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A108306
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Expansion of (3*x+1)/(1-3*x-3*x^2).
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8
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1, 6, 21, 81, 306, 1161, 4401, 16686, 63261, 239841, 909306, 3447441, 13070241, 49553046, 187869861, 712268721, 2700415746, 10238053401, 38815407441, 147160382526, 557927369901, 2115263257281, 8019571881546, 30404505416481, 115272231894081
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OFFSET
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0,2
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COMMENTS
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Binomial transform is A055271. May be seen as a ibasefor-transform of the zero-sequence A000004 with respect to the floretion given in the program code.
The sequence is the INVERT transform of (1, 5, 10, 20, 40, 80, 160, ...) and can be obtained by extracting the upper left terms of matrix powers of [(1,5); (1,2)]. These results are a case (a=5, b=2) of the conjecture: The INVERT transform of a sequence starting (1, a, a*b, a*b^2, a*b^3, ...) is equivalent to extracting the upper left terms of powers of the 2x2 matrix [(1,a); (1,b)]. - Gary W. Adamson, Jul 31 2016
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,3).
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FORMULA
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Recurrence: a(0)=1; a(1)=6; a(n) = 3a(n-1) + 3a(n-2) - N-E. Fahssi, Apr 20 2008
a(n) = (1/2)*(3/2 + (1/2)*sqrt(21))^n + (3/14)*(3/2 + (1/2)*sqrt(21))^n*sqrt(21) - (3/14)*sqrt(21)*(3/2 - (1/2)*sqrt(21))^n + (1/2)*(3/2 - (1/2)*sqrt(21))^n, with n >= 0. - Paolo P. Lava, Jun 12 2008
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MAPLE
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seriestolist(series((3*x+1)/(1-3*x-3*x^2), x=0, 25));
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MATHEMATICA
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CoefficientList[Series[(3 x + 1) / (1 - 3 x - 3 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 01 2016 *)
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PROG
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(Magma) I:=[1, 6]; [n le 2 select I[n] else 3*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 01 2016
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CROSSREFS
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Cf. A055271.
Cf. A084057.
Sequence in context: A053768 A255719 A134927 * A199115 A320649 A219596
Adjacent sequences: A108303 A108304 A108305 * A108307 A108308 A108309
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KEYWORD
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easy,nonn
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AUTHOR
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Creighton Dement, Jul 24 2005
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STATUS
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approved
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