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A125145
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a(n) = 3a(n-1) + 3a(n-2). a(0) = 1, a(1) = 4.
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20
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1, 4, 15, 57, 216, 819, 3105, 11772, 44631, 169209, 641520, 2432187, 9221121, 34959924, 132543135, 502509177, 1905156936, 7222998339, 27384465825, 103822392492, 393620574951, 1492328902329, 5657848431840, 21450532002507
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OFFSET
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0,2
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COMMENTS
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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)
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Joerg Arndt, Matters Computational (The Fxtbook)
Martin Burtscher, Igor Szczyrba, Rafał 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)
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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nn=23; CoefficientList[Series[(1+x)/(1-3x-3x^2), {x, 0, nn}], x] (* Geoffrey Critzer, Feb 09 2014 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,easy
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AUTHOR
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Tanya Khovanova, Jan 11 2007
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STATUS
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approved
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