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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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23
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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)
Number of 3-compositions of n+1 restricted to parts 1 and 2 (and allowed zeros); see Hopkins & Ouvry reference. - Brian Hopkins, Aug 16 2020
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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)
D. Birmajer, J. B. Gil, M. D. Weiner, n the Enumeration of Restricted Words over a Finite Alphabet , J. Int. Seq. 19 (2016) # 16.1.3, Example 7.
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.
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
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).
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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
a(n) = A030195(n)+A030195(n+1) . - R. J. Mathar, Feb 13 2022
E.g.f.: exp(3*x/2)*(21*cosh(sqrt(21)*x/2) + 5*sqrt(21)*sinh(sqrt(21)*x/2))/21. - Stefano Spezia, Aug 04 2022
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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
A125145 := proc(n)
option remember;
if n <= 1 then
op(n+1, [1, 4]) ;
else
3*(procname(n-1)+procname(n-2)) ;
end if;
end proc: # R. J. Mathar, Feb 13 2022
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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 *)
LinearRecurrence[{3, 3}, {1, 4}, 30] (* Harvey P. Dale, May 01 2022 *)
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PROG
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(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: A316592 A077823 A047108 * A242781 A346195 A277924
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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