Centered square number

Let C_k,n generally represent the nth centered k-gonal number. The nth centered square number is given by the formula:

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

That is, the nth centered square number is the sum of the nth and the (n – 1)th square numbers. The following pattern demonstrates this formula:

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

The formula can also be expressed as:

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

That is, the nth centered square number is half of the nth odd square number plus 1, as illustrated below:

${\displaystyle C_{4,1}={\frac {1+1}{2}}}$			${\displaystyle C_{4,2}={\frac {9+1}{2}}}$			${\displaystyle C_{4,3}={\frac {25+1}{2}}}$			${\displaystyle C_{4,4}={\frac {49+1}{2}}}$

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

${\displaystyle C_{4,n}=1+4\ T_{n-1}=1+2{n(n-1)},}$

where

${\displaystyle T_{n}={\frac {n(n+1)}{2}}={\binom {n+1}{2}}}$

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:

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

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

Another way the centered square numbers can be expressed is:

${\displaystyle C_{4,n}=1+4\dim(SO(n)),}$

where

${\displaystyle \dim(SO(n))={\frac {n(n-1)}{2}}.}$

Yet another way the centered square numbers can be expressed is in terms of the centered triangular numbers:

${\displaystyle C_{4,n}={\frac {4C_{3,n}-1}{3}},}$

where

${\displaystyle C_{3,n}=1+3{\frac {n(n-1)}{2}}.}$

The first centered square numbers (C_4,n < 4500) 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 digit follows the pattern 1-5-3-5-1.

All centered square numbers and their divisors have a remainder of 1 when divided by 4. Hence all centered square numbers and their divisors end with digit 1 or 5 in base 6, 8, and 12.

Every centered square number except 1 is the hypotenuse of a Pythagorean triple (3-4-5, 5-12-13, 7-24-25, ...). This is exactly the sequence of Pythagorean triples where the two longest sides differ by 1. (Example: 5² + 12² = 13².)

This is not to be confused with the relationship (n – 1)² + n² = C_4,n. (Example: 2² + 3² = 13.)

The generating function that gives the centered square numbers is:

${\displaystyle {\frac {(x+1)^{2}}{(1-x)^{3}}}=1+5x+13x^{2}+25x^{3}+41x^{4}+~...~.}$

• Alfred, U. (1962), "n and n + 1 consecutive integers with equal sums of squares", Mathematics Magazine, 35 (3): 155–164, doi:10.1080/0025570X.1962.11975326, 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.