looking at Sloane's A052169 (Kurepa hypothesis for left factorial) and A002467 (game of Mousetrap), is it self-evident that [[1]] a(n)=1+1/(x (k++) a(n-1)) , a(1)=x, for x->1 produces the same sequence as [[2]] a(2) = 1, a(3) = 2, a(n) = (n-2)*a(n-1) + (n-3)*a(n-2) ? Expanding [[1]], with the usual meaning of k++ as post-increment, we get: k=1;NestList[1+1/((k++)x #)&,x,3] {x, 1 + 1/x^2, 1 + 1/(2*(1 + 1/x^2)*x), 1 + 1/(3*(1 + 1/(2*(1 + 1/x^2)*x))*x)} Taking only the numerators gives: {x, 1 + x^2, 2 + x + 2*x^2, 2 + 6*x + 5*x^2 + 6*x^3, 14 + 27*x + 26*x^2 + 24*x^3, 8 + 94*x + 155*x^2 + 154*x^3 + 120*x^4, 118 + 699*x + 1060*x^2 + 1044*x^3 + 720*x^4, 48 + 1390*x + 5823*x^2 + 8344*x^3 + 8028*x^4 + 5040*x^5, ... set x to 1 and you get A052169 in a different way. As a 'limping' triangular table, the coefficient of the highest power in x stands out as n!, and the definition of A052169 as pairwise sums of A002467 (itself equal to n!-!n) is curious. {0,1}, {1,0,1}, {2,1,2}, {2,6,5,6}, {14,27,26,24}, {8,94,155,154,120}, {118,699,1060,1044,720}, {48,1390,5823,8344,8028,5040}, {1210,16013,54004,74060,69264,40320}, {384,22010,190701,552788,730764,663696,362880}, {14730,364217,2393046,6194420,7931016,6999840,3628800}, {3840,382130,5913397,31851386,75446260,93878136,80627040,39916800}, {208110,8591947,97284270,450355252,992596296,1203535872,1007441280,479001600 } maybe not math-funny to all, but quite amusing to at least one, W.