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

%I A113405 #62 Jul 04 2025 18:40:27
%S A113405 0,0,0,1,2,4,7,14,28,57,114,228,455,910,1820,3641,7282,14564,29127,
%T A113405 58254,116508,233017,466034,932068,1864135,3728270,7456540,14913081,
%U A113405 29826162,59652324,119304647,238609294,477218588,954437177,1908874354,3817748708
%N A113405 Expansion of x^3/(1 - 2*x + x^3 - 2*x^4) = x^3/( (1-2*x)*(1+x)*(1-x+x^2) ).
%C A113405 A transform of the Jacobsthal numbers. A059633 is the equivalent transform of the Fibonacci numbers.
%C A113405 _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).
%C A113405 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
%H A113405 Vincenzo Librandi, <a href="/A113405/b113405.txt">Table of n, a(n) for n = 0..1000</a>
%H A113405 M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, arXiv:math/0205301 [math.CO], 2002. [Link to arXiv version]
%H A113405 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H A113405 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A113405 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,2).
%F A113405 a(n) = 2a(n-1) - a(n-3) + 2a(n-4).
%F A113405 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*A001045(k).
%F A113405 a(n) = Sum_{k=0..n} binomial((n+k)/2,k)*A001045((n-k)/2)*(1+(-1)^(n-k))/2.
%F A113405 a(3n) = A015565(n), a(3n+1) = 2*A015565(n), a(3n+2) = 4*A015565(n). - _Paul Curtz_, Nov 30 2007
%F A113405 From _Paul Curtz_, Dec 16 2007: (Start)
%F A113405 a(n+1) - 2a(n) = A131531(n).
%F A113405 a(n) + a(n+3) = 2^n. (End)
%F A113405 a(n) = round(2^n/9). - _Ross Drewe_, Sep 03 2009
%F A113405 9*a(n) = 2^n + (-1)^n - 3*A010892(n). - _R. J. Mathar_, Mar 24 2018
%p A113405 A010892 := proc(n) op((n mod 6)+1,[1,1,0,-1,-1,0]) ; end proc:
%p A113405 A113405 := proc(n) (2^n-(-1)^n)/9 -A010892(n-1)/3; end proc: # _R. J. Mathar_, Dec 17 2010
%t A113405 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 *)
%o A113405 (PARI) a(n)=2^n\/9 \\ _Charles R Greathouse IV_, Jun 05 2011
%o A113405 (Magma) [Round(2^n/9): n in [0..40]]; // _Vincenzo Librandi_, Aug 11 2011
%o A113405 (Python)
%o A113405 def A113405(n): return ((1<<n)|4)//9 # _Chai Wah Wu_, Apr 17 2025
%Y A113405 From _Ross Drewe_, Sep 03 2009: (Start)
%Y A113405 Other sequences a(n) = round(b^n / c), where b and c are very small integers:
%Y A113405 A001045 b = 2; c = 3
%Y A113405 A007910 b = 2; c = 5
%Y A113405 A016029 b = 2; c = 5/3
%Y A113405 A077947 b = 2; c = 7
%Y A113405 abs(A078043) b = 2; c = 7/3
%Y A113405 A007051 b = 3; c = 2
%Y A113405 A015518 b = 3; c = 4
%Y A113405 A034478 b = 5; c = 2
%Y A113405 A003463 b = 5; c = 4
%Y A113405 A015531 b = 5; c = 6
%Y A113405 (End)
%K A113405 easy,nonn
%O A113405 0,5
%A A113405 _Paul Barry_, Oct 28 2005
%E A113405 Edited by _N. J. A. Sloane_, Dec 13 2007