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A069894 Centered square numbers: a(n) = 4*n^2 + 4*n + 2. 18

%I #122 Jul 28 2022 20:59:41

%S 2,10,26,50,82,122,170,226,290,362,442,530,626,730,842,962,1090,1226,

%T 1370,1522,1682,1850,2026,2210,2402,2602,2810,3026,3250,3482,3722,

%U 3970,4226,4490,4762,5042,5330,5626,5930,6242,6562,6890,7226,7570,7922,8282

%N Centered square numbers: a(n) = 4*n^2 + 4*n + 2.

%C Any number may be substituted for y to yield similar sequences. The number set used determines values given (i.e., integer yields integer). All centered square integers in the set of integers may be found by this formula.

%C 1/2 + 1/10 + 1/26 + ... = (Pi/4)*tanh(Pi/2) [Jolley]. - _Gary W. Adamson_, Dec 21 2006

%C For n > 0, a(n-1) is the number of triples (w, x, y) having all terms in {0, ..., n} and min(|w - x|, |x - y|) = 1. - _Clark Kimberling_, Jun 12 2012

%C Consider the primitive Pythagorean triples (x(n), y(n), z(n) = y(n) + 1) with n >= 0, and x(n) = 2*n + 1, y(n) = 2*n*(n + 1), z(n) = 2*n*(n + 1) + 1. The sequence, a(n), is 2*z(n). - _George F. Johnson_, Oct 22 2012

%C Ulam's spiral (SE corner). See the Wikipedia link. - _Kival Ngaokrajang_, Jul 25 2014

%C Conference matrix orders (A000952) of the form n-1 is a perfect square are all in this sequence. All values less than 1000 are conference matrices except for 226 which is still an open question (Balonin & Seberry 2014). - _Colin Hall_, Nov 21 2018

%C For n > 0, a(n-1) is the number of maximum number of regions into which the plane can be divided using n convex quadrilaterals. Related: A077588 A077591. - _Keyang Li_, Jun 17 2022

%D L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 176.

%H Ivan Panchenko, <a href="/A069894/b069894.txt">Table of n, a(n) for n = 0..1000</a>

%H N. A. Balonin and Jennifer Seberry, <a href="https://ro.uow.edu.au/eispapers/2748/">A Review and New Symmetric Conference Matrices</a>, Research Online, Faculty of Engineering and Information Sciences, University of Wollongong, 2014.

%H Keyang Li, <a href="/A069894/a069894.svg">figure for n=1,2,3,4,5</a>

%H Tintarn, <a href="https://artofproblemsolving.com/community/c6h377434p10984052">n convex quadrilaterals in the plane</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ulam_spiral#Construction">Ulam Spiral Construction</a>.

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

%F (y*(2*x + 1))^2 + (y*(2*x^2 + 2*x))^2 = (y*(2*x^2 + 2*x + 1))^2, where y = 2. If a^2 + b^2 = c^2, then c^2 = y^2*(4*x^4 + 8*x^3 + 8*x^2 + 4*x + 1). Also 2*A001844.

%F a(n) = (2*n + 1)^2 + 1. - _Vladimir Joseph Stephan Orlovsky_, Nov 10 2008 [Corrected by _R. J. Mathar_, Sep 16 2009]

%F a(n) = 8*n + a(n-1) for n > 0, a(0)=2. - _Vincenzo Librandi_, Aug 08 2010

%F From _George F. Johnson_, Oct 22 2012: (Start)

%F G.f.: 2*(1 + x)^2/(1 - x)^3, a(0) = 2, a(1) = 10.

%F a(n+1) = a(n) + 4 + 4*sqrt(a(n) - 1).

%F a(n-1) * a(n+1) = (a(n)-4)^2 + 16.

%F a(n) - 1 = (2*n+1)^2 = A016754(n) for n > 0.

%F (a(n+1) - a(n-1))/8 = sqrt(a(n) - 1).

%F a(n+1) = 2*a(n) - a(n-1) + 8 for n > 2, a(0)=2, a(1)=10, a(2)=26.

%F a(n+1) = 3*a(n) - 3*a(n-1) + a(n-2) for n > 3; a(0)=2, a(1)=10, a(2)=26, a(3)=50.

%F a(n) = A033996(n) + 2 = A002522(2n + 1).

%F a(n)^2 = A033996(n)^2 + A016825(n)^2. (End)

%F a(n) = A001105(n) + A001105(n+1). - _Bruno Berselli_, Jul 03 2017

%F E.g.f.: 2*(1 + 4*x + 2*x^2)*exp(x). - _G. C. Greubel_, Nov 21 2018

%F a(n) = A261327(4*n+2). - _Paul Curtz_, Dec 23 2021

%e If y = 3, then 81 + 144 = 225; if y = 4, then 12^2 + 16^2 = 20^2; 7^2 + 24^2 = 25^2 = 15^2 + 20^2.

%p A069894:=n->4*n^2+4*n+2: seq(A069894(n), n=0..50); # _Wesley Ivan Hurt_, Jul 26 2014

%t Table[4n(n + 1) + 2, {n, 0, 45}]

%o (Magma) [4*n^2+4*n+2 : n in [0..50]]; // _Wesley Ivan Hurt_, Jul 26 2014

%o (PARI) vector(100, n, (2*n-1)^2+1); \\ _Derek Orr_, Jul 27 2014

%o (Sage) [(2*n+1)^2 + 1 for n in range(50)] # _G. C. Greubel_, Nov 21 2018

%Y Cf. A001844.

%K nonn,easy

%O 0,1

%A Glenn B. Cox (igloos_r_us(AT)canada.com), Apr 10 2002

%E Edited by _Robert G. Wilson v_, Apr 11 2002

%E Equation 4*n^2 + 4*n + 2 = n^2 + 1 edited by _R. J. Mathar_, Sep 16 2009

%E Offset corrected by _Charles R Greathouse IV_, Jul 25 2010

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)