This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A016131 #54 Dec 27 2024 02:54:24
%S A016131 1,10,84,680,5456,43680,349504,2796160,22369536,178956800,1431655424,
%T A016131 11453245440,91625967616,733007749120,5864062009344,46912496107520,
%U A016131 375299968925696,3002399751536640,24019198012555264,192153584100966400
%N A016131 Expansion of 1/((1-2*x)*(1-8*x)).
%C A016131 "Numbral" powers of 10 (see A048888 for definition). - _John W. Layman_, Dec 18 2001
%C A016131 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).
%C A016131 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
%H A016131 Iain Fox, <a href="/A016131/b016131.txt">Table of n, a(n) for n = 0..1107</a>
%H A016131 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-16).
%F A016131 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
%F A016131 a(n) = binomial(2^n,2)*(2^n + 1)/3, n >= 1. - _Zerinvary Lajos_, Jan 07 2008
%F A016131 a(n) = (8^(n+1) - 2^(n+1))/6. - Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009
%F A016131 a(n) = Sum_{i=1...(2^n -1)} i*(i+1)/2. - _Ctibor O. Zizka_, Mar 03 2009
%F A016131 a(0) = 1, a(n) = 8*a(n-1) + 2^n. - _Vincenzo Librandi_, Feb 09 2011
%F A016131 a(n) = 10*a(n-1) - 16*a(n-2), n > 1. - _Vincenzo Librandi_, Feb 09 2011
%F A016131 a(n) = A248217(n+1)/6. - _Frank M Jackson_, Dec 28 2017
%F A016131 E.g.f.: e^(2*x) * (4*e^(6*x) - 1)/3. - _Iain Fox_, Dec 28 2017
%p A016131 seq(binomial(2^n,2)*(2^n + 1)/3,n=1..20); # _Zerinvary Lajos_, Jan 07 2008
%t A016131 CoefficientList[Series[1/((1 - 2x)(1 - 8x)), {x, 0, 100}], x] (* _Stefan Steinerberger_, Apr 21 2006 *)
%t A016131 a[n_] := (8^n-2^n)/6; Array[a, 20] (* _Frank M Jackson_, Dec 28 2017 *)
%o A016131 (Sage) [lucas_number1(n,10,16) for n in range(1, 21)] # _Zerinvary Lajos_, Apr 26 2009
%o A016131 (Sage) [(8^n - 2^n)/6 for n in range(1,21)] # _Zerinvary Lajos_, Jun 05 2009
%o A016131 (PARI) Vec(1/((1-2*x)*(1-8*x))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o A016131 (Magma) [Binomial(2^(n+1)+1,3): n in [0..40]]; // _G. C. Greubel_, Dec 26 2024
%o A016131 (Python)
%o A016131 def A016131(n): return binomial(pow(2,n+1) +1, 3)
%o A016131 print([A016131(n) for n in range(41)]) # _G. C. Greubel_, Dec 26 2024
%Y A016131 Cf. A000079, A002450, A016203, A048888, A059155, A120689, A248217.
%K A016131 nonn,easy
%O A016131 0,2
%A A016131 _N. J. A. Sloane_