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%I #20 Oct 08 2023 04:44:51
%S 3,2,15,6,35,12,63,20,99,30,143,42,195,56,255,72,323,90,399,110,483,
%T 132,575,156,675,182,783,210,899,240,1023,272,1155,306,1295,342,1443,
%U 380,1599,420,1763,462,1935,506,2115,552,2303,600,2499,650,2703,702
%N Third diagonal of triangle defined in A051537.
%H Colin Barker, <a href="/A070260/b070260.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F From _Vladeta Jovovic_, May 09 2002: (Start)
%F a(n) = n*(n+2)/4 if n is even else n*(n+2).
%F a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).
%F G.f.: x*(3 + 2*x + 6*x^2 - x^4)/(1 - x^2)^3. (End)
%F E.g.f.: (x/4)*((12 + x)*cosh(x) + (3 + 4*x)*sinh(x)). - _G. C. Greubel_, Jul 20 2017
%F From _Amiram Eldar_, Oct 08 2023: (Start)
%F Sum_{n>=1} 1/a(n) = 3/2.
%F Sum_{n>=1} (-1)^n/a(n) = 1/2.
%F Sum_{k=1..n} a(k) ~ (5/24) * n^3. (End)
%t Table[ LCM[i + 2, i] / GCD[i + 2, i], {i, 1, 60}]
%t LinearRecurrence[{0,3,0,-3,0,1},{3,2,15,6,35,12},60] (* _Harvey P. Dale_, Sep 14 2019 *)
%o (PARI) Vec(x*(3+2*x+6*x^2-x^4) / (1-x^2)^3 + O(x^60)) \\ _Colin Barker_, Mar 27 2017
%Y Bisections: A002378, A000466.
%Y Cf. A051537.
%K nonn,easy
%O 1,1
%A _Amarnath Murthy_, May 09 2002
%E More terms from _Vladeta Jovovic_, May 09 2002
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