

A108306


Expansion of (3*x+1)/(13*x3*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
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OFFSET

0,2


COMMENTS

Binomial transform is A055271. May be seen as a ibasefortransform of the zerosequence 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, RafaĆ Szczyrba, Analytic Representations of the nanacci 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(n1)+3a(n2)  NE. 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)/(13*x3*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(n1)+3*Self(n2): 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



