OFFSET
0,2
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..346
FORMULA
a(n) = floor(e^(1/6)*6^n*n!).
a(n) = n!*Sum_{k=0..n} 6^(n-k)/k!. E.g.f.: exp(x)/(1 - 6*x). - Philippe Deléham, Mar 14 2004
From Peter Bala, Mar 01 2017: (Start)
a(n) = Integral_{x = 0..inf} (6*x + 1)^n*exp(-x) dx.
The e.g.f. y = exp(x)/(1 - 6*x) satisfies the differential equation (1 - 6*x)*y' = (7 - 6*x)*y.
a(n) = (6*n + 1)*a(n-1) - 6*(n - 1)*a(n-2).
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)
EXAMPLE
a(2) = 6*2*a(1) + 1 = 12*7 + 1 = 85.
MATHEMATICA
nxt[{n_, a_}]:={n+1, 6a(n+1)+1}; NestList[nxt, {0, 1}, 20][[;; , 2]] (* Harvey P. Dale, Jul 17 2024 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, Jun 20 2000
EXTENSIONS
More terms from James A. Sellers, Jul 04 2000
STATUS
approved