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A194880 The numerators of the inverse Akiyama-Tanigawa algorithm from A001045(n).

%I A194880 #27 May 08 2024 02:28:09
%S A194880 0,-1,-1,-4,-5,-2,-7,-8,-3,-10,-11,-4,-13,-14,-5,-16,-17,-6,-19,-20,
%T A194880 -7,-22,-23,-8,-25,-26,-9,-28,-29,-10,-31,-32,-11,-34,-35,-12,-37,-38,
%U A194880 -13,-40,-41,-14,-43,-44,-15,-46,-47,-16,-49,-50,-17,-52,-53,-18,-55,-56,-19,-58,-59,-20
%N A194880 The numerators of the inverse Akiyama-Tanigawa algorithm from A001045(n).
%C A194880 0,  -1, -1, -4/3, -5/3, -2, -7/3, -8/3, -3, -10/3, -11/3, -4, -13/4, -14/3, -5,    = a(n)/b(n),
%C A194880 1,    0,  1,  4/3,  5/3,  2,  7/3,  8/3,  3,
%C A194880 1,   -2, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,
%C A194880 3,   -2,  1,  4/3,  5/3,  2,  7/3,  8/3,  3,
%C A194880 5,   -6, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,
%C A194880 11, -10,  1,  4/3,  5/3,  2,  7/3,  8/3,  3,
%C A194880 21, -22, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,
%C A194880 Vertical: A001045(n), -A078008(n), (-1)^(n+1)*A000012(n), (-1)^(n+1)*A010709(n)/A010701(n), (-1)^(n+1)*A010716(n+1)/A010701(n), A007395(n), .. .
%C A194880 a(n)=0, 1 before (-A145064(n+1)=-A051176(n+3).
%C A194880 b(n)=1, 1 before A169609(n). b(n)=1, 1, 1 before A144437(n+1).
%C A194880 a(n+5)-a(n+2)=b(n+5)  (=-1,-3,-3,=-A169609(n)).
%H A194880 G. C. Greubel, <a href="/A194880/b194880.txt">Table of n, a(n) for n = 0..5000</a>
%H A194880 D. Merlini, R. Sprugnoli, and M. C. Verri, <a href="http://www.emis.de/journals/INTEGERS/papers/f5/f5.Abstract.html">The Akiyama-Tanigawa Transformation</a>, Integers, 5 (1) (2005) #A05.
%H A194880 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).
%F A194880 a(3*n)=-3*n-1 except a(0)=0; a(3*n+1)=-3*n-2 except a(1)=-1; a(3*n+2)=-n-1.
%F A194880 From _Chai Wah Wu_, May 07 2024: (Start)
%F A194880 a(n) = 2*a(n-3) - a(n-6) for n > 7.
%F A194880 G.f.: x*(x^6 + x^5 - 3*x^3 - 4*x^2 - x - 1)/(x^6 - 2*x^3 + 1). (End)
%t A194880 a[0]=0; a[1]=-1; a[n_] := (-n-1)/Max[1, 2*Mod[n, 3]-1]; Table[a[n], {n, 0, 59}] (* _Jean-François Alcover_, Sep 18 2012 *)
%K A194880 sign,tabl,frac
%O A194880 0,4
%A A194880 _Paul Curtz_, Sep 07 2011