| | |} 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: where Yet another way the centered square numbers can be expressed is in terms of the centered triangular numbers: where List of centered square numbers The first centered square numbers (C4,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, … . 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 not to be confused with the relationship (n – 1)2 + n2 = C4,n. (Example: 22 + 32 = 13.) Generating function The generating function that gives the centered square numbers is: References . . . . Figurate numbers Quadrilaterals