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%I A082040 #52 Jul 16 2023 10:39:29
%S A082040 1,13,43,91,157,241,343,463,601,757,931,1123,1333,1561,1807,2071,2353,
%T A082040 2653,2971,3307,3661,4033,4423,4831,5257,5701,6163,6643,7141,7657,
%U A082040 8191,8743,9313,9901,10507,11131,11773,12433,13111,13807,14521,15253,16003
%N A082040 a(n) = 9*n^2 + 3*n + 1.
%C A082040 4th row of A082039, case k=3 of family T(n,k) = k^2n^2 + kn + 1.
%C A082040 a(n)^2 = 81n^4 + 54n^3 + 27n^2 + 6n + 1 = (24*((3*((3n^2 + n)/2)^2 + ((3n^2 + n)/2))/2) + 1). Therefore, (a(n)^2 - 1)/24 is a second pentagonal number (A005449) of index number equal to the n-th second pentagonal number. For example, a(30) = 8191 and (8191^2 - 1)/24 = (67092481 - 1)/24 = 2795520, the 1365th second pentagonal number. 1365 is the 30th second pentagonal number. - _Raphie Frank_, Sep 19 2012
%C A082040 For n >= 1, a(n) is the number of vertices in the hex derived network HDN1(n+1) from the Manuel et al. reference (see HFN1(4) in Fig. 8). - _Emeric Deutsch_, May 21 2018
%C A082040 4*a(n) - 3 is a square. - _Muniru A Asiru_, May 24 2018
%H A082040 Muniru A Asiru, <a href="/A082040/b082040.txt">Table of n, a(n) for n = 0..5000</a>
%H A082040 P. Manuel, R. Bharati, I. Rajasingh, and Chris Monica M, <a href="https://doi.org/10.1016/j.jda.2006.09.002">On minimum metric dimension of honeycomb networks</a>, J. Discrete Algorithms, 6, 2008, 20-27.
%H A082040 Leo Tavares, <a href="/A082040/a082040.jpg">Snowflake illustration</a>
%H A082040 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A082040 a(n) = 18*n + a(n-1) - 6 with n>0, a(0)=1. - _Vincenzo Librandi_, Aug 08 2010
%F A082040 a(n) = A045945(n) + 1: subsequence of A002061. - _Muniru A Asiru_, May 26 2018
%F A082040 a(n) = A003215(n) + 6*A000290(n). - _Leo Tavares_, Jul 14 2023
%p A082040 seq(9*n^2+3*n+1,n=0..50); # _Muniru A Asiru_, May 21 2018
%o A082040 (PARI) a(n)=9*n^2+3*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017
%o A082040 (GAP) List([0..50], n->9*n^2+3*n+1); # _Muniru A Asiru_, May 21 2018
%Y A082040 Cf. A045945, A005449, A054569, A002061, A082039.
%Y A082040 Partial sums of A019557.
%Y A082040 Cf. A003215, A000290.
%K A082040 nonn,easy
%O A082040 0,2
%A A082040 _Paul Barry_, Apr 02 2003

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