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 A063491 #46 Dec 20 2024 12:39:16
%S A063491 1,12,50,133,279,506,832,1275,1853,2584,3486,4577,5875,7398,9164,
%T A063491 11191,13497,16100,19018,22269,25871,29842,34200,38963,44149,49776,
%U A063491 55862,62425,69483,77054,85156,93807,103025,112828,123234,134261,145927,158250,171248,184939
%N A063491 a(n) = (2*n - 1)*(3*n^2 - 3*n + 2)/2.
%C A063491 A triangle has sides of lengths 6*n-3, 6*n^2-6*n+4, and 6*n^2-6*n+7; for n>2 its area is 6*sqrt(a(n)^2 - 1). - _J. M. Bergot_, Aug 30 2013
%C A063491 [The source of this is using (n,n+1), (n+1,n+2), and (n+2,n+3) as (a,b) in the creation of three Pythagorean triangles with sides b^2-a^2, 2*a*b, and a^2+b^2. Combine the three respective sides to create a new larger triangle, then find its area. It is not simply working backwards from the sequence. As well, the sequence has this as its first comment to show that the numbers are actually doing something to find a solution.]
%D A063491 T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
%H A063491 Harry J. Smith, <a href="/A063491/b063491.txt">Table of n, a(n) for n = 1..1000</a>
%H A063491 T. P. Martin, <a href="http://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Rep., 273 (1996), 199-241, eq. (10).
%H A063491 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A063491 G.f.: x*(1+x)*(1+7*x+x^2)/(1-x)^4. - _Colin Barker_, Apr 20 2012
%F A063491 a(n) = +4*a(n-1) -6*a(n-2) +4*a(n-3) -1*a(n-4) n > 3, a(1)=1, a(2)=12, a(3)=50, a(4)=133. - _Yosu Yurramendi_, Sep 04 2013
%F A063491 E.g.f.: (-2 + 4*x + 9*x^2 + 6*x^3)*exp(x)/2 + 1. - _G. C. Greubel_, Dec 01 2017
%F A063491 From _Bruce J. Nicholson_, Jun 17 2020: (Start)
%F A063491 a(n) = A005448(n) * A005408(n-1).
%F A063491 a(n) = A004188(n) + A004188(n-1). (End)
%t A063491 LinearRecurrence[{4,-6,4,-1},{1,12,50,133},40] (* _Harvey P. Dale_, Jun 05 2016 *)
%t A063491 Table[(2*n-1)*(3*n^2 -3*n +2)/2, {n,1,30}] (* _G. C. Greubel_, Dec 01 2017 *)
%o A063491 (PARI) a(n) = { (2*n - 1)*(3*n^2 - 3*n + 2)/2 } \\ _Harry J. Smith_, Aug 23 2009
%o A063491 (PARI) my(x='x+O('x^30)); Vec(serlaplace((-2 + 4*x + 9*x^2 + 6*x^3)*exp(x)/2 + 1)) \\ _G. C. Greubel_, Dec 01 2017
%o A063491 (R)
%o A063491 a <- c(0, 1, 9, 38, 110)
%o A063491 for(n in (length(a)+1):40)
%o A063491 a[n] <- +4*a[n-1]-6*a[n-2]+4*a[n-3]-a[n-4]
%o A063491 a [_Yosu Yurramendi_, Sep 04 2013]
%o A063491 (Magma) [(2*n-1)*(3*n^2 -3*n +2)/2: n in [1..30]]; // _G. C. Greubel_, Dec 01 2017
%Y A063491 1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.
%Y A063491 Cf. A005448, A004188.
%K A063491 nonn,easy
%O A063491 1,2
%A A063491 _N. J. A. Sloane_, Aug 01 2001