A heptagonal number is a figurate number that represents a heptagon. The n-th heptagonal number is given by the formula
.
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, 1918, 2059, 2205, 2356, 2512, 2673, 2839, 3010, 3186, 3367, 3553, 3744, 3940, 4141, 4347, 4558, 4774, 4995, 5221, 5452, 5688 (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
where Tn 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]
References
- ^ B. Srinivasa Rao, "Heptagonal Numbers in the Pell Sequence and Diophantine equations
" Fib. Quart. 43 3: 194
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