A078043 -id:A078043 - cp's OEIS frontend

A113405 Expansion of x^3/(1 - 2x + x^3 - 2x^4) = x^3/( (1-2x)(1+x)*(1-x+x^2) ).

0, 0, 0, 1, 2, 4, 7, 14, 28, 57, 114, 228, 455, 910, 1820, 3641, 7282, 14564, 29127, 58254, 116508, 233017, 466034, 932068, 1864135, 3728270, 7456540, 14913081, 29826162, 59652324, 119304647, 238609294, 477218588, 954437177, 1908874354, 3817748708

Offset: 0

Paul Barry, Oct 28 2005

A transform of the Jacobsthal numbers. A059633 is the equivalent transform of the Fibonacci numbers.
Paul Curtz, Aug 05 2007, observes that the inverse binomial transform of 0,0,0,1,2,4,7,14,28,57,114,228,455,910,1820,... gives the same sequence up to signs. That is, the extended sequence is an eigensequence for the inverse binomial transform (an autosequence).
The round() function enables the closed (non-recurrence) formula to take a very simple form: see Formula section. This can be generalized without loss of simplicity to a(n) = round(b^n/c), where b and c are very small, incommensurate integers (c may also be an integer fraction). Particular choices of small integers for b and c produce a number of well-known sequences which are usually defined by a recurrence - see Cross Reference. - Ross Drewe, Sep 03 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

From Ross Drewe, Sep 03 2009: (Start)
Other sequences a(n) = round(b^n / c), where b and c are very small integers:
A001045 b = 2; c = 3
A007910 b = 2; c = 5
A016029 b = 2; c = 5/3
A077947 b = 2; c = 7
abs(A078043) b = 2; c = 7/3
A007051 b = 3; c = 2
A015518 b = 3; c = 4
A034478 b = 5; c = 2
A003463 b = 5; c = 4
A015531 b = 5; c = 6
(End)

[Round(2^n/9): n in [0..40]]; // Vincenzo Librandi, Aug 11 2011

A010892 := proc(n) op((n mod 6)+1,[1,1,0,-1,-1,0]) ; end proc:
A113405 := proc(n) (2^n-(-1)^n)/9 -A010892(n-1)/3; end proc: # R. J. Mathar, Dec 17 2010

CoefficientList[Series[x^3/(1-2x+x^3-2x^4),{x,0,40}],x] (* or *) LinearRecurrence[{2,0,-1,2},{0,0,0,1},40] (* Harvey P. Dale, Apr 30 2011 *)

a(n)=2^n\/9 \\ Charles R Greathouse IV, Jun 05 2011

def A113405(n): return ((1<Chai Wah Wu, Apr 17 2025

a(n) = 2a(n-1) - a(n-3) + 2a(n-4).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*A001045(k).
a(n) = Sum_{k=0..n} binomial((n+k)/2,k)*A001045((n-k)/2)*(1+(-1)^(n-k))/2.
a(3n) = A015565(n), a(3n+1) = 2*A015565(n), a(3n+2) = 4*A015565(n). - Paul Curtz, Nov 30 2007
From Paul Curtz, Dec 16 2007: (Start)
a(n+1) - 2a(n) = A131531(n).
a(n) + a(n+3) = 2^n. (End)
a(n) = round(2^n/9). - Ross Drewe, Sep 03 2009
9*a(n) = 2^n + (-1)^n - 3*A010892(n). - R. J. Mathar, Mar 24 2018

Edited by N. J. A. Sloane, Dec 13 2007

A033129 Base-2 digits are, in order, the first n terms of the periodic sequence with initial period [1,1,0].

Original entry on oeis.org

0, 1, 3, 6, 13, 27, 54, 109, 219, 438, 877, 1755, 3510, 7021, 14043, 28086, 56173, 112347, 224694, 449389, 898779, 1797558, 3595117, 7190235, 14380470, 28760941, 57521883, 115043766, 230087533, 460175067, 920350134, 1840700269

Offset: 0

Clark Kimberling

Number of moves to separate a Hanoi Tower into two towers of even resp. odd stones. - Martin von Gagern, May 26 2004
From Reinhard Zumkeller, Feb 22 2010: (Start)
Terms of A173593 with initial digits '11' in binary representation: a(n) = A173593(2*n-3) for n>0;
for n>0: a(3*n-1) = A083713(n);
a(n+1) - a(n) = abs(A078043(n)). (End)

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Mohammad Sajjad Hossain, reArrange.
James Metz, Twists on the Tower of Hanoi, Math. Teacher, Vol. 107, No. 9 (2014), 712-715.
Index entries for sequences related to Towers of Hanoi
Index entries for linear recurrences with constant coefficients, signature (2,0,1,-2).

Cf. A011655 (repeat 0,1,1), A289006 (the same in octal).

Cf. A057744, A294627 (first differences).

