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 A108306 Expansion of (3*x+1)/(1-3*x-3*x^2). 8
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Martin Burtscher, Igor Szczyrba, 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). FORMULA 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 MAPLE seriestolist(series((3*x+1)/(1-3*x-3*x^2), x=0, 25)); -or- Floretion Algebra Multiplication Program, FAMP Code: 4ibaseforseq[ + .25'i + .25i' + 1.25'ii' + 1.25'jj' + .25'kk' + .25'jk' + .25'kj' + .25e], 1vesfor = A000004 MATHEMATICA CoefficientList[Series[(3 x + 1) / (1 - 3 x - 3 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 01 2016 *) PROG (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 CROSSREFS Cf. A055271. Cf. A084057. Sequence in context: A053768 A255719 A134927 * A199115 A320649 A219596 Adjacent sequences:  A108303 A108304 A108305 * A108307 A108308 A108309 KEYWORD easy,nonn AUTHOR Creighton Dement, Jul 24 2005 STATUS approved

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Last modified January 19 13:17 EST 2021. Contains 340270 sequences. (Running on oeis4.)