OFFSET
0,2
COMMENTS
"Numbral" powers of 10 (see A048888 for definition). - John W. Layman, Dec 18 2001
For n > 1, a(n-1) is the (2^n-3)rd coefficient in the expansion of th(0)=y, th(n+1)=th(n)*(th(n)+1).
If 2^(n+1) is the length of the even leg of a primitive Pythagorean triangle (PPT) then it constrains the odd leg to have a length of 4^n-1 and the hypotenuse to have a length of 4^n+1. The resulting triangle has a semiperimeter of 4^n+2^n, an area of 8^n-2^n and an inradius of 2^n-1. Now consider the term 8^n-2^n: it must at least be divisible by 6 because it is the area of a PPT. a(n) is 1/6 the area of such triangles. - Frank M Jackson, Dec 28 2017
LINKS
Iain Fox, Table of n, a(n) for n = 0..1107
Index entries for linear recurrences with constant coefficients, signature (10,-16).
FORMULA
a(0) = 1, a(n) = (2^(3n+2) - 2^n)/3 = A059155(n)/12 = A000079(n)*A002450(n+1) = A016203(n+1) - A016203(n). - Ralf Stephan, Aug 14 2003
a(n) = binomial(2^n,2)*(2^n + 1)/3, n >= 1. - Zerinvary Lajos, Jan 07 2008
a(n-1) = ((5+sqrt 9)^n-(5-sqrt 9)^n)/6. - Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009
a(n) = Sum_{i=1...(2^n -1)} i*(i+1)/2. - Ctibor O. Zizka, Mar 03 2009
a(0) = 1, a(n) = 8*a(n-1) + 2^n. - Vincenzo Librandi, Feb 09 2011
a(n) = 10*a(n-1) - 16*a(n-2), n > 1. - Vincenzo Librandi, Feb 09 2011
a(n-1) = (8^n-2^n)/6 = A248217(n)/6 - Frank M Jackson, Dec 28 2017
E.g.f.: e^(2*x) * (4*e^(6*x) - 1)/3. - Iain Fox, Dec 28 2017
MAPLE
seq(binomial(2^n, 2)*(2^n + 1)/3, n=1..20); # Zerinvary Lajos, Jan 07 2008
MATHEMATICA
CoefficientList[Series[1/((1 - 2x)(1 - 8x)), {x, 0, 100}], x] (* Stefan Steinerberger, Apr 21 2006 *)
a[n_] := (8^n-2^n)/6; Array[a, 20] (* Frank M Jackson, Dec 28 2017 *)
PROG
(Sage) [lucas_number1(n, 10, 16) for n in range(1, 21)] # Zerinvary Lajos, Apr 26 2009
(Sage) [(8^n - 2^n)/6 for n in range(1, 21)] # Zerinvary Lajos, Jun 05 2009
(PARI) Vec(1/((1-2*x)*(1-8*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved