A070262 5th diagonal of triangle defined in A051537.

5, 3, 21, 2, 45, 15, 77, 6, 117, 35, 165, 12, 221, 63, 285, 20, 357, 99, 437, 30, 525, 143, 621, 42, 725, 195, 837, 56, 957, 255, 1085, 72, 1221, 323, 1365, 90, 1517, 399, 1677, 110, 1845, 483, 2021, 132, 2205, 575, 2397, 156, 2597, 675, 2805, 182, 3021, 783

Offset: 1

Author

Amarnath Murthy, May 09 2002

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Crossrefs

Cf. A061037. [From R. J. Mathar, Sep 29 2008]

Cf. A010873, A051537.

Cf. A000466, A002378, A003185, A085027.

Programs

Magma

[LCM(n + 4, n)/GCD(n + 4, n): n in [1..50]]; // G. C. Greubel, Sep 20 2018

Mathematica

Table[ LCM[i + 4, i] / GCD[i + 4, i], {i, 1, 60}]
LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1},{5,3,21,2,45,15,77,6,117,35,165,12},90] (* Harvey P. Dale, Jul 13 2019 *)

PARI

Vec(x*(5 + 3*x + 21*x^2 + 2*x^3 + 30*x^4 + 6*x^5 + 14*x^6 - 3*x^8 - x^9 - 3*x^10) / ((1 - x)^3*(1 + x)^3*(1 + x^2)^3) + O(x^60)) \\ Colin Barker, Mar 27 2017

PARI

a(n) = lcm(n+4,n)/gcd(n+4,n); \\ Altug Alkan, Sep 20 2018

Formula

a(n) = lcm(n + 4, n) / gcd(n + 4, n).

From Colin Barker, Mar 27 2017: (Start)

G.f.: x*(5 + 3*x + 21*x^2 + 2*x^3 + 30*x^4 + 6*x^5 + 14*x^6 - 3*x^8 - x^9 - 3*x^10) / ((1 - x)^3*(1 + x)^3*(1 + x^2)^3).

a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12) for n>12. (End)

From Luce ETIENNE, May 10 2018: (Start)

a(n) = n*(n+4)*4^((5*(n mod 4)^3 - 24*(n mod 4)^2 + 31*(n mod 4)-12)/6).

a(n) = n*(n+4)*(37-27*cos(n*Pi)-6*cos(n*Pi/2))/64. (End)

From Amiram Eldar, Oct 08 2023: (Start)

Sum_{n>=1} 1/a(n) = 11/6.

Sum_{n>=1} (-1)^n/a(n) = 7/6.

Sum_{k=1..n} a(k) ~ (37/192) * n^3. (End)

Extensions

Edited by Robert G. Wilson v, May 10 2002