%I #71 Nov 16 2024 17:33:16

%S 1,9,21,37,57,81,109,141,177,217,261,309,361,417,477,541,609,681,757,

%T 837,921,1009,1101,1197,1297,1401,1509,1621,1737,1857,1981,2109,2241,

%U 2377,2517,2661,2809,2961,3117,3277,3441,3609,3781,3957,4137,4321,4509,4701,4897

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

%C Nonnegative integers m such that 2*m + 7 is a square. - _Vincenzo Librandi_, Mar 01 2013

%C 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

%C 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

%D 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

%H Harry J. Smith, <a href="/A059993/b059993.txt">Table of n, a(n) for n = 0..1000</a>

%H Author?, <a href="https://web.archive.org/web/20020810200217/http://www.geocities.co.jp/Technopolis/1793/figure.txt">figure</a>. [Wayback Machine link]

%H Ângela Mestre and José Agapito, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Mestre/mestre2.html">Square Matrices Generated by Sequences of Riordan Arrays</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 4*n + a(n-1) + 4 for n > 0, a(0)=1. - _Vincenzo Librandi_, Aug 07 2010

%F G.f.: (1 + 6*x - 3*x^2)/(1-x)^3. - _Arkadiusz Wesolowski_, Dec 24 2011

%F a(n) = 2*a(n-1) - a(n-2) + 4. - _Vincenzo Librandi_, Mar 01 2013

%F a(n) = Hyper2F1([-2, n], [1], -2). - _Peter Luschny_, Aug 02 2014

%F Sum_{n>=0} 1/a(n) = 1/3 + Pi*tan(sqrt(7)*Pi/2)/(2*sqrt(7)). - _Amiram Eldar_, Dec 13 2022

%F From _Elmo R. Oliveira_, Nov 16 2024: (Start)

%F E.g.f.: exp(x)*(1 + 8*x + 2*x^2).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

%t Table[2 n^2 + 6 n + 1, {n, 0, 46}] (* _Zerinvary Lajos_, Jul 10 2009 *)

%t LinearRecurrence[{3,-3,1},{1,9,21},50] (* _Harvey P. Dale_, Oct 01 2018 *)

%o (PARI) { for (n=0, 1000, write("b059993.txt", n, " ", 2*n^2 + 6*n + 1); ) } \\ _Harry J. Smith_, Jul 01 2009

%o (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

%Y Cf. numbers n such that 2*n + 2*k + 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).

%K nonn,easy,changed

%O 0,2

%A _Naohiro Nomoto_, Mar 14 2001