%I A113405
%S A113405 0,0,0,1,2,4,7,14,28,57,114,228,455,910,1820,3641,7282,14564,29127,58254,
%T A113405 116508,233017,466034,932068,1864135,3728270,7456540,14913081,29826162,
%U A113405 59652324,119304647,238609294,477218588,954437177,1908874354,3817748708
%N A113405 G.f.: x^3/(1-2x+x^3-2x^4).
%C A113405 A transform of the Jacobsthal numbers. A059633 is the equivalent transform 
               of the Fibonacci numbers.
%C A113405 Paul Curtz (bpcrtz(AT)free.fr), 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 eigen-sequence for the inverse binomial transform.
%C A113405 Contribution from Ross Drewe (rd(AT)labyrinth.net.au), Sep 03 2009: (Start)
%C A113405 The round() function enables the closed (non-recurrence) formula to take 
               a
%C A113405 very simple form: see Formula section. This can be generalised without 
               loss of
%C A113405 simplicity to a(n) = round(b^n/c), where b and c are very small, incommensurate
%C A113405 integers (c may also be an integer fraction). Particular choices of small
%C A113405 integers for b and c produce a number of well-known sequences which are 
               usually
%C A113405 defined by a recurrence - see Cross Reference. (End)
%H A113405 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A113405 M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/
               0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 
               226-228 (1995), 57-72; erratum 320 (2000), 210.
%H A113405 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%F A113405 a(n)=2a(n-1)-a(n-3)+2a(n-4); a(n)=sum{k=0..floor(n/2), C(n-k, k)A001045(k)}; 
               a(n)=sum{k=0..n, C((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 (bpcrtz(AT)free.fr), Nov 30 2007
%F A113405 a(n+1)-2a(n)=hexaperiodic 0, 0, 1, 0, 0, -1, A131531. a(n)+a(n+3)=2^n, 
               A000079. - Paul Curtz (bpcrtz(AT)free.fr), Dec 16 2007
%F A113405 a(n)=round(2^n/9) [From Ross Drewe (rd(AT)labyrinth.net.au), Sep 03 2009]
%Y A113405 Contribution from Ross Drewe (rd(AT)labyrinth.net.au), 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)
%Y A113405 Sequence in context: A018330 A068060 A057744 this_sequence A119340 A119341 
               A119342
%Y A113405 Adjacent sequences: A113402 A113403 A113404 this_sequence A113406 A113407 
               A113408
%K A113405 easy,nonn
%O A113405 0,5
%A A113405 Paul Barry (pbarry(AT)wit.ie), Oct 28 2005
%E A113405 Edited by N. J. A. Sloane (njas(AT)research.att.com), Dec 13 2007