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A015565
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a(n) = 7*a(n-1) + 8*a(n-2), a(0) = 0, a(1) = 1.
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26
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0, 1, 7, 57, 455, 3641, 29127, 233017, 1864135, 14913081, 119304647, 954437177, 7635497415, 61083979321, 488671834567, 3909374676537, 31274997412295, 250199979298361, 2001599834386887, 16012798675095097, 128102389400760775
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OFFSET
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0,3
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COMMENTS
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A linear 2nd order recurrence. A Jacobsthal number sequence.
Second binomial transform of A080424. Binomial transform of A053573, with leading zero. Binomial transform is 0,1,9,81,729,....(9^n - 0^n)/9. Second binomial transform is 0,1,11,111,1111,... (A002275: repunits). - Paul Barry, Mar 14 2004
Number of walks of length n between any two distinct nodes of the complete graph K_9. Example: a(2)=7 because the walks of length 2 between the nodes A and B of the complete graph ABCDEFGHI are: ACB, ADB, AEB, AFB, AGB, AHB and AIB. - Emeric Deutsch, Apr 01 2004
General form: k=8^n-k. Also: A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
The ratio a(n+1)/a(n) converges to 8 as n approaches infinity. - Felix P. Muga II, Mar 09 2014
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LINKS
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FORMULA
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a(n) = (8^n - (-1)^n)/9.
a(n) = J(3*n)/3 = A001045(3*n)/3. (End)
a(n) = 8^(n-1) - a(n-1).
G.f.: x/(1-7*x-8*x^2). (End)
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EXAMPLE
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G.f. = x + 7*x^2 + 57*x^3 + 455*x^4 + 3641*x^5 + 29127*x^6 + 233017*x^7 + ...
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MAPLE
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MATHEMATICA
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LinearRecurrence[{7, 8}, {0, 1}, 30] (* Harvey P. Dale, Mar 04 2016 *)
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PROG
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(Sage) [lucas_number1(n, 7, -8) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009
(PARI) x='x+O('x^30); concat([0], Vec(x/(1-7*x-8*x^2))) \\ G. C. Greubel, Dec 30 2017
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CROSSREFS
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Cf. A001045, A078008, A097073, A115341, A015518, A054878, A015521, A109499, A015531, A109500, A109501, A015552. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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