A194767 Denominator of the fourth increasing diagonal of the autosequence of second kind from (-1)^n / (n+1).

2, 2, 12, 20, 10, 42, 56, 24, 90, 110, 44, 156, 182, 70, 240, 272, 102, 342, 380, 140, 462, 506, 184, 600, 650, 234, 756, 812, 290, 930, 992, 352, 1122, 1190, 420, 1332, 1406, 494, 1560, 1640, 574, 1806, 1892, 660, 2070, 2162, 752, 2352, 2450, 850, 2652, 2756, 954, 2970, 3080, 1064, 3306, 3422, 1180, 3660

Offset: 0

Author

Paul Curtz, Sep 02 2011

Keywords

• nonn
• frac
• easy

Comments

The autosequence of first kind from (-1)^n/(n+1) is A189733.

For the second kind (the second increasing diagonal is (-1)^n/(n+1), half of the main one):

2, 1, 0, -1/2, -1/3, 1/6, 1/2, 5/12,

-1, -1, -1/2, 1/6, 1/2, 1/3, -1/12, -7/20,

0, 1/2, 2/3, 1/3, -1/6, -5/12, -4/15, 1/12,

1/2, 1/6, -1/3, -1/2, -1/4, 3/20, 7/20, 13/60,

-1/3, -1/2, -1/6, 1/4, 2/5, 1/5, -2/15, -3/10,

-1/6, 1/3, 5/12, 3/20, -1/5, -1/3, -1/6, 5/42,

1/2, 1/12, -4/15, -7/20, -2/15, 1/6, 2/7, 1/7,

-5/12, -7/20, -1/12, 13/60, 3/10, 5/42, -1/7, -1/4.

Main diagonal: (period 2:repeat 2, -1)/A026741(n+1).

Second (increasing) diagonal: (-1)^n / (n+1).

Third (increasing) diagonal: (-1)^(n+1)*A026741(n) / A045896(n).

Fourth (increasing) diagonal: (-1)^(n+1)*A146535(n)/ a(n).

Links

• OEIS Wiki, Autosequence.
• Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Crossrefs

Cf. A001504 = 2*A060544, A049450 = 2*A000326, A045945 = 6*A005449.

Cf. A026741, A045896, A051176, A146535, A189733, A306368.

Programs

Mathematica

c = Table[1/9 (7 n + 7 n^2 + 2 n Cos[2 n *Pi/3] + 2 n^2 Cos[2 n *Pi/3] + 2 Sqrt[3] n Sin[2 n *Pi/3] + 2 Sqrt[3] n^2 Sin[2 n *Pi/3]), {n, 1, 50}] (* Roger Bagula, Mar 25 2012 *)
a[n_] := (n+1) * Numerator[(n+2)/3]; Array[a, 60, 0] (* Amiram Eldar, Sep 17 2023 *)
LinearRecurrence[{0,0,3,0,0,-3,0,0,1},{2,2,12,20,10,42,56,24,90},60] (* Harvey P. Dale, May 15 2025 *)

Formula

a(3*n) = (3*n+1)*(3*n+2), a(3*n+1) = (n+1)*(3*n+2), a(3*n+2) = 3*(n+1)*(3*n+4).

G.f.: 2*(1+x+6*x^2+7*x^3+2*x^4+3*x^5+x^6)/(1-x^3)^3. - Jean-François Alcover, Nov 11 2016

a(n+2) = 2 * A306368(n) for n >= 0. - Joerg Arndt, Aug 25 2023

a(n) = (n+1) * A051176(n+2) for n >= 0. - Paul Curtz, Sep 13 2023

Sum_{n>=0} 1/a(n) = 1 + log(3) - Pi/(3*sqrt(3)). - Amiram Eldar, Sep 17 2023