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 A186505 Antidiagonal sums of triangle A186084. 3
 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, 571, 780, 1071, 1466, 2010, 2754, 3775, 5175, 7092, 9724, 13329, 18274, 25052, 34347, 47091, 64562, 88522, 121369, 166411, 228168, 312848, 428959, 588163 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Triangle A186084 is the number of 1-dimensional sandpiles with n grains and base length k. 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). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 FORMULA G.f.: (1 - x/(1 - 1/B(x))))/x^3 where B(x) equals the g.f. of the row sums of triangle A186084. 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). 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). 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 a(n) ~ c * d^n, where d = 1.3712018040437285..., c = 0.154355235026898... . - Vaclav Kotesovec, Sep 10 2014 EXAMPLE G.f.: 1 + x^2 + x^4 + x^5 + x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 4*x^10 +... MAPLE b:= proc(n, i) option remember; `if`(n=0, `if`(i=1, 1, 0), `if`(n<0 or i<1, 0, expand(x*add(b(n-i, i+j), j=-1..1)) )) end: a:= n-> add(coeff(b(n-k, 1), x, k), k=0..n): seq(a(n), n=0..70); # Alois P. Heinz, Jul 24 2013 MATHEMATICA m = 100; f[i_] := If[i == 0, 1, -x^(2i+3)]; g[i_] := 1 - x^(i+2); 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 *) PROG (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)} (PARI) N = 66; x = 'x + O('x^N); Q(k) = if(k>N, 1, 1/x^(k+1) - 1 - 1/Q(k+1) ); gf = 1/x^3 - (Q(0) + 1)/x^2; Vec(gf) \\ Joerg Arndt, May 07 2013 CROSSREFS Cf. A186084, A186085. Sequence in context: A240011 A211228 A338463 * A228693 A116676 A240575 Adjacent sequences: A186502 A186503 A186504 * A186506 A186507 A186508 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 23 2011 STATUS approved

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Last modified December 5 00:12 EST 2023. Contains 367565 sequences. (Running on oeis4.)