login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A125145 a(n) = 3a(n-1) + 3a(n-2). a(0) = 1, a(1) = 4. 20
1, 4, 15, 57, 216, 819, 3105, 11772, 44631, 169209, 641520, 2432187, 9221121, 34959924, 132543135, 502509177, 1905156936, 7222998339, 27384465825, 103822392492, 393620574951, 1492328902329, 5657848431840, 21450532002507 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of aa-avoiding words of length n on the alphabet {a,b,c,d}.

Equals row 3 of the array shown in A180165, the INVERT transform of A028859 and the INVERTi transform of A086347. - Gary W. Adamson, Aug 14 2010

From Tom Copeland, Nov 08 2014: (Start)

This array is one of a family related by compositions of C(x)= [1-sqrt(1-4x)]/2, an o.g.f. for A000108; its inverse Cinv(x) = x(1-x); and the special Mobius transformation P(x,t) = x / (1+t*x) with inverse P(x,-t) in x. Cf. A091867.

O.g.f.: G(x) = P[P[P[-Cinv(-x),-1],-1],-1] = P[-Cinv(-x),-3] = x*(1+x)/[1-3x(1-x)]= x*A125145(x).

Ginv(x) = -C[-P(x,3)] = [-1 + sqrt(1+4x/(1+3x))]/2 = x*A104455(-x).

G(-x) = -x(1-x) * [ 1 - 3*[x*(1+x)] + 3^2*[x*(1+x)]^2 - ...] , and so this array is related to finite differences in the row sums of A030528 * Diag((-3)^1,3^2,(-3)^3,..). (Cf. A146559.)

The inverse of -G(-x) is C[-P(-x,3)]= [1 - sqrt(1-4x/(1-3x))]/2 = x*A104455(x). (End)

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Joerg Arndt, Matters Computational (The Fxtbook)

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.

M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

Tanya Khovanova, Recursive Sequences

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

FORMULA

G.f.: (1+z)/(1-3z-3z^2). - Emeric Deutsch, Feb 27 2007

a(n) = (5*sqrt(21)/42 + 1/2)*(3/2 + sqrt(21)/2))^(n-1) + (-5*sqrt(21)/42 + 1/2)*(3/2 - sqrt(21)/2))^(n-1). - Antonio Alberto Olivares, Mar 20 2008

MAPLE

a[0]:=1: a[1]:=4: for n from 2 to 27 do a[n]:=3*a[n-1]+3*a[n-2] od: seq(a[n], n=0..27); # Emeric Deutsch, Feb 27 2007

MATHEMATICA

nn=23; CoefficientList[Series[(1+x)/(1-3x-3x^2), {x, 0, nn}], x] (* Geoffrey Critzer, Feb 09 2014 *)

PROG

(Haskell)

a125145 n = a125145_list !! n

a125145_list =

   1 : 4 : map (* 3) (zipWith (+) a125145_list (tail a125145_list))

-- Reinhard Zumkeller, Oct 15 2011

(MAGMA) I:=[1, 4]; [n le 2 select I[n] else 3*Self(n-1)+3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Nov 10 2014

CROSSREFS

Cf. A028859 = a(n+2) = 2 a(n+1) + 2 a(n); A086347 = On a 3 X 3 board, number of n-move routes of chess king ending at a given side cell. a(n) = 4a(n-1) + 4a(n-2).

Cf. A128235.

Cf. A180165, A028859, A086347. - Gary W. Adamson, Aug 14 2010

Cf. A002605, A026150, A030195, A080040, A083337, A106435, A108898.

Cf. A000108, A091867, A125145, A104455, A030528, A146559.

Sequence in context: A244824 A077823 A047108 * A242781 A277924 A095930

Adjacent sequences:  A125142 A125143 A125144 * A125146 A125147 A125148

KEYWORD

nonn,easy

AUTHOR

Tanya Khovanova, Jan 11 2007

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified July 20 20:57 EDT 2017. Contains 289629 sequences.