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, 2014; http://matinf.pmfbl.org/wp-content/uploads/2015/01/za-arhiv-18.-1.pdf

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

(C++) #include <iostream.h> #include <stdlib.h> #include <math.h> int main(int argc, char *argv[]) { int i; // counter for for loop double j; // for (i=1; i < 12; i++) // change 9 to whatever number you want if desired { j = ( 5.0*sqrt(21.0)/42 + 1.0/2.0)*pow((3.0/2.0 + sqrt(21)/2), (i-1))+ (-5.0*sqrt(21.0)/42 + 1.0/2.0)*pow((3.0/2.0 - sqrt(21)/2), (i-1)) ; // cout << i << ' ' << j << " "; } return EXIT_SUCCESS; } /* Antonio Alberto Olivares, Mar 20 2008 */

(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 March 24 10:04 EDT 2017. Contains 283985 sequences.