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:


$C_{4,1} = 0 + 1$		$C_{4,2} = 1 + 4$		$C_{4,3} = 4 + 9$		$C_{4,4} = 9 + 16$

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 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).

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 the OEIS).

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 .

Classes of natural numbers

Powers and
related numbers

Of the form
a × 2^b ± 1

Other polynomial
numbers

Recursively defined
numbers

Possessing a
specific set
of other numbers

Expressible via
specific sums

Generated
via a sieve

Lucky

Code related

Meertens

Figurate numbers

2-dimensional

centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star

non-centered	Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional

centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral

non-centered	Tetrahedral Octahedral Dodecahedral Icosahedral Stella octangula

pyramidal	Square pyramidal Pentagonal pyramidal Hexagonal pyramidal Heptagonal pyramidal

4-dimensional

centered	Centered pentachoric Squared triangular

non-centered	Pentatope

Pseudoprimes

Combinatorial
numbers

Arithmetic functions

By properties of σ(n)	Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect

By properties of Ω(n)	Almost prime Semiprime

By properties of φ(n)	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient

By properties of s(n)	Amicable Betrothed Deficient Semiperfect

Dividing a quotient

Other prime factor
or divisor related
numbers

Recreational
mathematics

Base-dependent
numbers

This article is issued from Wikipedia - version of the 10/18/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.