

A108300


a(n+2) = 3*a(n+1) + a(n), a(0) = 1, a(1) = 5.


5



1, 5, 16, 53, 175, 578, 1909, 6305, 20824, 68777, 227155, 750242, 2477881, 8183885, 27029536, 89272493, 294847015, 973813538, 3216287629, 10622676425, 35084316904, 115875627137, 382711198315, 1264009222082, 4174738864561
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OFFSET

0,2


COMMENTS

Binomial transform is A109114 (Comment: KekulĂ© numbers for certain benzenoids). Invert transform is A109115 (Comment: KekulĂ© numbers for certain benzenoids.) Inverse invert transform is: A016777 (Comment: Ignoring the first term, this sequence represents the number of bonds in a hydrocarbon: a(# of carbon atoms) = number of bonds.  Nathan Savir (thoobik(AT)yahoo.com), Jul 03 2003.) Inverse binomial transform is A006130. Program "Superseeker" finds (incomplete): A052924(n+1)  A052924(n) = a(n). May be seen as a transform of the zerosequence A000004 (see "force transforms" link).
From Gary W. Adamson, Sep 06 2008: (Start)
Equals right border of triangle A143972.
(1, 5, 16, 53, 175,...) = row sums of triangle A143972 and INVERT transform of A016777: (1, 4, 7, 10,...). (End)


LINKS

Table of n, a(n) for n=0..24.
Sergio Falcon, The kFibonacci difference sequences, Chaos, Solitons & Fractals, Volume 87, June 2016, Pages 153157.
Tanya Khovanova, Recursive Sequences
Vincent Vatter, Growth rates of permutation classes: from countable to uncountable, arXiv:1605.04297 [math.CO], 2016. (Mentions a signed version.)
Index entries for linear recurrences with constant coefficients, signature (3,1).


FORMULA

G.f. (2*x1)/(x^21+3*x).
a(n)=(7/26)*[3/2(1/2)*sqrt(13)]^n*sqrt(13)+(7/26)*sqrt(13)*[3/2+(1/2)*sqrt(13)]^n+(1/2)*[3/2 (1/2)*sqrt(13)]^n+(1/2)*[3/2+(1/2)*sqrt(13)]^n, with n>=0.  Paolo P. Lava, Sep 19 2008
a(n)*a(n2) = a(n1)^2+9*(1)^n.  Roger L. Bagula, May 17 2010


MAPLE

seriestolist(series((2*x1)/(x^21+3*x), x=0, 25)); or Floretion Algebra Multiplication Program, FAMP Code: 4ibaseforseq[ + .25'i + .25i' + 1.25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj' + .25e], 1vesfor = A000004
with(combinat): a:=n>2*fibonacci(n1, 3)+fibonacci(n, 3): seq(a(n), n=1..25); # Zerinvary Lajos, Apr 04 2008


MATHEMATICA

LinearRecurrence[{3, 1}, {1, 5}, 40] (* Harvey P. Dale, Jul 04 2013 *)


CROSSREFS

Cf. A109114, A109115, A016777, A006130, A000004, A052924, A228916.
Cf. A143972, A016777.  Gary W. Adamson, Sep 06 2008
Sequence in context: A274492 A147536 A173871 * A041469 A089102 A098912
Adjacent sequences: A108297 A108298 A108299 * A108301 A108302 A108303


KEYWORD

nonn,easy


AUTHOR

Creighton Dement, Jul 24 2005


STATUS

approved



