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A038754 a(2n) = 3^n, a(2n+1) = 2*3^n. 84
1, 2, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486, 729, 1458, 2187, 4374, 6561, 13122, 19683, 39366, 59049, 118098, 177147, 354294, 531441, 1062882, 1594323, 3188646, 4782969, 9565938, 14348907, 28697814, 43046721, 86093442, 129140163, 258280326, 387420489 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

In general, for the recurrence a(n) = a(n-1)*a(n-2)/a(n-3), all terms are integers iff a(0) divides a(2) and first three terms are positive integers, since a(2n+k) = a(k)*(a(2)/a(0))^n for all nonnegative integers n and k.

a(n) = A140740(n+2,2). - Reinhard Zumkeller, May 26 2008

Equals eigensequence of triangle A070909; (1, 1, 2, 3, 6, 9, 18, ...) shifts to the left with multiplication by triangle A070909. - Gary W. Adamson, May 15 2010

The a(n) represent all paths of length (n+1), n >= 0, starting at the initial node on the path graph P_5, see the second Maple program. - Johannes W. Meijer, May 29 2010

a(n) is the difference between numbers of multiple of 3 evil (A001969) and odious (A000069) numbers in interval [0, 2^(n+1)). - Vladimir Shevelev, May 16 2012

A "half-geometric progression": to obtain a term (beginning with the third one) we multiply the before previous one by 3. - Vladimir Shevelev, May 21 2012

Pisano period lengths: 1, 2, 1, 4, 8, 2, 12, 4, 1, 8, 10, 4, 6, 12, 8, 8, 32, 2, 36, 8, ... . - R. J. Mathar, Aug 10 2012

Sum_(n>=0) 1/a(n) = 9/4. - Alexander R. Povolotsky, Aug 24 2012

Numbers n such that the n-th cyclotomic polynomial has a root mod 3. - Eric M. Schmidt, Jul 31 2013

Range of row n of the circular Pascal array of order 6. - Shaun V. Ault, Jun 05 2014

a(2*n) = A000244(n), a(2*n+1) = A008776(n). - Reinhard Zumkeller, Oct 19 2015

LINKS

T. D. Noe and Indranil Ghosh, Table of n, a(n) for n = 0..1500, (first 401 terms from T. D. Noe)

S. V. Ault and C. Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Mathematics (2014).

Paul Barry, Three √Čtudes on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.

Nachum Dershowitz, Between Broadway and the Hudson, arXiv:2006.06516 [math.CO], 2020.

Richard L. Ollerton and Anthony G. Shannon, Some properties of generalized Pascal squares and triangles, Fib. Q., 36 (1998), 98-109. See pages 106-7.

Vladimir Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.

M. B. Wells, Elements of Combinatorial Computing, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237-240]

Index entries for linear recurrences with constant coefficients, signature (0,3).

FORMULA

a(n) = a(n-1)*a(n-2)/a(n-3) with a(0)=1, a(1)=2, a(2)=3.

a(2n) = (3/2)*a(2n-1) = 3^n, a(2n+1) = 2*a(2n) = 2*3^n.

a(1)=1, a(n)=2*a(n-1) if a(n-1) is odd, or a(n) = (3/2)*a(n-1) if a(n-1) is even. - Benoit Cloitre, Apr 27 2003

a(n) = (1/6)*(5-(-1)^n)*3^floor(n/2); a(2n) = a(2n-1) + a(2n-2) + a(2n-3); a(2n+1) = a(2n) + a(2n-1). - Benoit Cloitre, Apr 27 2003

