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A186446
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Expansion of 1/(1 - 7*x + 2*x^2).
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6
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1, 7, 47, 315, 2111, 14147, 94807, 635355, 4257871, 28534387, 191224967, 1281505995, 8588092031, 57553632227, 385699241527, 2584787426235, 17322113500591, 116085219651667, 777952310560487, 5213495734620075, 34938565521219551
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OFFSET
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0,2
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COMMENTS
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The first differences are in A122074.
a(n+1) equals the number of words of length n over {0,1,2,3,4,5,6} avoiding 01 and 02. - Milan Janjic, Dec 17 2015
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LINKS
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Bruno Berselli, Table of n, a(n) for n = 0..800
Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607.
Index entries for linear recurrences with constant coefficients, signature (7,-2).
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FORMULA
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G.f.: 1/(1-7*x+2*x^2).
a(n) = ((7+sqrt(41))^(1+n)-(7-sqrt(41))^(1+n))/(2^(1+n)*sqrt(41)).
a(n) = 7*a(n-1)-2*a(n-2), with a(0)=1, a(1)=7.
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MATHEMATICA
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CoefficientList[Series[1 / (1 - 7 x + 2 x^2), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{7, -2}, {1, 7}, 30] (* Harvey P. Dale, Aug 06 2017 *)
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PROG
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(Magma) m:=21; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-7*x+2*x^2)));
(Magma) I:=[1, 7]; [n le 2 select I[n] else 7*Self(n-1)-2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 19 2013
(PARI) Vec(1/(1-7*x+2*x^2) + O(x^100)) \\ Altug Alkan, Dec 17 2015
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CROSSREFS
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For similar closed formulas: A015446 [((1+sqrt(41))^(1+n)-(1-sqrt(41))^(1+n))/(2^(1+n)*sqrt(41))], A015525 [((3+sqrt(41))^n-(3-sqrt(41))^n)/(2^n*sqrt(41))], A015537 [((5+sqrt(41))^n-(5-sqrt(41))^n)/(2^n*sqrt(41))], A178869 [((9+sqrt(41))^n-(9-sqrt(41))^n)/(2^n*sqrt(41))].
Same recurrence as in A122074, A003771.
Sequence in context: A085352 A125370 A163346 * A244830 A126528 A214992
Adjacent sequences: A186443 A186444 A186445 * A186447 A186448 A186449
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KEYWORD
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nonn,easy
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AUTHOR
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Bruno Berselli, Feb 21 2011
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STATUS
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approved
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