Table[(1/14)*(-9 - 2*(-1)^Floor[(2 n)/3] + (-1)^(1 + Floor[(1/3)*(7 + 2 n)]) + 3*2^(2 + n)), {n, 0, 100}] (* John M. Campbell, Dec 26 2016 *)
Table[FromDigits[PadRight[{},n,{1,1,0}],2],{n,0,40}] (* Harvey P. Dale, Oct 02 2022 *)

A033129(n)=3<<(n+1)\7 \\ M. F. Hasler, Jun 23 2017

print([(6*2**n//7) for n in range(50)]) # Karl V. Keller, Jr., Jul 11 2022

From Paul Barry, Jan 23 2004: (Start)
Partial sums of abs(A078043).
G.f.: x*(1+x)/((1-x)*(1-2*x)*(1+x+x^2)) = x*(1+x)/(1-2*x-x^3+2*x^4).
a(n) = (6/7)*2^n - (4/21)*cos(2*Pi*n/3) - (2/21)*sqrt(3)*sin(2*Pi*n/3) - 2/3. (End)
a(n) = a(n-3) + 3 * 2^(n-3). - Martin von Gagern, May 26 2004
a(n+1) = 2*a(n) + 1 - 0^((a(n)+1) mod 4). - Reinhard Zumkeller, Feb 22 2010
a(n) = floor(2^(n+1)*3/7). - Jean-Marie Madiot, Oct 05 2012
a(n) = (1/14)*(-9 - 2*(-1)^floor((2n)/3) + (-1)^(floor((2*n + 7)/3) + 1) + 3*2^(n + 2)). - John M. Campbell, Dec 26 2016

A294627 Expansion of x(1 + x)/((1-2x)*(1+x+x^2)).

Original entry on oeis.org

0, 1, 2, 3, 7, 14, 27, 55, 110, 219, 439, 878, 1755, 3511, 7022, 14043, 28087, 56174, 112347, 224695, 449390, 898779, 1797559, 3595118, 7190235, 14380471, 28760942, 57521883, 115043767, 230087534, 460175067, 920350135, 1840700270, 3681400539, 7362801079, 14725602158, 29451204315

Offset: 0

Wojciech Florek, Feb 12 2018

A generalized tribonacci (A001590) sequence.
For n > 2, 6*a(n) is the number of quaternary sequences of length n in which all triples (q(i),q(i+1),q(i+2)) contain two (arbitrarily chosen) digits, e.g., 0 and 3.
Similarly, recurrences a(n) = a(n-1) + a(n-2) + k*a(n-3) are related to binary (k=0, the Fibonacci sequence A000045), ternary (k=1, the tribonacci sequence A001590), quinary (k=3) and so on sequences with all triples (t(i),t(i+1),t(i+2)) containing two (arbitrarily chosen) digits (usually 0 and k+1).
For n > 0, a(n) is the number of ways to tile a strip of length n with squares, dominoes, and two colors of trominoes, with the restriction that the first tile cannot be a tromino. - Greg Dresden and Bora Bursalı, Aug 31 2023
For n > 1, a(n) is the number of ways to tile a strip of length n-2 with squares, dominoes, and two colors of trominoes, where the strip begins with an upper level of two cells. For example, when n=7 we have this strip of length 5:
 _
|||_____
|||_|||. - Guanji Chen and Greg Dresden, Jun 17 2024

			For n=4 there are 6*7=42 quaternary sequences of length 4 such that each triple (i.e., exactly two of them: q1,q2,q3 and q2,q3,q4) contain both 0 and 3. They are 003x, 030x, 03y0, 0330, 330x, 303x, 30y3, 3003, 0y30, 3y03, y03x, y30x, where x=0,1,2,3 and y=1,2.

Wojciech Florek, Table of n, a(n) for n = 0..500
Wojciech Florek, A class of generalized Tribonacci sequences applied to counting problems, Appl. Math. Comput., 338 (2018), 809-821.
Index entries for linear recurrences with constant coefficients, signature (1,1,2).

Cf. A000045, A001590, A007040, A033129 (partial sums), A078043, A167373.

LinearRecurrence[{1, 1, 2}, {0, 1, 2}, 50] (* Paolo Xausa, Aug 28 2024 *)

my(x='x+O('x^99)); concat(0, Vec(x*(1+x)/(1-x-x^2-2*x^3))) \\ Altug Alkan, Mar 03 2018

a(n) = a(n-1) + a(n-2) + 2*a(n-3) for n > 2.
a(n+1)/a(n) tends to 2, the unique real root of x^3 - x^2 - x - 2 = 0.
a(n+1) = abs(A078043(n)).
7*a(n) = 3*2^n - A167373(n+1). - R. J. Mathar, Mar 24 2018
E.g.f.: exp(-x/2)*(9*exp(5*x/2) - 9*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))/21. - Stefano Spezia, Aug 29 2024

cp's OEIS Frontend

A113405 Expansion of x^3/(1 - 2x + x^3 - 2x^4) = x^3/( (1-2x)(1+x)*(1-x+x^2) ).

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A033129 Base-2 digits are, in order, the first n terms of the periodic sequence with initial period [1,1,0].

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A294627 Expansion of x(1 + x)/((1-2x)*(1+x+x^2)).

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cp's OEIS Frontend

A113405 Expansion of x^3/(1 - 2*x + x^3 - 2*x^4) = x^3/( (1-2*x)*(1+x)*(1-x+x^2) ).

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A033129 Base-2 digits are, in order, the first n terms of the periodic sequence with initial period [1,1,0].

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Mathematica

PARI

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A294627 Expansion of x*(1 + x)/((1-2*x)*(1+x+x^2)).

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Programs

Mathematica

PARI

Formula

A113405 Expansion of x^3/(1 - 2x + x^3 - 2x^4) = x^3/( (1-2x)(1+x)*(1-x+x^2) ).

A294627 Expansion of x(1 + x)/((1-2x)*(1+x+x^2)).