login
This site is supported by donations to The OEIS Foundation.

 

Logo

The October issue of the Notices of the Amer. Math. Soc. has an article about the OEIS.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A069894 Centered square numbers: a(n) = 4*n^2 + 4*n + 2. 12
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

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

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 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

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

CROSSREFS

Cf. A001844.

Sequence in context: A167386 A027719 A254709 * A045605 A294871 A212969

Adjacent sequences:  A069891 A069892 A069893 * A069895 A069896 A069897

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

Edited by Robert G. Wilson v, Apr 11 2002

Edited the equation 4n^2 + 4n + 2 = n^2 + 1. - R. J. Mathar, Sep 16 2009

Offset corrected by Charles R Greathouse IV, Jul 25 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 23 19:43 EDT 2018. Contains 315285 sequences. (Running on oeis4.)