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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

figure [broken link]

Ângela Mestre, 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

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 December 7 10:26 EST 2022. Contains 358656 sequences. (Running on oeis4.)