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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, RafaƂ 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 A219596 A182251

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 June 22 12:24 EDT 2017. Contains 288613 sequences.