A182757 -id:A182757 - cp's OEIS frontend

A038754 a(2n) = 3^n, a(2n+1) = 2*3^n.

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

Offset: 0

Henry Bottomley, May 03 2000

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.
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 periods: 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
Numbers k such that the k-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
Also, the number of walks of length n on the graph 0--1--2--3--4 starting at vertex 1. - Sean A. Irvine, Jun 03 2025

			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

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, Vol. 332 (2014), pp. 45-54.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Nachum Dershowitz, Between Broadway and the Hudson, arXiv:2006.06516 [math.CO], 2020.
Sean A. Irvine, Walks on Graphs.
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).

Cf. A000045, A000069, A000079, A000120, A000244, A001969, A006720, A006721.
Cf. A006722, A006723, A008776, A028495, A030436, A037124, A048328, A061551.
Cf. A068911, A070909, A087503, A124302, A128588, A133464, A140730, A140740.
Cf. A178381, A182751, A182752, A182753, A182754, A182755, A182756, A182757.
Fifth row of the array A094718.

import Data.List (transpose)
a038754 n = a038754_list !! n
a038754_list = concat $ transpose [a000244_list, a008776_list]
-- Reinhard Zumkeller, Oct 19 2015

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

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

LinearRecurrence[{0,3},{1,2},40] (* Harvey P. Dale, Jan 26 2014 *)
CoefficientList[Series[(1+2x)/(1-3x^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 *)

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

[2^(n%2)*3^((n-(n%2))/2) for n in range(61)] # G. C. Greubel, Oct 10 2022

a(n) = a(n-1)*a(n-2)/a(n-3) with a(0)=1, a(1)=2, a(2)=3.
a(2*n) = (3/2)*a(2*n-1) = 3^n, a(2*n+1) = 2*a(2*n) = 2*3^n.
From Benoit Cloitre, Apr 27 2003: (Start)
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.
a(n) = (1/6)*(5-(-1)^n)*3^floor(n/2).
a(2*n) = a(2*n-1) + a(2*n-2) + a(2*n-3).
a(2*n+1) = a(2*n) + a(2*n-1). (End)
G.f.: (1+2*x)/(1-3*x^2). - Paul Barry, Aug 25 2003
From Reinhard Zumkeller, Sep 11 2003: (Start)
a(n) = (1 + n mod 2) * 3^floor(n/2).
a(n) = A087503(n) - A087503(n-1). (End)
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
From Reinhard Zumkeller, May 26 2008: (Start)
a(n) = A140740(n+2,2).
a(n+1) = a(n) + a(n - n mod 2). (End)
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
Sum_(n>=0) 1/a(n) = 9/4. - Alexander R. Povolotsky, Aug 24 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
From Reinhard Zumkeller, Oct 19 2015: (Start)
a(2*n) = A000244(n), a(2*n+1) = A008776(n).
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. (End)
E.g.f.: (7*cosh(sqrt(3)*x) + 4*sqrt(3)*sinh(sqrt(3)*x) - 4)/3. - Stefano Spezia, Feb 17 2022
Sum_{n>=0} (-1)^n/a(n) = 3/4. - Amiram Eldar, Dec 02 2022

A182751 a(1)=1, a(2)=3, a(3)=6; a(n) = 3*a(n-2) for n > 3.

Original entry on oeis.org

1, 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

Offset: 1

Jaroslav Krizek, Nov 27 2010

For n >= 3: a(n) = the smallest number > a(n-1) such that ((a(n-2) + a(n-1))*(a(n-2) + a(n))*(a(n-1) + a(n)))/(a(n-2)*a(n-1)*a(n)) is an integer (= 10 for n >= 4).

Number of necklaces with n-1 beads and 3 colors that are the same when turned over and hence have reflection symmetry. Example: For n=4 there are 9 necklaces with the colors A, B and C: AAA, AAB, AAC, ABB, ACC, BBB, BBC, BCC, CCC. The only necklaces without reflection symmetry are ABC and ACB. - Herbert Kociemba, Nov 24 2016

			For n = 5; a(3) = 6, a(4) = 9, a(5) = 18 before ((6+9)*(6+18)*(9+18)) / (6*9*18) = 10.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,3).

Essentially the same as A038754 (cf. formula).

Cf. A182752 - A182757.

