This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A082040 #61 Jul 07 2025 00:46:35 %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 8191,8743,9313,9901,10507,11131,11773,12433,13111,13807,14521,15253,16003,16771,17557 %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^2*n^2 + k*n + 1. %C A082040 a(n)^2 = 81*n^4 + 54*n^3 + 27*n^2 + 6*n + 1 = (24*((3*((3*n^2 + n)/2)^2 + ((3*n^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 Nicolay Avilov, <a href="/A082040/a082040_1.jpg">Graphic illustration of sequence members</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>, Journal of Discrete Algorithms, Vol. 6 (2008), pp. 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 %F A082040 From _Elmo R. Oliveira_, Oct 23 2024: (Start) %F A082040 G.f.: (1 + 10*x + 7*x^2)/(1 - x)^3. %F A082040 E.g.f.: (1 + 12*x + 9*x^2)*exp(x). %F A082040 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End) %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. A002061, A005449, A045945, A054569, A082039. %Y A082040 Partial sums of A019557. %Y A082040 Cf. A000290, A003215. %K A082040 nonn,easy %O A082040 0,2 %A A082040 _Paul Barry_, Apr 02 2003