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

Author

Paul Barry, Oct 28 2005

Keywords

• easy
• nonn

Comments

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

Links

• 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).

Crossrefs

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)

Programs

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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

Extensions

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