A059993 Pinwheel numbers: a(n) = 2*n^2 + 6*n + 1.

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

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

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

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

Adjacent sequences: A059990 A059991 A059992 * A059994 A059995 A059996

Keyword

nonn,easy

Author

Naohiro Nomoto, Mar 14 2001

Status

approved