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, 1071, 1177, 1288, 1404, 1525, 1651, 1782, … (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.
Generalized heptagonal numbers
A generalized heptagonal number is obtained by the formula
where Tn is the nth triangular number. The first few generalized heptagonal numbers are:
Sum of reciprocals
A formula for the sum of the reciprocals of the heptagonal numbers is given by:
Test for heptagonal numbers
Let y be a positive integer.
In analogy to the square root of x, one can calculate the heptagonal roots of x.
The positive heptagonal root of x are given by the formula:
Derivation of heptagonal root formula
The heptagonal roots n of x are derived by:
- (use quadratic formula to solve for n)
Rearrange this to:
Properties regarding the heptagonal roots
If , and
Now substitute with
- B. Srinivasa Rao, "Heptagonal Numbers in the Pell Sequence and Diophantine equations " Fib. Quart. 43 3: 194
- Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers