A128019 -id:A128019 - cp's OEIS frontend

A162436 a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 3.

1, 3, 3, 9, 9, 27, 27, 81, 81, 243, 243, 729, 729, 2187, 2187, 6561, 6561, 19683, 19683, 59049, 59049, 177147, 177147, 531441, 531441, 1594323, 1594323, 4782969, 4782969, 14348907, 14348907, 43046721, 43046721, 129140163, 129140163, 387420489, 387420489, 1162261467

Offset: 1

Klaus Brockhaus, Jul 03 2009, Jul 05 2009

Interleaving of A000244 and 3*A000244.
Unsigned version of A128019.
Partial sums are in A164123.
Apparently a(n) = A056449(n-1) for n > 1. a(n) = A108411(n) for n >= 1.
Binomial transform is A026150 without initial 1, second binomial transform is A001834, third binomial transform is A030192, fourth binomial transform is A161728, fifth binomial transform is A162272.

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

Cf. A000244 (powers of 3), A128019 (expansion of (1-3x)/(1+3x^2)), A164123, A026150, A001834, A030192, A161728, A162272.

Essentially the same as A056449 (3^floor((n+1)/2)) and A108411 (powers of 3 repeated).

[ n le 2 select 2*n-1 else 3*Self(n-2): n in [1..35] ];

CoefficientList[Series[(-3*x - 1)/(3*x^2 - 1), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
Transpose[NestList[{Last[#],3*First[#]}&,{1,3},40]][[1]] (* or *) With[{c= 3^Range[20]},Join[{1},Riffle[c,c]]](* Harvey P. Dale, Feb 17 2012 *)

a(n)=3^(n>>1) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = 3^((1/4)*(2*n - 1 + (-1)^n)).
G.f.: x*(1 + 3*x)/(1 - 3*x^2).
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
E.g.f.: cosh(sqrt(3)*x) - 1 + sinh(sqrt(3)*x)/sqrt(3). - Stefano Spezia, Dec 31 2022

G.f. corrected, formula simplified, comments added by Klaus Brockhaus, Sep 18 2009

A128018 Expansion of (1-4x)/(1-2x+4*x^2).

Original entry on oeis.org

1, -2, -8, -8, 16, 64, 64, -128, -512, -512, 1024, 4096, 4096, -8192, -32768, -32768, 65536, 262144, 262144, -524288, -2097152, -2097152, 4194304, 16777216, 16777216, -33554432, -134217728, -134217728, 268435456, 1073741824, 1073741824, -2147483648, -8589934592

Offset: 0

Paul Barry, Feb 11 2007

Hankel transform of A128014(n+1). Binomial transform of A128019.
Hankel transform of A002426(n+1). - Paul Barry, Mar 15 2008
Hankel transform of A007971(n+1). - Paul Barry, Sep 30 2009
Hankel transform of A103970 is a(n)/4^C(n+1,2). - Paul Barry, Nov 20 2009
The real part of Q^(n+1), where Q is the quaternion 1+i+j+k. - Stanislav Sykora, Jun 11 2012.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-4).

Cf. A002426, A007971, A128014, A103970, A128019, A138340.

CoefficientList[Series[(1 - 4*x)/(1 - 2*x + 4*x^2), {x,0,50}], x] (* or *) LinearRecurrence[{2,-4},{1,-2},50] (* G. C. Greubel, Feb 28 2017 *)

x='x+O('x^50); Vec((1-4*x)/(1-2*x+4*x^2)) \\ G. C. Greubel, Feb 28 2017

a(n) = A138340(n)/2^n. - Philippe Deléham, Nov 14 2008
a(n) = 2^(n+1)*cos(Pi*(n+1)/3). - Richard Choulet, Nov 19 2008
From Paul Barry, Oct 21 2009: (Start)
a(n) = Sum_{k=0..floor((n+1)/2)} C(n+1,2*k)*(-3)^k.
a(n) = ((1+i*sqrt(3))^(n+1) + (1-i*sqrt(3))^(n+1))/2, i=sqrt(-1). (End)
G.f.: G(0)/(2*x)-1/x, where G(k)= 1 + 1/(1 - x*(3*k+1)/(x*(3*k+4) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013
a(n) = 2^n*A057079(n+2). - R. J. Mathar, Mar 04 2018
Sum_{n>=0} 1/a(n) = 1/3. - Amiram Eldar, Feb 14 2023

A141533 The first subdiagonal of the array of A141425 and its higher order differences.