I:=[3,6]; [1] cat [n le 2 select I[n] else 3*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 11 2018

Join[{1},RecurrenceTable[{a[2]==3,a[3]==6,a[n]==3a[n-2]},a[n],{n,50}]] (* or *) Transpose[NestList[{#[[2]],#[[3]],3#[[2]]}&,{1,3,6},49]][[1]] (* Harvey P. Dale, Oct 19 2011 *)
Rest@ CoefficientList[Series[x (1 + 3 x + 3 x^2)/(1 - 3 x^2), {x, 0, 34}], x] (* Michael De Vlieger, Nov 24 2016 *)
Join[{1}, LinearRecurrence[{0, 3}, {3, 6}, 30]] (* Vincenzo Librandi, Nov 25 2016 *)

x='x+O('x^30); Vec(x*(1+3*x+3*x^2)/(1-3*x^2)) \\ G. C. Greubel, Jan 11 2018

a(n) = A038754(n) for n >= 2.
a(2*k) = (3/2)*a(2*k-1) for k >= 2, a(2*k+1) = 2*a(2*k).
G.f.: x*(1 + 3*x + 3*x^2)/(1 - 3*x^2). - Herbert Kociemba, Nov 24 2016

A182752 a(1) = 1, a(2) = 6, for n >= 3; a(n) = the smallest number greater than a(n-1) such that [[a(n-2) + a(n-1)] * [a(n-2) + a(n)] * [a(n-1) + a(n)]] / [a(n-2) * a(n-1) * a(n)] is an integer.

Original entry on oeis.org

1, 6, 14, 84, 196, 1176, 2744, 16464, 38416, 230496, 537824, 3226944, 7529536, 45177216, 105413504, 632481024, 1475789056, 8854734336, 20661046784

Offset: 1

Jaroslav Krizek, Nov 27 2010

			a(5)=196 since (14+84)*(14+x)*(84+x)/(14*84*x) is an integer for x=196, but not an integer for any x satisfying 85 <= x <= 195.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,14).

Cf. A182751, A182753, A182754, A182755, A182756, A182757.

nxt[{n_,a_}]:={n+1,If[OddQ[n],6*a,7/3 a]}; NestList[nxt,{1,1},20][[All,2]] (* Harvey P. Dale, Aug 14 2020 *)

a(2n) = 6 * a(2n-1), a(2n+1) = 7/3 * a(2n).

G.f.: (1 + 6*x)/(1 - 14*x^2). - Georg Fischer, Nov 17 2022

A182754 a(1) = 1, a(2) = 21, a(n) = 77*a(n-2) for n>=3.

Original entry on oeis.org

1, 21, 77, 1617, 5929, 124509, 456533, 9587193, 35153041, 738213861, 2706784157, 56842467297, 208422380089, 4376869981869, 16048523266853, 337018988603913, 1235736291547681, 25950462122501301, 95151694449171437, 1998185583432600177, 7326680472586200649

Offset: 1

Jaroslav Krizek, Nov 27 2010

For n >= 3, a(n) = the smallest number h > a(n-1) such that [[a(n-2) + a(n-1)] * [a(n-2) + a(n)] * [a(n-1) + a(n)]] / [a(n-2) * a(n-1) * a(n)] is an integer (= 104).

			For n = 4; a(2) = 21, a(3) = 77, a(4) = 1617 before [(21+77)*(21+1617)*(77+1617)]  / (21*77*1617) = 104.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,77).

Cf. A182751 - A182757, A038754.

I:=[1,21]; [n le 2 select I[n] else 77*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 11 2018

LinearRecurrence[{0,77},{1,21},30] (* Harvey P. Dale, Sep 05 2013 *)

A182754(n)=if(n%2,77^(n\2),77^(n\2-1)*21)

Vec(x*(1 + 21*x) / (1 - 77*x^2) + O(x^40)) \\ Colin Barker, Jan 11 2018

def aupton(nn):
  dmo = [1, 21, 77]
  for n in range(3, nn+1): dmo.append(77*dmo[-2])
  return dmo[:nn]
print(aupton(21)) # Michael S. Branicky, Jan 21 2021

a(2*n) = 21*a(2*n-1), a(2*n+1) = (11/3)*a(2*n).
G.f.: x*(1+21*x) / ( 1 - 77*x^2 ).
From Colin Barker, Jan 11 2018: (Start)
a(n) = 3*7^(n/2)*11^(n/2-1) for n even.
a(n) = 77^((n-1)/2) for n odd. (End)

More terms from Harvey P. Dale, Sep 05 2013

A182753 Expansion of (1 + 14x)/(1 - 35x^2).

