OFFSET
0,2
COMMENTS
Nonnegative integers m such that 2*m + 7 is a square. - Vincenzo Librandi, Mar 01 2013
Numbers of the form 4*(h+1)*(2*h-1) + 1, where h = 0, -1, 1, -2, 2, -3, 3, -4, 4, ... . - Bruno Berselli, Feb 03 2017
a(n) is also the number of vertices of the Aztec diamond AZ(n) (see Lemma 2.1 of the Imran et al. paper). - Emeric Deutsch, Sep 23 2017
REFERENCES
M. Imran and S. Hayat, On computation of topological indices of Aztec diamonds, Sci. Int. (Lahore), Vol. 26(4), 2014, pp. 1407-1412. - Emeric Deutsch, Sep 23 2017
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..1000
Author?, figure. [Wayback Machine link]
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 4*n + a(n-1) + 4 for n > 0, a(0)=1. - Vincenzo Librandi, Aug 07 2010
G.f.: (1 + 6*x - 3*x^2)/(1-x)^3. - Arkadiusz Wesolowski, Dec 24 2011
a(n) = 2*a(n-1) - a(n-2) + 4. - Vincenzo Librandi, Mar 01 2013
a(n) = Hyper2F1([-2, n], [1], -2). - Peter Luschny, Aug 02 2014
Sum_{n>=0} 1/a(n) = 1/3 + Pi*tan(sqrt(7)*Pi/2)/(2*sqrt(7)). - Amiram Eldar, Dec 13 2022
From Elmo R. Oliveira, Nov 16 2024: (Start)
E.g.f.: exp(x)*(1 + 8*x + 2*x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
MATHEMATICA
Table[2 n^2 + 6 n + 1, {n, 0, 46}] (* Zerinvary Lajos, Jul 10 2009 *)
LinearRecurrence[{3, -3, 1}, {1, 9, 21}, 50] (* Harvey P. Dale, Oct 01 2018 *)
PROG
(PARI) { for (n=0, 1000, write("b059993.txt", n, " ", 2*n^2 + 6*n + 1); ) } \\ Harry J. Smith, Jul 01 2009
(Magma) [2*n^2+6*n+1: n in [0..50]]; /* or */ I:=[1, 9]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2)+4: n in [1..50]]; // Vincenzo Librandi, Mar 01 2013
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Naohiro Nomoto, Mar 14 2001
STATUS
approved