Centered square number

Relationships with other figurate numbers

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

: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:

:{| |- align="center" valign="middle" style="line-height: 0;" |[[Image:GrayDot.svg|16px]] |
|
|[[Image:RedDot.svg|16px]] |
|
|[[Image:GrayDot.svg|16px]] |
|
|[[Image:RedDot.svg|16px]] |- align="center" valign="top" | 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} = \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:

:{| |- align="center" valign="bottom" style="line-height: 0;" |[[Image:GrayDot.svg|16px]] |
|
|[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]] |
|
|[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]] |
|
|[[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]][[Image:GrayDot.svg|16px]] |- align="center" valign="top" | C_{4,1} = \frac{1 + 1}{2} |
|
| C_{4,2} = \frac{9 + 1}{2} |
|
| C_{4,3} = \frac{25 + 1}{2} |
|
| C_{4,4} = \frac{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} = 1 + 2{n(n-1)},

where

: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:

:{| |- align="center" valign="middle" style="line-height: 0;" |[[Image:BlackDot.svg|16px]] |
|
|[[Image:RedDot.svg|16px]] |
|
|[[Image:RedDot.svg|16px]] |
|
|[[Image:RedDot.svg|16px]] |- align="center" valign="top" | 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).

Another way the centered square numbers can be expressed is:

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

where

:\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:

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

where

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

List of centered square numbers

The first centered square numbers (C4,n

: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, … .

Properties

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 a consequence of (2n − 1)2 + (2n2 − 2n)2 = (2n2 − 2n + 1)2.

Generating function

The generating function that gives the centered square numbers is: :\frac{(x+1)^2}{(1-x)^3}= 1+5x+13x^2+25x^3+41x^4+....

References

• {{citation
• .
• {{citation
• {{citation | author1-link = John Horton Conway | author2-link = Richard K. Guy