A115067 - OEIS

A115067

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

%I #109 Jul 08 2024 10:37:18

%S 0,4,11,21,34,50,69,91,116,144,175,209,246,286,329,375,424,476,531,

%T 589,650,714,781,851,924,1000,1079,1161,1246,1334,1425,1519,1616,1716,

%U 1819,1925,2034,2146,2261,2379,2500,2624,2751,2881,3014,3150,3289,3431,3576

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

%C 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

%C 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

%C 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

%C 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

%H Vincenzo Librandi, <a href="/A115067/b115067.txt">Table of n, a(n) for n = 1..1000</a>

%H Sela Fried, <a href="https://arxiv.org/abs/2406.18923">Counting r X s rectangles in nondecreasing and Smirnov words</a>, arXiv:2406.18923 [math.CO], 2024. See p. 5.

%H Alfred Hoehn, <a href="/A000326/a000326.pdf">Illustration of initial terms of A000326, A005449, A045943, A115067</a>.

%H Leo Tavares, <a href="/A115067/a115067.jpg">Illustration: Trapezoids (A115067)</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AndrasfaiGraph.html">Andrásfai Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Clique.html">Clique</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

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

%F a(n+1) = n*(3*n + 5)/2. - _Omar E. Pol_, May 21 2008

%F a(n) = 3*n + a(n-1) - 2 for n>1, a(1)=0. - _Vincenzo Librandi_, Nov 13 2010

%F a(n) = A095794(-n). - _Bruno Berselli_, Sep 02 2011

%F G.f.: x^2*(4-x) / (1-x)^3. - _R. J. Mathar_, Sep 02 2011

%F a(n) = A055998(2*n-2) - A055998(n-1). - _Bruno Berselli_, Sep 23 2016

%F E.g.f.: exp(x)*x*(8 + 3*x)/2. - _Stefano Spezia_, May 19 2021

%F From _Amiram Eldar_, Feb 22 2022: (Start)

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

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

%F 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

%e Illustrations for n = 2..7 from _Stefano Spezia_, Jun 05 2021:

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%e a(2) = 4 a(3) = 11 a(4) = 21

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%e a(5) = 34 a(6) = 50 a(7) = 69

%t Table[n (3 n - 1)/2 - 1, {n, 50}] (* _Vincenzo Librandi_, Jun 11 2017 *)

%t LinearRecurrence[{3, -3, 1}, {0, 4, 11}, 20] (* _Eric W. Weisstein_, Nov 29 2017 *)

%t CoefficientList[Series[(-4 + x) x/(-1 + x)^3, {x, 0, 20}], x] (* _Eric W. Weisstein_, Nov 29 2017 *)

%o (PARI) a(n)=n*(3*n-1)/2-1 \\ _Charles R Greathouse IV_, Jan 27 2012

%o (Magma) [n*(3*n-1)/2-1: n in [1..50]]; // _Vincenzo Librandi_, Jun 11 2017

%Y 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.

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

%Y Cf. A055998, A095794.

%K nonn,easy

%O 1,2

%A _Roger L. Bagula_, Mar 01 2006

%E Edited by _N. J. A. Sloane_, Mar 05 2006