Centered square number

25 spheres in the form of nested squares

A centered square number is a number that is the sum of two consecutive square numbers . For example, is a centered square number. The first centered square numbers are ${\ displaystyle 13 = 4 + 9 = 2 ^ {2} + 3 ^ {2}}$

1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, ... (sequence A001844 in OEIS )

A centered square number indicates a number of stones, so that a stone in the middle is surrounded by other stones so that they form a square.


1 stone	5 stones	13 stones	25 stones

The -th centered square number is calculated using the formula ${\ displaystyle n}$ ${\ displaystyle ZQ_ {n}}$

{\ displaystyle ZQ_ {n} = 2n ^ {2} -2n + 1}

Relationships to other figured numbers

Triangular numbers

The -th centered square number is an odd number that is one greater than four times the -th triangular number . ${\ displaystyle n}$ ${\ displaystyle (n-1)}$

{\ displaystyle ZQ_ {n} = 1 + 4 \ Delta _ {n-1} = 1 + 4 \ cdot {\ frac {n (n-1)} {2}}}

Every -th centered square number can also be generated as the sum of four triangular numbers with three consecutive indices: ${\ displaystyle n}$

{\ displaystyle ZQ_ {n} = \ Delta _ {n-2} +2 \ Delta _ {n + 1} + \ Delta _ {n} = {\ frac {(n-2) (n-1)} { 2}} + 2 \ cdot {\ frac {(n-1) n} {2}} + {\ frac {n (n + 1)} {2}}}

Apart from the 1, no centered square number is also a triangular number.

Square numbers

Each centered square number is the sum of two consecutive (decentralized) square numbers. This becomes obvious when you rearrange the calculation formula for the centered square numbers.

{\ displaystyle 2n ^ {2} -2n + 1 = (n-1) ^ {2} + n ^ {2}}

This fact can also be recognized by the pattern on which a centered square number is based:

Some centered square numbers are also square numbers, such as 25, 841, 28561, and 970225.

Others

Compared to the square numbers , which are always compound, there are also some prime numbers in the sequence of the centered square numbers . The first prime centered square numbers are

5, 13, 41, 61, 113, 181, 313, 421, 613, 761, 1013, ... (sequence A027862 in OEIS ).

Modifications and generalizations

The centered triangle , centered pentagon , and centered hexagon numbers .

The decentralized square and hexagonal numbers or the polygonal numbers .

Octahedral numbers are the sums of the first centered square numbers (see figured number ).

Web links

Eric W. Weisstein : Centered Square Number . In: MathWorld (English).