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A069894 Centered square numbers: a(n) = 4*n^2 + 4*n + 2. 13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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 series. 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

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, Jennifer Seberry, A Review and New Symmetric Conference Matrices, Research Online, Faculty of Engineering and Information Sciences, University of Wollongong, 2014.

Wikipedia, Ulam Spiral Construction.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

EXAMPLE

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.

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}]

PROG

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

(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

CROSSREFS

Cf. A001844.

Sequence in context: A167386 A027719 A254709 * A045605 A294871 A212969

Adjacent sequences:  A069891 A069892 A069893 * A069895 A069896 A069897

KEYWORD

nonn,easy,changed

AUTHOR

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

EXTENSIONS

Edited by Robert G. Wilson v, Apr 11 2002

Equation 4*n^2 + 4*n + 2 = n^2 + 1 edited by R. J. Mathar, Sep 16 2009

Offset corrected by Charles R Greathouse IV, Jul 25 2010

STATUS

approved

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Last modified October 21 08:47 EDT 2019. Contains 328292 sequences. (Running on oeis4.)