The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A059993 Pinwheel numbers: a(n) = 2*n^2 + 6*n + 1. 20
1, 9, 21, 37, 57, 81, 109, 141, 177, 217, 261, 309, 361, 417, 477, 541, 609, 681, 757, 837, 921, 1009, 1101, 1197, 1297, 1401, 1509, 1621, 1737, 1857, 1981, 2109, 2241, 2377, 2517, 2661, 2809, 2961, 3117, 3277, 3441, 3609, 3781, 3957, 4137, 4321, 4509 (list; graph; refs; listen; history; text; internal format)
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), 26 (4), 1407-1412, 2014. - Emeric Deutsch, Sep 23 2017
LINKS
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.
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
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
Cf. numbers n such that 2n+2k+1 is a square: A046092 (k=0), A142463 (k=1), A090288 (k=2), this sequence (k=3), A139570 (k=4), A222182 (k=5), A181510 (k=6).
Sequence in context: A146069 A140673 A186294 * A036704 A338911 A107890
KEYWORD
nonn,easy
AUTHOR
Naohiro Nomoto, Mar 14 2001
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 25 06:28 EDT 2024. Contains 372782 sequences. (Running on oeis4.)