A056547 - OEIS

%I #20 Jul 17 2024 13:26:52

%S 1,7,85,1531,36745,1102351,39684637,1666754755,80004228241,

%T 4320228325015,259213699500901,17108104167059467,1231783500028281625,

%U 96079113002205966751,8070645492185301207085,726358094296677108637651

%N a(n) = 6*n*a(n-1) + 1 with a(0)=1.

%H Harvey P. Dale, <a href="/A056547/b056547.txt">Table of n, a(n) for n = 0..346</a>

%F a(n) = floor(e^(1/6)*6^n*n!).

%F a(n) = n!*Sum_{k=0..n} 6^(n-k)/k!. E.g.f.: exp(x)/(1 - 6*x). - _Philippe Deléham_, Mar 14 2004

%F From _Peter Bala_, Mar 01 2017: (Start)

%F a(n) = Integral_{x = 0..inf} (6*x + 1)^n*exp(-x) dx.

%F The e.g.f. y = exp(x)/(1 - 6*x) satisfies the differential equation (1 - 6*x)*y' = (7 - 6*x)*y.

%F a(n) = (6*n + 1)*a(n-1) - 6*(n - 1)*a(n-2).

%F The sequence b(n) := 6^n*n! also satisfies the same recurrence with b(0) = 1, b(1) = 6. This leads to the continued fraction representation a(n) = 6^n*n!*( 1 + 1/(6 - 6/(13 - 12/(19 - ... - (6*n - 6)/(6*n + 1) )))) for n >= 2. Taking the limit gives the continued fraction representation exp(1/6) = 1 + 1/(6 - 6/(13 - 12/(19 - ... - (6*n - 6)/((6*n + 1) - ... )))). Cf. A010844. (End)

%e a(2) = 6*2*a(1) + 1 = 12*7 + 1 = 85.

%t nxt[{n_,a_}]:={n+1,6a(n+1)+1}; NestList[nxt,{0,1},20][[;;,2]] (* _Harvey P. Dale_, Jul 17 2024 *)

%Y Cf. A000522, A010844, A010845, A056545, A056546 for analogs. A056547/(A000142*A000400) is an increasingly good approximation to 6th root of e.

%K nonn,easy

%O 0,2

%A _Henry Bottomley_, Jun 20 2000

%E More terms from _James A. Sellers_, Jul 04 2000