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A227310 G.f.: 1/G(0) where G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ).

%I A227310 #54 Mar 11 2022 07:57:29
%S A227310 1,1,0,0,1,0,0,1,0,1,1,0,2,1,1,3,2,3,4,4,6,7,8,11,13,16,20,24,31,37,
%T A227310 46,58,70,88,108,133,167,204,252,315,386,479,594,731,909,1122,1386,
%U A227310 1720,2124,2628,3254,4022,4980,6160,7618,9432,11665,14433,17860,22093,27341,33824,41847,51785,64065,79267
%N A227310 G.f.: 1/G(0) where G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ).
%C A227310 Number of rough sandpiles: 1-dimensional sandpiles (see A186085) with n grains without flat steps (no two successive parts of the corresponding composition equal), see example. - _Joerg Arndt_, Mar 08 2014
%C A227310 The sequence of such sandpiles by base length starts (n>=0) 1, 1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, ... (A097331, essentially A000108 with interlaced zeros).  This is a consequence of the obvious connection to Dyck paths, see example. - _Joerg Arndt_, Mar 09 2014
%C A227310 a(n>=1) are the Dyck paths with area n between the x-axis and the path which return to the x-axis only once (at their end), whereas A143951 includes paths with intercalated touches of the x-axis. - _R. J. Mathar_, Aug 22 2018
%H A227310 Seiichi Manyama, <a href="/A227310/b227310.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Joerg Arndt)
%H A227310 A. M. Odlyzko and H. S. Wilf, <a href="http://www.jstor.org/stable/2322898">The editor's corner: n coins in a fountain</a>, Amer. Math. Monthly, 95 (1988), 840-843.
%F A227310 a(0) = 1 and a(n) = abs(A049346(n)) for n>=1.
%F A227310 G.f.: 1/ (1-q/(1+q/ (1+q^2/(1-q^2/ (1-q^3/(1+q^3/ (1+q^4/(1-q^4/ (1-q^5/(1+q^5/ (1+-...) )) )) )) )) )).
%F A227310 G.f.: 1 + q * F(q,q) where F(q,y) = 1/(1 - y * q^2 * F(q, q^2*y) ); cf. A005169 and p. 841 of the Odlyzko/Wilf reference; 1/(1 - q * F(q,q)) is the g.f. of A143951. - _Joerg Arndt_, Mar 09 2014
%F A227310 G.f.: 1 + q/(1 - q^3/(1 - q^5/(1 - q^7/ (...)))) (from formulas above). - _Joerg Arndt_, Mar 09 2014
%F A227310 G.f.: F(x, x^2) where F(x,y) is the g.f. of A239927. - _Joerg Arndt_, Mar 29 2014
%F A227310 a(n) ~ c * d^n, where d = 1.23729141259673487395949649334678514763130846902468... and c = 0.0773368373684184197215007198148835507944051447907... - _Vaclav Kotesovec_, Sep 05 2017
%F A227310 G.f.: A(x) = 2 -1/A143951(x). - _R. J. Mathar_, Aug 23 2018
%e A227310 From _Joerg Arndt_, Mar 08 2014: (Start)
%e A227310 The a(21) = 7 rough sandpiles are:
%e A227310 :
%e A227310 :   1:      [ 1 2 1 2 1 2 1 2 1 2 3 2 1 ]  (composition)
%e A227310 :
%e A227310 :           o
%e A227310 :  o o o o ooo
%e A227310 : ooooooooooooo  (rendering of sandpile)
%e A227310 :
%e A227310 :
%e A227310 :   2:      [ 1 2 1 2 1 2 1 2 3 2 1 2 1 ]
%e A227310 :
%e A227310 :         o
%e A227310 :  o o o ooo o
%e A227310 : ooooooooooooo
%e A227310 :
%e A227310 :
%e A227310 :   3:      [ 1 2 1 2 1 2 3 2 1 2 1 2 1 ]
%e A227310 :
%e A227310 :       o
%e A227310 :  o o ooo o o
%e A227310 : ooooooooooooo
%e A227310 :
%e A227310 :
%e A227310 :   4:      [ 1 2 1 2 3 2 1 2 1 2 1 2 1 ]
%e A227310 :
%e A227310 :     o
%e A227310 :  o ooo o o o
%e A227310 : ooooooooooooo
%e A227310 :
%e A227310 :
%e A227310 :   5:      [ 1 2 3 2 1 2 1 2 1 2 1 2 1 ]
%e A227310 :
%e A227310 :   o
%e A227310 :  ooo o o o o
%e A227310 : ooooooooooooo
%e A227310 :
%e A227310 :
%e A227310 :   6:      [ 1 2 3 2 3 4 3 2 1 ]
%e A227310 :
%e A227310 :      o
%e A227310 :   o ooo
%e A227310 :  ooooooo
%e A227310 : ooooooooo
%e A227310 :
%e A227310 :
%e A227310 :   7:      [ 1 2 3 4 3 2 3 2 1 ]
%e A227310 :
%e A227310 :    o
%e A227310 :   ooo o
%e A227310 :  ooooooo
%e A227310 : ooooooooo
%e A227310 (End)
%e A227310 From _Joerg Arndt_, Mar 09 2014: (Start)
%e A227310 The A097331(9) = 14 such sandpiles with base length 9 are:
%e A227310 01:  [ 1 2 1 2 1 2 1 2 1 ]
%e A227310 02:  [ 1 2 1 2 1 2 3 2 1 ]
%e A227310 03:  [ 1 2 1 2 3 2 3 2 1 ]
%e A227310 04:  [ 1 2 1 2 3 2 1 2 1 ]
%e A227310 05:  [ 1 2 1 2 3 4 3 2 1 ]
%e A227310 06:  [ 1 2 3 2 1 2 3 2 1 ]
%e A227310 07:  [ 1 2 3 2 1 2 1 2 1 ]
%e A227310 08:  [ 1 2 3 2 3 2 1 2 1 ]
%e A227310 09:  [ 1 2 3 2 3 2 3 2 1 ]
%e A227310 10:  [ 1 2 3 4 3 2 1 2 1 ]
%e A227310 11:  [ 1 2 3 2 3 4 3 2 1 ]
%e A227310 12:  [ 1 2 3 4 3 2 3 2 1 ]
%e A227310 13:  [ 1 2 3 4 3 4 3 2 1 ]
%e A227310 14:  [ 1 2 3 4 5 4 3 2 1 ]
%e A227310 (End)
%o A227310 (PARI) N = 66;  q = 'q + O('q^N);
%o A227310 G(k) = if(k>N, 1, 1 + (-q)^(k+1) / (1 - (-q)^(k+1) / G(k+1) ) );
%o A227310 gf = 1 / G(0);
%o A227310 Vec(gf)
%o A227310 (PARI)
%o A227310 N = 66;  q = 'q + O('q^N);
%o A227310 F(q,y,k) = if(k>N, 1, 1/(1 - y*q^2 * F(q, q^2*y, k+1) ) );
%o A227310 Vec( 1 + q * F(q,q,0) ) \\ _Joerg Arndt_, Mar 09 2014
%Y A227310 Cf. A049346 (g.f.: 1 - 1/G(0), where G(k)= 1 + q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
%Y A227310 Cf. A226728 (g.f.: 1/G(0), where G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
%Y A227310 Cf. A226729 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
%Y A227310 Cf. A006958 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
%Y A227310 Cf. A227309 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ) ).
%Y A227310 Cf. A173258, A291874.
%K A227310 nonn
%O A227310 0,13
%A A227310 _Joerg Arndt_, Jul 06 2013