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 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 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

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