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A345207
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Number of (unlabeled) 7-paths with n vertices.
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6
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1, 1, 2, 4, 11, 32, 117, 468, 2151, 10722, 58071, 333774, 2018321, 12678506, 82035085, 542520052, 3646124339, 24791545874, 169986552195, 1172526610674, 8122332718341, 56435590886610, 392969320828713, 2740480494041976, 19132214719583207, 133671249471111626
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OFFSET
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9,3
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COMMENTS
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A k-path with order n at least k+2 is a k-tree with exactly two k-leaves (vertices of degree k). It can be constructed from a clique with k+1 vertices by iteratively adding a new degree k vertex adjacent to an existing clique containing an existing k-leaf.
Also, the number of equivalence classes of strings of length n-9 using a maximum of seven different numbers that are equivalent when they can be made the same by permutation of their numbers and possible reversal of the string.
Recurrences appear in the papers by Bickle, Eckhoff, and Markenzon et al.
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REFERENCES
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M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2.]
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (20,-134,200,1502,-6120,-200,35440,-41269,-66380,141454,840,-135912,70560).
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FORMULA
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a(n) = (7^(n-9) + 21*5^(n-9) + 70*4^(n-9) + 315*3^(n-9) + 924*2^(n-9) + 232*7^((n-9)/2) + 700*4^((n-9)/2) + 840*3^((n-9)/2) + 1008*2^((n-9)/2) + 2975)/10080 for n>9 odd;
a(n) = (7^(n-9) + 21*5^(n-9) + 70*4^(n-9) + 315*3^(n-9) + 924*2^(n-9) + 76*7^((n-8)/2) + 280*4^((n-8)/2) + 420*3^((n-8)/2) + 504*2^((n-8)/2) + 2975)/10080 for n even.
a(n) = 20*a(n-1) - 134*a(n-2) + 200*a(n-3) + 1502*a(n-4) - 6120*a(n-5) - 200*a(n-6) + 35440*a(n-7) - 41269*a(n-8) - 66380*a(n-9) + 141454*a(n-10) + 840*a(n-11) - 135912*a(n-12) + 70560*a(n-13) for n > 22. - Stefano Spezia, Aug 01 2021
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MATHEMATICA
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LinearRecurrence[{20, -134, 200, 1502, -6120, -200, 35440, -41269, -66380, 141454, 840, -135912, 70560}, {1, 1, 2, 4, 11, 32, 117, 468, 2151, 10722, 58071, 333774, 2018321, 12678506}, 26] (* Stefano Spezia, Aug 01 2021 *)
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CROSSREFS
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The sequences above converge to A103293(n+1).
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KEYWORD
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easy,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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