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A007901
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Number of minimal unavoidable n-celled pebbling configurations.
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2
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0, 0, 0, 0, 4, 22, 98, 412, 1700, 6974, 28576, 117146, 480722, 1974914, 8122084, 33435390, 137757480, 567998152, 2343472004, 9674252070, 39956606552, 165099840920, 682446679582, 2821858504062, 11671572244666, 48287711006032, 199822535773958, 827069530795224, 3423890026639184, 14176516741276534
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OFFSET
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1,5
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.50.
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LINKS
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F. R. K. Chung, R. L. Graham, J. A. Morrison and A. M. Odlyzko, Pebbling a chessboard, Amer. Math. Monthly 102 (1995), pp. 113-123.
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FORMULA
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G.f.: x^3*((1-3*x+x^2)*sqrt(1-4*x)-1+5*x-x^2-6*x^3)/(1-7*x+14*x^2-9*x^3) [from the Stanley reference]. - Joerg Arndt, Apr 20 2011
Conjecture: (n-3)*(n-8)*a(n) +(-11*n^2+127*n-324)*a(n-1) +42*(n^2-12*n+34)*a(n-2) +(-65*n^2+799*n-2400)*a(n-3) +18*(n-6)*(2*n-13)*a(n-4)=0. - R. J. Mathar, Aug 14 2012
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MAPLE
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The Maple snippet provides an alternative solution to the Google congratulations puzzle at http://www.7427466391.com. After running the Maple code, f(1) to f(4) match the puzzle, with f(5) being 1510865746 and f(6) being 6171783928.
Digits:=2000: E:=evalf(exp(1)): g:=n->trunc((E-(10^(-n)*trunc(E*10^n)))*10^(10+n)): h:=[0, 0, 0, 0, 4, 22, 98, 412, 1700]: f:=k->g(h[k+3]):
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MATHEMATICA
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p[k_] := If[k < 7, {0, 4, -14, 22, -20, 6}[[k]], 0]; h[n_] := Sum[ k*p[k] * Binomial[2*n-k-1, n-k], {k, 1, n}]/n; u[n_] := Sum[ Sum[Binomial[j, n-k-j]*7^(2*k-n+j)*Binomial[k, j]*(-2)^(-n+k+2*j)*3^(2*(n-k-j)), {j, 0, k}], {k, 0, n}]; b[n_] := Sum[h[i]*u[n-i], {i, 1, n}]; a[n_] := If[n<2, 0 , b[n-2]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jan 14 2015, after Vladimir Kruchinin *)
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PROG
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(PARI) x='x+O('x^44);
gf=x^3*((1-3*x+x^2)*sqrt(1-4*x)-1+5*x-x^2-6*x^3)/(1-7*x+14*x^2-9*x^3);
(Maxima)
Polynom:[0, 4, -14, 22, -20, 6];
p(k):=if k<7 then Polynom[k] else 0;
h(n):=sum(k*p(k)*binomial(2*n-k-1, n-k), k, 1, n)/n;
u(n):=sum(sum(binomial(j, n-k-j)*7^(2*k-n+j)*binomial(k, j)*(-2)^(-n+k+2*j)*3^(2*(n-k-j)), j, 0, k), k, 0, n);
b(n):=sum(h(i)*u(n-i), i, 1, n);
a(n):=if n<2 then 0 else b(n-2);
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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odlyzko(AT)dtc.umn.edu (A. M. Odlyzko)
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STATUS
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approved
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