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%I #43 Sep 08 2022 08:45:04
%S 1,12,50,133,279,506,832,1275,1853,2584,3486,4577,5875,7398,9164,
%T 11191,13497,16100,19018,22269,25871,29842,34200,38963,44149,49776,
%U 55862,62425,69483,77054,85156,93807,103025,112828,123234,134261
%N a(n) = (2*n - 1)*(3*n^2 - 3*n + 2)/2.
%C 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 [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 T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
%H Harry J. Smith, <a href="/A063491/b063491.txt">Table of n, a(n) for n = 1..1000</a>
%H 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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: x*(1+x)*(1+7*x+x^2)/(1-x)^4. - _Colin Barker_, Apr 20 2012
%F 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 E.g.f.: (-2 + 4*x + 9*x^2 + 6*x^3)*exp(x)/2 + 1. - _G. C. Greubel_, Dec 01 2017
%F From _Bruce J. Nicholson_, Jun 17 2020: (Start)
%F a(n) = A005448(n) * A005408(n-1).
%F a(n) = A004188(n) + A004188(n-1). (End)
%t LinearRecurrence[{4,-6,4,-1},{1,12,50,133},40] (* _Harvey P. Dale_, Jun 05 2016 *)
%t Table[(2*n-1)*(3*n^2 -3*n +2)/2, {n,1,30}] (* _G. C. Greubel_, Dec 01 2017 *)
%o (PARI) { for (n=1, 1000, write("b063491.txt", n, " ", (2*n - 1)*(3*n^2 - 3*n + 2)/2) ) } \\ _Harry J. Smith_, Aug 23 2009
%o (PARI) 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 (R)
%o a <- c(0, 1, 9, 38, 110)
%o for(n in (length(a)+1):40)
%o a[n] <- +4*a[n-1]-6*a[n-2]+4*a[n-3]-a[n-4]
%o a [_Yosu Yurramendi_, Sep 04 2013]
%o (Magma) [(2*n-1)*(3*n^2 -3*n +2)/2: n in [1..30]]; // _G. C. Greubel_, Dec 01 2017
%Y 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 Cf. A005448, A004188.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_, Aug 01 2001
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