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 A016131 Expansion of 1/((1-2x)(1-8x)). 11
 1, 10, 84, 680, 5456, 43680, 349504, 2796160, 22369536, 178956800, 1431655424, 11453245440, 91625967616, 733007749120, 5864062009344, 46912496107520, 375299968925696, 3002399751536640, 24019198012555264, 192153584100966400 (list; graph; refs; listen; history; text; internal format)
 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 xrange(1, 21)] # Zerinvary Lajos, Apr 26 2009 (Sage) [(8^n - 2^n)/6 for n in xrange(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 Cf. A120689, A248217. Sequence in context: A289163 A092718 A090763 * A027310 A287826 A155593 Adjacent sequences:  A016128 A016129 A016130 * A016132 A016133 A016134 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified November 12 12:43 EST 2018. Contains 317109 sequences. (Running on oeis4.)