login
Number of minimal unavoidable n-celled pebbling configurations.
2

%I #32 May 19 2024 14:15:19

%S 0,0,0,0,4,22,98,412,1700,6974,28576,117146,480722,1974914,8122084,

%T 33435390,137757480,567998152,2343472004,9674252070,39956606552,

%U 165099840920,682446679582,2821858504062,11671572244666,48287711006032,199822535773958,827069530795224,3423890026639184,14176516741276534

%N Number of minimal unavoidable n-celled pebbling configurations.

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.50.

%H Alois P. Heinz, <a href="/A007901/b007901.txt">Table of n, a(n) for n = 1..1000</a>

%H F. R. K. Chung, R. L. Graham, J. A. Morrison and A. M. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/doc/pebbling.pdf">Pebbling a chessboard</a>, Amer. Math. Monthly 102 (1995), pp. 113-123.

%H Google Labs, <a href="http://www.7427466391.com/">Google Labs congratulations puzzle</a>

%H Marcus Kazmierczak, <a href="http://www.mkaz.com/math/google/">Google Labs Puzzles</a>, Jul 29, 2004.

%H Slashdot (CmdrTaco), <a href="http://developers.slashdot.org/article.pl?sid=04/09/16/139232&amp;tid=217&amp;tid=8">Google's Math Puzzle</a>, Thu Sep 16, 2004.

%F 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

%F 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

%p 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.

%p 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]):

%t 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_ *)

%t CoefficientList[Series[x^3((1-3x+x^2)Sqrt[1-4x]-1+5x-x^2-6x^3)/(1-7x+14x^2-9x^3),{x,0,30}],x] (* _Harvey P. Dale_, May 19 2024 *)

%o (PARI) x='x+O('x^44);

%o 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);

%o Vec(gf) \\ _Joerg Arndt_, Apr 20 2011

%o (Maxima)

%o Polynom:[0,4,-14,22,-20,6];

%o p(k):=if k<7 then Polynom[k] else 0;

%o h(n):=sum(k*p(k)*binomial(2*n-k-1,n-k),k,1,n)/n;

%o 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);

%o b(n):=sum(h(i)*u(n-i),i,1,n);

%o a(n):=if n<2 then 0 else b(n-2);

%o makelist(a(n),n,0,40); /* _Vladimir Kruchinin_, Sep 20 2014 */

%Y Cf. A007902.

%K nonn,easy,nice

%O 1,5

%A odlyzko(AT)dtc.umn.edu (A. M. Odlyzko)