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 A115067 a(n) = (3*n^2 - n - 2)/2. 37
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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)-Andrasfai 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Alfred Hoehn, Illustration of initial terms of A000326, A005449, A045943, A115067. Leo Tavares, Illustration: Trapezoids (A115067) Eric Weisstein's World of Mathematics, Andrasfai Graph. Eric Weisstein's World of Mathematics, Clique. Index entries for linear recurrences with constant coefficients, signature (3,-3,1). 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 EXAMPLE 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 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 *) PROG (PARI) a(n)=n*(3*n-1)/2-1 \\ Charles R Greathouse IV, Jan 27 2012 (Magma) [n*(3*n-1)/2-1: n in [1..50]]; // Vincenzo Librandi, Jun 11 2017 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. Sequence in context: A038427 A323625 A301096 * A298787 A009893 A027369 Adjacent sequences: A115064 A115065 A115066 * A115068 A115069 A115070 KEYWORD nonn,easy AUTHOR Roger L. Bagula, Mar 01 2006 EXTENSIONS Edited by N. J. A. Sloane, Mar 05 2006 STATUS approved

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Last modified December 9 15:09 EST 2023. Contains 367691 sequences. (Running on oeis4.)