G.f.: (1+2x)/(1-3x^2). - Paul Barry, Aug 25 2003

a(n) = (1 + n mod 2) * 3^floor(n/2). a(n) = A087503(n) - A087503(n-1). - Reinhard Zumkeller, Sep 11 2003

a(n) = sqrt(3)*(2+sqrt(3))*(sqrt(3))^n/6 - sqrt(3)*(2-sqrt(3))*(-sqrt(3))^n/6. - Paul Barry, Sep 16 2003

a(n+1) = a(n) + a(n - n mod 2). - Reinhard Zumkeller, May 26 2008

If p(i) = Fibonacci(i-3) and if A is the Hessenberg matrix of order n defined by A(i,j) = p(j-i+1), (i<=j), A(i,j)=-1, (i=j+1), and A(i,j)=0 otherwise. Then, for n>=1, a(n-1) = (-1)^n det A. - Milan Janjic, May 08 2010

a(n) = A182751(n) for n >= 2. - Jaroslav Krizek, Nov 27 2010

a(n) = Sum_{i=0..2^(n+1), i==0 (mod 3)} (-1)^A000120(i). - Vladimir Shevelev, May 16 2012

a(0)=1, a(1)=2, for n>=3, a(n)=3*a(n-2). - Vladimir Shevelev, May 21 2012

a(n) = sqrt(3*a(n-1)^2 + (-3)^(n-1)). - Richard R. Forberg, Sep 04 2013

a(n) = 2^((1-(-1)^n)/2)*3^((2*n-1+(-1)^n)/4). - Luce ETIENNE, Aug 11 2014

For n > 0: a(n+1) = a(n) + if a(n) odd then min{a(n), a(n-1)} else max{a(n), a(n-1)}, see also A128588. - Reinhard Zumkeller, Oct 19 2015

E.g.f.: (7*cosh(sqrt(3)*x) + 4*sqrt(3)*sinh(sqrt(3)*x) - 4)/3. - Stefano Spezia, Feb 17 2022

EXAMPLE

In the interval [0,2^5) we have 11 multiples of 3 numbers, from which 10 are evil and only one (21) is odious. Thus a(4) = 10 - 1 = 9. - Vladimir Shevelev, May 16 2012

MAPLE

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=3*a[n-2]+2 od: seq(a[n]+1, n=0..34); # Zerinvary Lajos, Mar 20 2008

with(GraphTheory): P:=5: G:=PathGraph(P): A:= AdjacencyMatrix(G): nmax:=35; for n from 1 to nmax do B(n):=A^n; a(n):=add(B(n)[1, k], k=1..P) od: seq(a(n), n=1..nmax); # Johannes W. Meijer, May 29 2010

MATHEMATICA

LinearRecurrence[{0, 3}, {1, 2}, 40] (* Harvey P. Dale, Jan 26 2014 *)

CoefficientList[Series[(1 + 2 x) / (1 - 3 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2016 *)

Module[{nn=20, c}, c=3^Range[0, nn]; Riffle[c, 2c]] (* Harvey P. Dale, Aug 21 2021 *)

PROG

(PARI) a(n)=(1/6)*(5-(-1)^n)*3^floor(n/2)

(PARI) a(n)=3^(n>>1)<<bittest(n, 0)

(Haskell)

import Data.List (transpose)

a038754 n = a038754_list !! n

a038754_list = concat $ transpose [a000244_list, a008776_list]

-- Reinhard Zumkeller, Oct 19 2015

(Magma) [n le 2 select n else 3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 18 2016

CROSSREFS

Cf. A006720, A006721, A006722, A006723.

Fifth row of the array A094718.

Cf. A000045, A000079, A028495, A030436, A037124, A048328, A061551, A068911, A070909, A124302, A133464, A140730, A140740, A178381, A182751, A182752, A182753, A182754, A182755, A182756, A182757.

Cf. A000069, A000120, A000244, A001969, A008776, A087503, A128588.

Sequence in context: A035522 A018311 A018481 * A182522 A165647 A191398

Adjacent sequences:  A038751 A038752 A038753 * A038755 A038756 A038757

KEYWORD

easy,nice,nonn

AUTHOR

Henry Bottomley, May 03 2000

STATUS

approved

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Last modified May 19 12:20 EDT 2022. Contains 353833 sequences. (Running on oeis4.)