Original entry on oeis.org

1, -1, -2, 23, 28, -7, 22, 251, 376, 149, 658, 3143, 5188, 4913, 13102, 42611, 75376, 101549, 232618, 612863, 1137148, 1831433, 3928582, 9185771, 17574376, 31162949, 64717378, 141392183, 275609908, 515347553, 1052218462, 2212053731, 4359537376, 8396224349

Offset: 1

Paul Curtz, Aug 12 2008

sign

			A141425 and its first, second, third differences etc. in followup rows define an array T(n,m):
..1...2...4...5...7...8...1...2...4...5...
..1...2...1...2...1..-7...1...2...1...2...
..1..-1...1..-1..-8...8...1..-1...1..-1...
.-2...2..-2..-7..16..-7..-2...2..-2..-7...
..4..-4..-5..23.-23...5...4..-4..-5..23...
.-8..-1..28.-46..28..-1..-8..-1..28.-46...
..7..29.-74..74.-29..-7...7..29.-74..74...
.22.-103.148.-103..22..14..22.-103.148.-103...
-125.251.-251.125..-8...8.-125.251.-251.125...
376.-502.376.-133..16.-133.376.-502.376.-133...
Then a(n) = T(n+1,n) .

a(2*n)+a(2*n+1)= 0, 21, 21, 273, 525, 3801,... (multiples of 21).
a(n)= +a(n-1) -a(n-2) +3*a(n-3) +6*a(n-4). G.f.: x*(1-2*x+21*x^3)/((1-2*x) * (1+x) * (3*x^2+1)). [R. J. Mathar, Nov 22 2009]
a(n)= (3*(-1)^n+2^n-A128019(n+1))/2. [R. J. Mathar, Nov 22 2009]

Edited and extended by R. J. Mathar, Nov 22 2009

A287479 Expansion of g.f. (x + x^2)/(1 + 3*x^2).

Original entry on oeis.org

0, 1, 1, -3, -3, 9, 9, -27, -27, 81, 81, -243, -243, 729, 729, -2187, -2187, 6561, 6561, -19683, -19683, 59049, 59049, -177147, -177147, 531441, 531441, -1594323, -1594323, 4782969, 4782969, -14348907, -14348907, 43046721, 43046721, -129140163, -129140163, 387420489

Offset: 0

Jean-François Alcover and Paul Curtz, Jul 19 2017

This is the inverse binomial transform of A157241.
Successive differences of A157241 begin:
0,   1,   3,   3,   -5,  -21,  -21,    43,   171,   171, ... = A157241
1,   2,   0,  -8,  -16,    0,   64,   128,     0,  -512, ... = A088138
1,  -2,  -8,  -8,   16,   64,   64,  -128,  -512,  -512, ... = A138230
-3, -6,   0,  24,   48,    0, -192,  -384,     0,  1536, ...
-3,  6,  24,  24,  -48, -192, -192,   384,  1536,  1536, ...
9,  18,   0, -72, -144,    0,  576,  1152,     0, -4608, ...
9, -18, -72  -72,  144,  576,  576, -1152, -4608, -4608, ...
...
a(n) is the n-th term of the first column.
Successive differences of a(n) begin:
0,     1,    1,   -3,   -3,    9,     9,   -27,   -27,    81, ...
1,     0,   -4,    0,   12,    0,   -36,     0,   108,     0, ...
-1,   -4,    4,   12,  -12,  -36,    36,   108,  -108,  -324, ...
-3,    8,    8,  -24,  -24,   72,    72,  -216,  -216,   648, ...
11,    0,  -32,    0,   96,    0,  -288,     0,   864,     0, ...
-11, -32,   32,   96,  -96, -288,   288,   864,  -864, -2592, ...
-21,  64,   64, -192, -192,  576,   576, -1728, -1728,  5184, ...
85,    0, -256,    0,  768,    0, -2304,     0,  6912,     0, ...
...
First column appears to be a subsequence of Jacobsthal numbers A001045 (the trisection A082311 is missing), second column is A104538, and third column is A137717.
a(n) = A128019(n-2) for n > 2. - Georg Fischer, Oct 23 2018

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

Cf. A001045, A082311, A088138, A104538, A108411, A128019, A137717, A138230, A140429, A157241.

Join[{0}, LinearRecurrence[{0, -3}, {1, 1}, 40]]
(* or, computation from b = A157241 : *)
b[n_] := (Switch[Mod[n, 3], 0, (-1)^((n + 3)/3), 1, (-1)^((n + 5)/3), 2, (-1)^((n + 4)/3)*2]*2^n + 1)/3; tb = Table[b[n], {n, 0, 40}]; Table[ Differences[tb, n], {n, 0, 40}][[All, 1]]

concat([0], Vec((x + x^2)/(1 + 3*x^2) + O(x^40))) \\ Felix Fröhlich, Oct 23 2018

a(n) = -3*a(n-2) for n > 2.

E.g.f.: (1 - cos(sqrt(3)*x) + sqrt(3)*sin(sqrt(3)*x))/3. - Stefano Spezia, Jul 15 2024

cp's OEIS Frontend

A162436 a(n) = 3*a(n-2) for n > 2; a(1) = 1, a(2) = 3.

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A128018 Expansion of (1-4*x)/(1-2*x+4*x^2).

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A141533 The first subdiagonal of the array of A141425 and its higher order differences.

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A287479 Expansion of g.f. (x + x^2)/(1 + 3*x^2).

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A128018 Expansion of (1-4x)/(1-2x+4*x^2).