Centered triangular number

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A centered triangular number is a centered figurate number that represents a triangle with a dot in the center and all other dots surrounding the center in successive triangular layers. The centered triangular number for n is given by the formula

${{3n^2 + 3n + 2} \over 2}.$

The following image shows the building of the centered triangular numbers using the associated figures: at each step the previous figure, shown in red, is surrounded by a triangle of new points, in blue.

The first few centered triangular numbers (sequence A005448 in OEIS) are

1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, 1786, 1891, 1999, 2110, 2224, 2341, 2461, 2584, 2710, 2839, 2971

Each centered triangular number from 10 onwards is the sum of three consecutive regular triangular numbers. Also each centred triangular number has a remainder of 1 when divided by three and the quotient (if positive) is the previous regular triangular number.

The sum of the first n centered triangular numbers is the magic constant for an n by n normal magic square for n > 2.

[edit] Centered triangular prime

A centered triangular prime is a centered triangular number that is prime. The first few centered triangular primes are (sequence A125602 in OEIS)

19, 31, 109, 199, 409, ...

(corresponding to n: 3, 4, 8, 11, 16, ...)