Original entry on oeis.org

1, 14, 35, 490, 1225, 17150, 42875, 600250, 1500625, 21008750, 52521875, 735306250, 1838265625, 25735718750, 64339296875, 900750156250, 2251875390625, 31526255468750, 78815638671875, 1103418941406250, 2758547353515625, 38619662949218750, 96549157373046875

Offset: 1

Jaroslav Krizek, Nov 27 2010

a(1) = 1, a(2) = 14, for n >= 3; a(n) = the smallest number h > a(n-1) such that [[a(n-2) + a(n-1)] * [a(n-2) + h] * [a(n-1) + h]] / [a(n-2) * a(n-1) * h] is an integer (= 54).

5^(floor((n - 1)/2)) | a(n), n>=1. - G. C. Greubel, Jan 11 2018

			For n = 5; a(3) = 35, a(4) = 490, a(5) = 1225 before [(35+490)*(35+1225)*(490+1225)]  / (35*490*1225) = 54.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,35).

Cf. A182751, A182752, A182754, A182755, A182756, A182757.

I:=[1,14]; [n le 2 select I[n] else 35*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 11 2018

LinearRecurrence[{0, 35}, {1, 14}, 30] (* or *) CoefficientList[Series[(1 + 14*x)/(1-35*x^2), {x,0,50}], x] (* G. C. Greubel, Jan 11 2018 *)

Vec((1+14*x)/(1-35*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

a(2n) = 14 * a(2n-1), a(2n+1) = (5/2) * a(2n).

a(2n) = 14*35^(n-1), a(2n+1) = 35^n.

Terms a(15) onward added by G. C. Greubel, Jan 11 2018

A182755 Expansion of (1+35x)/(1-90x^2).

Original entry on oeis.org

1, 35, 90, 3150, 8100, 283500, 729000, 25515000, 65610000, 2296350000, 5904900000, 206671500000, 531441000000, 18600435000000, 47829690000000, 1674039150000000, 4304672100000000, 150663523500000000, 387420489000000000, 13559717115000000000, 34867844010000000000

Offset: 1

Jaroslav Krizek, Nov 27 2010

a(1) = 1, a(2) = 35, for n >= 3; a(n) = the smallest number h > a(n-1) such that [[a(n-2) + a(n-1)] * [a(n-2) + h] * [a(n-1) + h]] / [a(n-2) * a(n-1) * h] is an integer (= 130). (conjectured)

10^(floor((n - 1)/2)) | a(n), for n>=1. - G. C. Greubel, Jan 11 2018

			For n = 4; a(2) = 35, a(3) = 90, a(4) = 3150 before [(35+90)*(35+3150)*(90+3150)]  / (35*90*3150) = 130.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,90).

Cf. A182751, A182752, A182753, A182754, A182756, A182757, A038754.

I:=[1,35]; [n le 2 select I[n] else 90*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 11 2018

LinearRecurrence[{0,90}, {1,35}, 50] (* or *) CoefficientList[Series[(1 + 35*x)/(1-90*x^2), {x,0,50}], x] (* G. C. Greubel, Jan 11 2018 *)

Vec((1+35*x)/(1-90*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

a(2n) = 35* a(2n-1), a(2n+1) = (18/7) * a(2n).

a(2n) = 35*90^(n-1), a(2n+1) = 90^n.

Terms a(12) onward added by G. C. Greubel, Jan 11 2018

cp's OEIS Frontend

A038754 a(2n) = 3^n, a(2n+1) = 2*3^n.

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A182751 a(1)=1, a(2)=3, a(3)=6; a(n) = 3*a(n-2) for n > 3.

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A182752 a(1) = 1, a(2) = 6, for n >= 3; a(n) = the smallest number greater than a(n-1) such that [[a(n-2) + a(n-1)] * [a(n-2) + a(n)] * [a(n-1) + a(n)]] / [a(n-2) * a(n-1) * a(n)] is an integer.

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A182754 a(1) = 1, a(2) = 21, a(n) = 77*a(n-2) for n>=3.

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A182753 Expansion of (1 + 14*x)/(1 - 35*x^2).

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A182755 Expansion of (1+35*x)/(1-90*x^2).

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A182753 Expansion of (1 + 14x)/(1 - 35x^2).

A182755 Expansion of (1+35x)/(1-90x^2).