|
%I #56 Sep 08 2022 08:45:23
%S 0,0,0,1,2,4,7,14,28,57,114,228,455,910,1820,3641,7282,14564,29127,
%T 58254,116508,233017,466034,932068,1864135,3728270,7456540,14913081,
%U 29826162,59652324,119304647,238609294,477218588,954437177,1908874354,3817748708
%N Expansion of x^3/(1 - 2*x + x^3 - 2*x^4) = x^3/( (1-2*x)*(1+x)*(1-x+x^2) ).
%C A transform of the Jacobsthal numbers. A059633 is the equivalent transform of the Fibonacci numbers.
%C _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 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 Vincenzo Librandi, <a href="/A113405/b113405.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 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 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,2).
%F a(n) = 2a(n-1) - a(n-3) + 2a(n-4).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*A001045(k).
%F a(n) = Sum_{k=0..n} binomial((n+k)/2,k)*A001045((n-k)/2)*(1+(-1)^(n-k))/2.
%F a(3n) = A015565(n), a(3n+1) = 2*A015565(n), a(3n+2) = 4*A015565(n). - _Paul Curtz_, Nov 30 2007
%F From _Paul Curtz_, Dec 16 2007: (Start)
%F a(n+1) - 2a(n) = A131531(n).
%F a(n) + a(n+3) = 2^n. (End)
%F a(n) = round(2^n/9). - _Ross Drewe_, Sep 03 2009
%F 9*a(n) = 2^n + (-1)^n - 3*A010892(n). - _R. J. Mathar_, Mar 24 2018
%p A010892 := proc(n) op((n mod 6)+1,[1,1,0,-1,-1,0]) ; end proc:
%p A113405 := proc(n) (2^n-(-1)^n)/9 -A010892(n-1)/3; end proc: # _R. J. Mathar_, Dec 17 2010
%t 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 (PARI) a(n)=2^n\/9 \\ _Charles R Greathouse IV_, Jun 05, 2011
%o (Magma) [Round(2^n/9): n in [0..40]]; // _Vincenzo Librandi_, Aug 11 2011
%Y From _Ross Drewe_, Sep 03 2009: (Start)
%Y Other sequences a(n) = round(b^n / c), where b and c are very small integers:
%Y A001045 b = 2; c = 3
%Y A007910 b = 2; c = 5
%Y A016029 b = 2; c = 5/3
%Y A077947 b = 2; c = 7
%Y abs(A078043) b = 2; c = 7/3
%Y A007051 b = 3; c = 2
%Y A015518 b = 3; c = 4
%Y A034478 b = 5; c = 2
%Y A003463 b = 5; c = 4
%Y A015531 b = 5; c = 6
%Y (End)
%K easy,nonn
%O 0,5
%A _Paul Barry_, Oct 28 2005
%E Edited by _N. J. A. Sloane_, Dec 13 2007
|
