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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Binomial transform is A109114 (Comment: Kekulé numbers for certain benzenoids). Invert transform is A109115 (Comment: Kekulé 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 zero-sequence 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)

REFERENCES

Sergio Falcon, The k-Fibonacci difference sequences, Chaos, Solitons & Fractals, Volume 87, June 2016, Pages 153-157.

Vincent Vatter, Growth rates of permutation classes: from countable to uncountable, arXiv:1605.04297 (Mentions a signed version.)

LINKS

Table of n, a(n) for n=0..24.

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (3,1).

FORMULA

G.f. (-2*x-1)/(x^2-1+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(n-2) = a(n-1)^2+9*(-1)^n. - Roger L. Bagula, May 17 2010

MAPLE

seriestolist(series((-2*x-1)/(x^2-1+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(n-1, 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

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Last modified June 28 17:13 EDT 2017. Contains 288839 sequences.