A115067 a(n) = (3*n^2 - n - 2)/2.

0, 4, 11, 21, 34, 50, 69, 91, 116, 144, 175, 209, 246, 286, 329, 375, 424, 476, 531, 589, 650, 714, 781, 851, 924, 1000, 1079, 1161, 1246, 1334, 1425, 1519, 1616, 1716, 1819, 1925, 2034, 2146, 2261, 2379, 2500, 2624, 2751, 2881, 3014, 3150, 3289, 3431, 3576

Offset: 1

Author

Roger L. Bagula, Mar 01 2006

Keywords

• nonn
• easy

Comments

Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 6720. - Philippe A.J.G. Chevalier, Dec 28 2015

a(n) is the sum of the numerator and denominator of the reduced fraction resulting from the sum A000217(n-2)/A000217(n-1) + A000217(n-1)/A000217(n), n>1. - J. M. Bergot, Jun 10 2017

For n > 1, a(n) is also the number of (not necessarily maximal) cliques in the (n-1)-Andrásfai graph. - Eric W. Weisstein, Nov 29 2017

a(n+1) is the sum of the lengths of all the segments used to draw a square of side n representing the most classic pattern for walls made of 2 X 1 bricks, known as a 1-over-2 pattern, where each joint between neighboring bricks falls over the center of the brick below. - Stefano Spezia, Jun 05 2021

Examples

			Illustrations for n = 2..7 from _Stefano Spezia_, Jun 05 2021:
       _           _ _          _ _ _
      |_|         |_|_|        |_|_ _|
                  |_ _|        |_ _|_|
                               |_|_ _|
   a(2) = 4     a(3) = 11     a(4) = 21
    _ _ _ _     _ _ _ _ _    _ _ _ _ _ _
   |_ _|_ _|   |_ _|_ _|_|  |_ _|_ _|_ _|
   |_|_ _|_|   |_|_ _|_ _|  |_|_ _|_ _|_|
   |_ _|_ _|   |_ _|_ _|_|  |_ _|_ _|_ _|
   |_|_ _|_|   |_|_ _|_ _|  |_|_ _|_ _|_|
               |_ _|_ _|_|  |_ _|_ _|_ _|
                            |_|_ _|_ _|_|
   a(5) = 34    a(6) = 50     a(7) = 69
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 5.
• Alfred Hoehn, Illustration of initial terms of A000326, A005449, A045943, A115067.
• Leo Tavares, Illustration: Trapezoids (A115067)
• Eric Weisstein's World of Mathematics, Andrásfai Graph.
• Eric Weisstein's World of Mathematics, Clique.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

Orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A008585, A005843, A001477, A000217.

Cf. A055998, A095794.

Programs

Magma

[n*(3*n-1)/2-1: n in [1..50]]; // Vincenzo Librandi, Jun 11 2017

Mathematica

Table[n (3 n - 1)/2 - 1, {n, 50}] (* Vincenzo Librandi, Jun 11 2017 *)
LinearRecurrence[{3, -3, 1}, {0, 4, 11}, 20] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[(-4 + x) x/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)

PARI

a(n)=n*(3*n-1)/2-1 \\ Charles R Greathouse IV, Jan 27 2012

Formula

a(n) = (3*n+2)*(n-1)/2.

a(n+1) = n*(3*n + 5)/2. - Omar E. Pol, May 21 2008

a(n) = 3*n + a(n-1) - 2 for n>1, a(1)=0. - Vincenzo Librandi, Nov 13 2010

a(n) = A095794(-n). - Bruno Berselli, Sep 02 2011

G.f.: x^2*(4-x) / (1-x)^3. - R. J. Mathar, Sep 02 2011

a(n) = A055998(2*n-2) - A055998(n-1). - Bruno Berselli, Sep 23 2016

E.g.f.: exp(x)*x*(8 + 3*x)/2. - Stefano Spezia, May 19 2021

From Amiram Eldar, Feb 22 2022: (Start)

Sum_{n>=2} 1/a(n) = Pi/(5*sqrt(3)) - 3*log(3)/5 + 21/25.

Sum_{n>=2} (-1)^n/a(n) = 4*log(2)/5 - 2*Pi/(5*sqrt(3)) + 9/25. (End)

a(n) = Sum_{j=0..n-2} (2*n-j) = Sum_{j=0..n-2} (n+2+j), for n>=1. See the trapezoid link. - Leo Tavares, May 20 2022

Extensions

Edited by N. J. A. Sloane, Mar 05 2006