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

2, 10, 26, 50, 82, 122, 170, 226, 290, 362, 442, 530, 626, 730, 842, 962, 1090, 1226, 1370, 1522, 1682, 1850, 2026, 2210, 2402, 2602, 2810, 3026, 3250, 3482, 3722, 3970, 4226, 4490, 4762, 5042, 5330, 5626, 5930, 6242, 6562, 6890, 7226, 7570, 7922, 8282

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

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.

1/2 + 1/10 + 1/26 + ... = (Pi/4)*tanh(Pi/2) [Jolley]. - Gary W. Adamson, Dec 21 2006

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

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

Ulam's spiral (SE corner). See the Wikipedia link. - Kival Ngaokrajang, Jul 25 2014

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

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

Examples

			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.
		

References

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

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• N. A. Balonin and Jennifer Seberry, A Review and New Symmetric Conference Matrices, Research Online, Faculty of Engineering and Information Sciences, University of Wollongong, 2014.
• Keyang Li, Figure for n=1,2,3,4,5
• Tintarn, n convex quadrilaterals in the plane
• Wikipedia, Ulam Spiral Construction.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A001105, A001844, A002522, A016754, A016825, A033996, A261377.

Programs

Magma

[4*n^2+4*n+2 : n in [0..50]]; // Wesley Ivan Hurt, Jul 26 2014

Maple

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

Mathematica

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

PARI

vector(100, n, (2*n-1)^2+1); \\ Derek Orr, Jul 27 2014

Sage

[(2*n+1)^2 + 1 for n in range(50)] # G. C. Greubel, Nov 21 2018

Formula

(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.

a(n) = (2*n + 1)^2 + 1. - Vladimir Joseph Stephan Orlovsky, Nov 10 2008 [Corrected by R. J. Mathar, Sep 16 2009]

a(n) = 8*n + a(n-1) for n > 0, a(0)=2. - Vincenzo Librandi, Aug 08 2010

From George F. Johnson, Oct 22 2012: (Start)

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

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

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

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

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

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

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.

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

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

a(n) = A001105(n) + A001105(n+1). - Bruno Berselli, Jul 03 2017

E.g.f.: 2*(1 + 4*x + 2*x^2)*exp(x). - G. C. Greubel, Nov 21 2018

a(n) = A261327(4*n+2). - Paul Curtz, Dec 23 2021

a(n) = 2*A001844(n) = 4*A000217(n) + 2*A002061(n+1). - Klaus Purath, Aug 13 2025

Extensions

Edited by Robert G. Wilson v, Apr 11 2002

Offset corrected by Charles R Greathouse IV, Jul 25 2010