A186505 - OEIS

%I #25 Feb 12 2025 03:14:14

%S 1,0,1,0,1,1,1,2,2,3,4,5,8,9,14,18,25,34,46,64,86,119,162,222,304,416,

%T 571,780,1071,1466,2010,2754,3775,5175,7092,9724,13329,18274,25052,

%U 34347,47091,64562,88522,121369,166411,228168,312848,428959,588163

%N Antidiagonal sums of triangle A186084.

%C Triangle A186084 is the number of 1-dimensional sandpiles with n grains and base length k.

%C The g.f. T(x,y) of triangle A186084 satisfies: T(x,y) = 1/(1 - x*y - x^3*y^2*T(x,x*y)); therefore, the g.f. of this sequence is T(x,x).

%H Alois P. Heinz, <a href="/A186505/b186505.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (1 - x/(1 - 1/B(x)))/x^3 where B(x) equals the g.f. of the row sums of triangle A186084.

%F G.f.: 1/(1-x^2 - x^5/(1-x^3 - x^7/(1-x^4 - x^9/(1-x^5 - x^11/(1-x^6 - x^13/(1-...)))))) (continued fraction).

%F G.f.: 1/(1-x^2/(1-x^3/(1-x^7/(1-x^4/(1-x^5/(1-x^11/(1-x^6/(1 -x^7/(1-x^15/(1-...)))))))))) (continued fraction).

%F G.f.: 1/x^3 - (Q(0) + 1)/x^2, where Q(k)= 1/x^(k+1) - 1 - 1/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 07 2013

%F a(n) ~ c * d^n, where d = 1.3712018040437285..., c = 0.154355235026898... . - _Vaclav Kotesovec_, Sep 10 2014

%e G.f.: 1 + x^2 + x^4 + x^5 + x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 4*x^10 +...

%p b:= proc(n, i) option remember; `if`(n=0, `if`(i=1, 1, 0),

%p       `if`(n<0 or i<1, 0, expand(x*add(b(n-i, i+j), j=-1..1)) ))

%p     end:

%p a:= n-> add(coeff(b(n-k, 1), x, k), k=0..n):

%p seq(a(n), n=0..70);  # _Alois P. Heinz_, Jul 24 2013

%t m = 100;

%t f[i_] := If[i == 0, 1, -x^(2i+3)];

%t g[i_] := 1 - x^(i+2);

%t ContinuedFractionK[f[i], g[i], {i, 0, Sqrt[m] // Ceiling}] + O[x]^m // CoefficientList[#, x]& (* _Jean-François Alcover_, Oct 14 2019, after _Sergei N. Gladkovskii_ *)

%o (PARI) {a(n)=local(Txy=1+x*y); for(i=1, n, Txy=1/(1-x*y-x^3*y^2*subst(Txy, y, x*y+x*O(x^n)))); polcoeff(subst(Txy,y,x), n, x)}

%o (PARI) N = 66; x = 'x + O('x^N);

%o Q(k) = if(k>N, 1, 1/x^(k+1) - 1 - 1/Q(k+1) );

%o gf = 1/x^3 - (Q(0) + 1)/x^2;

%o Vec(gf) \\ _Joerg Arndt_, May 07 2013

%Y Cf. A186084, A186085.

%K nonn,changed

%O 0,8

%A _Paul D. Hanna_, Feb 23 2011