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4th diagonal of triangle defined in A051537.
2

%I #11 Oct 08 2023 04:44:45

%S 4,10,2,28,40,6,70,88,12,130,154,20,208,238,30,304,340,42,418,460,56,

%T 550,598,72,700,754,90,868,928,110,1054,1120,132,1258,1330,156,1480,

%U 1558,182,1720,1804,210,1978,2068,240,2254,2350,272,2548,2650,306,2860

%N 4th diagonal of triangle defined in A051537.

%H Colin Barker, <a href="/A070261/b070261.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,3,0,0,-3,0,0,1).

%F a(n) = lcm(n + 3, n) / gcd(n + 3, n).

%F From _Colin Barker_, Mar 27 2017: (Start)

%F G.f.: 2*x*(2 + 5*x + x^2 + 8*x^3 + 5*x^4 - x^6 - x^7) / ((1 - x)^3*(1 + x + x^2)^3).

%F a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) for n>9.

%F (End)

%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) = 22*log(2)/9 - 7/6.

%F Sum_{k=1..n} a(k) ~ (19/81) * n^3. (End)

%t Table[ LCM[i + 3, i] / GCD[i + 3, i], {i, 1, 60}]

%o (PARI) Vec(2*x*(2 + 5*x + x^2 + 8*x^3 + 5*x^4 - x^6 - x^7) / ((1 - x)^3*(1 + x + x^2)^3) + O(x^60)) \\ _Colin Barker_, Mar 27 2017

%Y Cf. A051537.

%K nonn,easy

%O 1,1

%A _Amarnath Murthy_, May 09 2002

%E Edited by _Robert G. Wilson v_, May 10 2002