»
Arabic Bulgarian Chinese Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hindi Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Malagasy Norwegian Persian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Thai Turkish Vietnamese
Arabic Bulgarian Chinese Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hindi Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Malagasy Norwegian Persian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Thai Turkish Vietnamese

# definition - centered square number

definition of Wikipedia

# Centered square number

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:

 $C_{4,1} = 1$ $C_{4,2} = 5$ $C_{4,3} = 13$ $C_{4,4} = 25$

## Relationships with other figurate numbers

The nth centered square number is given by the formula

$C_{4,n} = n^2 + (n - 1)^2.\,$

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

 $C_{4,1} = 1$ $C_{4,2} = 1 + 4$ $C_{4,3} = 4 + 9$ $C_{4,4} = 9 + 16$

The formula can also be expressed as

$C_{4,n} = {(2n-1)^2 + 1 \over 2};$

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

 $C_{4,1} = (1 + 1) / 2$ $C_{4,2} = (9 + 1) / 2$ $C_{4,3} = (25 + 1) / 2$ $C_{4,4} = (49 + 1) / 2$

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

$C_{4,n} = 1 + 4\, T_{n-1},\,$

where

$T_n = {n(n + 1) \over 2} = {n^2 + n \over 2} = {n+1 \choose 2}$

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:

 $C_{4,1} = 1$ $C_{4,2} = 1 + 4 \times 1$ $C_{4,3} = 1 + 4 \times 3$ $C_{4,4} = 1 + 4 \times 6.$

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 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.

All centered square numbers except 1 are the third term of a Leg-Hypotenuse Pythagorean triple (for example, 3-4-5, 5-12-13).

### Centered square prime

A centered square prime is a centered square number that is prime. Unlike regular square numbers, which are never prime, quite a few of the centered square numbers are prime. The first few centered square primes are:

5, 13, 41, 61, 113, 181, 313, 421, 613, 761, 1013, 1201, 1301, 1741, 1861, 2113, 2381, 2521, 3121, 3613, … (sequence A027862 in OEIS). A striking example can be seen in the 10th century al-Antaakii magic square.

## References

sensagent's content

• definitions
• synonyms
• antonyms
• encyclopedia

Dictionary and translator for handheld

New : sensagent is now available on your handheld

sensagent's office

Shortkey or widget. Free.

Windows Shortkey: . Free.

Vista Widget : . Free.

Webmaster Solution

Alexandria

A windows (pop-into) of information (full-content of Sensagent) triggered by double-clicking any word on your webpage. Give contextual explanation and translation from your sites !

Try here  or   get the code

SensagentBox

With a SensagentBox, visitors to your site can access reliable information on over 5 million pages provided by Sensagent.com. Choose the design that fits your site.

WordGame

The English word games are:
○   Anagrams
○   Wildcard, crossword
○   Lettris
○   Boggle.

Lettris

Lettris is a curious tetris-clone game where all the bricks have the same square shape but different content. Each square carries a letter. To make squares disappear and save space for other squares you have to assemble English words (left, right, up, down) from the falling squares.

boggle

Boggle gives you 3 minutes to find as many words (3 letters or more) as you can in a grid of 16 letters. You can also try the grid of 16 letters. Letters must be adjacent and longer words score better. See if you can get into the grid Hall of Fame !

English dictionary
Main references

Most English definitions are provided by WordNet .
English thesaurus is mainly derived from The Integral Dictionary (TID).
English Encyclopedia is licensed by Wikipedia (GNU).

The wordgames anagrams, crossword, Lettris and Boggle are provided by Memodata.
The web service Alexandria is granted from Memodata for the Ebay search.
The SensagentBox are offered by sensAgent.

Translation

Change the target language to find translations.
Tips: browse the semantic fields (see From ideas to words) in two languages to learn more.

last searches on the dictionary :

3844 online visitors

computed in 0.063s

I would like to report:
section :
a spelling or a grammatical mistake
an offensive content(racist, pornographic, injurious, etc.)