To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Centered square number

From Wikipedia, the free encyclopedia

In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a given city block distance of the center dot on a regular square lattice. While centered square numbers, like figurate numbers in general, have few if any direct practical applications, they are sometimes studied in recreational mathematics for their elegant geometric and arithmetic properties.

The figures for the first four centered square numbers are shown below:

GrayDot.svg
   
GrayDot.svg

GrayDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
   
GrayDot.svg

GrayDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
   
GrayDot.svg

GrayDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
           

YouTube Encyclopedic

  • 1/5
    Views:
    18 098
    227 245
    148 769
    199 063
    111 725
  • ✪ ESTIMATE THE ROOT with linear approximation (KristaKingMath)
  • ✪ Finding the Center-Radius Form of a Circle by Completing the Square - Example 3
  • ✪ How to find the center and radius of a circle in standard form
  • ✪ How to find the center, foci and vertices of an ellipse
  • ✪ How to find the Centre of Rotation

Transcription

Relationships with other figurate numbers

The nth centered square number is given by the formula[notation clarification needed]

In other words, a centered square number is the sum of two consecutive square numbers. The following pattern demonstrates this formula:

GrayDot.svg
   
RedDot.svg

RedDot.svg
GrayDot.svg
RedDot.svg

RedDot.svg
   
GrayDot.svg

GrayDot.svg
RedDot.svg
GrayDot.svg

GrayDot.svg
RedDot.svg
GrayDot.svg
RedDot.svg
GrayDot.svg

GrayDot.svg
RedDot.svg
GrayDot.svg

GrayDot.svg
   
RedDot.svg

RedDot.svg
GrayDot.svg
RedDot.svg

RedDot.svg
GrayDot.svg
RedDot.svg
GrayDot.svg
RedDot.svg

RedDot.svg
GrayDot.svg
RedDot.svg
GrayDot.svg
RedDot.svg
GrayDot.svg
RedDot.svg

RedDot.svg
GrayDot.svg
RedDot.svg
GrayDot.svg
RedDot.svg

RedDot.svg
GrayDot.svg
RedDot.svg

RedDot.svg
           

The formula can also be expressed as

that is, n th centered square number is half of n th odd square number plus one, as illustrated below:

GrayDot.svg
   
GrayDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
GrayDot.svg
MissingDot.svg

MissingDot.svg
MissingDot.svg
MissingDot.svg
   
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
GrayDot.svg
GrayDot.svg
MissingDot.svg
MissingDot.svg

MissingDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg

MissingDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg
   
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg

MissingDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg

MissingDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg

MissingDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg
MissingDot.svg
           

Like all centered polygonal numbers, centered square numbers can also be expressed in terms of triangular numbers:

where

is the nth triangular number. This can be easily seen by removing the center dot and dividing the rest of the figure into four triangles, as below:

BlackDot.svg
   
RedDot.svg

GrayDot.svg
BlackDot.svg
GrayDot.svg

RedDot.svg
   
RedDot.svg

RedDot.svg
RedDot.svg
GrayDot.svg

GrayDot.svg
GrayDot.svg
BlackDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
RedDot.svg
RedDot.svg

RedDot.svg
   
RedDot.svg

RedDot.svg
RedDot.svg
GrayDot.svg

RedDot.svg
RedDot.svg
RedDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
GrayDot.svg
GrayDot.svg
BlackDot.svg
GrayDot.svg
GrayDot.svg
GrayDot.svg

GrayDot.svg
GrayDot.svg
RedDot.svg
RedDot.svg
RedDot.svg

GrayDot.svg
RedDot.svg
RedDot.svg

RedDot.svg
           

The difference between two consecutive octahedral numbers is a centered square number (Conway and Guy, p.50).

Properties

The first few centered square numbers are:

1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4325, … (sequence A001844 in the OEIS).

All centered square numbers are odd, and in base 10 one can notice the one's digits follows the pattern 1-5-3-5-1.

All centered square numbers and their divisors have a remainder of one when divided by four. Hence all centered square numbers and their divisors end with digits 1 or 5 in base 6, 8 or 12.

Every centered square number except 1 is the third term of a leg–hypotenuse Pythagorean triple (for example, 3-4-5, 5-12-13).

References

  • Alfred, U. (1962), "n and n + 1 consecutive integers with equal sums of squares", Mathematics Magazine, 35 (3): 155–164, JSTOR 2688938, MR 1571197.
  • Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001.
  • Beiler, A. H. (1964), Recreations in the Theory of Numbers, New York: Dover, p. 125.
  • Conway, John H.; Guy, Richard K. (1996), The Book of Numbers, New York: Copernicus, pp. 41–42, ISBN 0-387-97993-X, MR 1411676.
This page was last edited on 31 December 2018, at 22:33
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.