A082040 a(n) = 9*n^2 + 3*n + 1.

1, 13, 43, 91, 157, 241, 343, 463, 601, 757, 931, 1123, 1333, 1561, 1807, 2071, 2353, 2653, 2971, 3307, 3661, 4033, 4423, 4831, 5257, 5701, 6163, 6643, 7141, 7657, 8191, 8743, 9313, 9901, 10507, 11131, 11773, 12433, 13111, 13807, 14521, 15253, 16003

Offset

0,2

Comments

4th row of A082039, case k=3 of family T(n,k) = k^2n^2 + kn + 1.

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

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

4*a(n) - 3 is a square. - Muniru A Asiru, May 24 2018

Links

Muniru A Asiru, Table of n, a(n) for n = 0..5000

P. Manuel, R. Bharati, I. Rajasingh, and Chris Monica M, On minimum metric dimension of honeycomb networks, J. Discrete Algorithms, 6, 2008, 20-27.

Leo Tavares, Snowflake illustration

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = 18*n + a(n-1) - 6 with n>0, a(0)=1. - Vincenzo Librandi, Aug 08 2010

a(n) = A045945(n) + 1: subsequence of A002061. - Muniru A Asiru, May 26 2018

a(n) = A003215(n) + 6*A000290(n). - Leo Tavares, Jul 14 2023

Maple

seq(9*n^2+3*n+1, n=0..50); # Muniru A Asiru, May 21 2018

Prog

(PARI) a(n)=9*n^2+3*n+1 \\ Charles R Greathouse IV, Jun 17 2017
(GAP) List([0..50], n->9*n^2+3*n+1); # Muniru A Asiru, May 21 2018

Crossrefs

Cf. A045945, A005449, A054569, A002061, A082039.

Partial sums of A019557.

Cf. A003215, A000290.

Sequence in context: A282322 A031382 A187677 * A106734 A066465 A023262

Adjacent sequences: A082037 A082038 A082039 * A082041 A082042 A082043

Keyword

nonn,easy

Author

Paul Barry, Apr 02 2003

Status

approved