 * k - 1) * r + 4 * k * (k - 2)) * 3^(r - k - 2)/(k! * (r - k)!) + (2 + (-1)^k - (-1)^r) * (binomial(floor(r/2), floor(k/2)) + 2 * binomial(floor(r/2) - 1, floor(k/2) - 1)) * 3^(floor(r/2) - floor(k/2) - 1));
else
-1;
end if;
end proc:
CyvinI(n, 3) ;
end proc:
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MATHEMATICA
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CyvinI[r_, k_] := If[r >= k, 1/4 * (Binomial[r, k] + (r-2)! * (r^2 + (4k - 1) * r + 4k * (k-2)) * 3^(r-k-2)/(k! * (r-k)!) + (2 + (-1)^k - (-1)^r) * (Binomial[Floor[r/2], Floor[k/2]] + 2 Binomial[Floor[r/2]-1, Floor[k/2]-1]) * 3^(Floor[r/2] - Floor[k/2] - 1)), -1];
a[n_] := CyvinI[n, 3];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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