Heptagonal number

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A heptagonal number is a figurate number that represents a heptagon. The n-th heptagonal number is given by the formula

$\frac{5n^2 - 3n}{2}$ .

The first few heptagonal numbers are:

1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970 (sequence A000566 in OEIS)

The parity of heptagonal numbers follows the pattern odd-odd-even-even. Like square numbers, the digital root in base 10 of a heptagonal number can only be 1, 4, 7 or 9. Five times a heptagonal number, plus 1 equals a triangular number.

A generalized heptagonal number is obtained by the formula

$T_n + T_{\lfloor \frac{n}{2} \rfloor},$

where T_n is the nth triangular number. The first few generalized heptagonal numbers are:

1, 4, 7, 13, 18, 27, 34, 46, 55, 70, 81, 99, 112 A085787

Every other generalized heptagonal number is a regular heptagonal number. Besides 1 and 70, no generalized heptagonal numbers are also Pell numbers.^[1]