Centered square number

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Each centered square number is the sum of successive squares. Example: as shown in the following figure of Floyd's triangle, 25 is a centered square number, and is the sum of the square 16 (yellow rhombus formed by shearing a square) and of the next smaller square, 9 (sum of two blue triangles):

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:

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 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.

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,
  MR 1571197
  .
• Zbl 0335.10001
  .
• Beiler, A. H. (1964), Recreations in the Theory of Numbers, New York: Dover, p. 125.
• MR 1411676
  .