A081251 Numbers n such that A081249(m)/m^2 has a local maximum for m = n.

2, 6, 20, 60, 182, 546, 1640, 4920, 14762, 44286, 132860, 398580, 1195742, 3587226, 10761680, 32285040, 96855122, 290565366, 871696100, 2615088300, 7845264902, 23535794706, 70607384120, 211822152360, 635466457082, 1906399371246

Offset: 1

Author

Klaus Brockhaus, Mar 17 2003

Keywords

• nonn

Comments

The limit of the local maxima, lim A081249(n)/n^2 = 1/6. For local minima cf. A081250.

Also the number of different 4- and 3-colorings for the vertices of all triangulated planar polygons on a base with n+2 vertices, if the colors of the two base vertices are fixed. - Patrick Labarque, Mar 23 2010

From Toby Gottfried, Apr 18 2010: (Start)

a(n) = the number of ternary sequences of length n+1 where the numbers of (0's, 1's) are both odd.

A015518 covers the (odd, even) and (even, odd) cases, and A122983 covers (even, even). (End)

Examples

			6 is a term since A081249(5)/5^2 = 4/25 = 0.160, A081249(6)/6^2 = 7/36 = 0.194, A081249(7)/7^2 = 9/49 = 0.184.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Klaus Brockhaus, Illustration for A081134, A081249, A081250 and A081251
• Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Crossrefs

Cf. A008776, A033113, A054880, A081134, A081249, A081250.

Programs

GAP

List([1..30], n-> (9*3^(n-1) -(-1)^n -2)/4); # G. C. Greubel, Jul 14 2019

Magma

[Floor(3^(n+1)/4) : n in [1..30]]; // Vincenzo Librandi, Jun 25 2011

Maple

seq(floor(3^(n+1)/4), n=1..30). # Mircea Merca, Dec 27 2010

Mathematica

a[n_]:= Floor[3^(n+1)/4]; Array[a, 30]
Table[(9*3^(n-1) -(-1)^n -2)/4, {n, 1, 30}] (* G. C. Greubel, Jul 14 2019 *)

PARI

vector(30, n, (9*3^(n-1) -(-1)^n -2)/4) \\ G. C. Greubel, Jul 14 2019

Sage

[(9*3^(n-1) -(-1)^n -2)/4 for n in (1..30)] # G. C. Greubel, Jul 14 2019

Formula

G.f.: 2/((1-x)*(1+x)*(1-3*x)).

a(n) = a(n-2) + 2*3^(n) for n > 1.

a(n+2) - a(n) = A008776(n).

a(n) = 2*A033113(n+1).

a(2*n+1) = A054880(n+1).

a(n) = floor(3^(n+1)/4). - Mircea Merca, Dec 26 2010

From G. C. Greubel, Jul 14 2019: (Start)

a(n) = (9*3^(n-1) -(-1)^n -2)/4.

E.g.f.: (3*exp(3*x) - 2*exp(x) - exp(-x))/4. (End)