mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a heptagonal number is a

figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The ancient Greek mathemat ...

that is constructed by combining

heptagon In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using ''Wikt:septa-, septa-'' (an elision of ''Wikt:septua-, septua-''), a Latin-derived numerical prefix, rather than ...

s with ascending size. The ''n''-th heptagonal number is given by the formula :

H_n=\frac

The first few heptagonal numbers are: : 0, 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, …

Parity

The parity of heptagonal numbers follows the pattern odd-odd-even-even. Like

square number In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...

s, the

digital root The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit su ...

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 triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...

Additional properties

* The heptagonal numbers have several notable formulas: :

H_=H_m+H_n+5mn

H_=H_m+H_n-5mn+3n

H_m-H_n=\frac

40H_n+9=(10n-3)^2

Sum of reciprocals

A formula for the sum of the reciprocals of the heptagonal numbers is given by: :

\begin\sum_^\infty \frac 
&= \frac+\frac\ln(5)+\frac\ln\left(\frac\sqrt\right)+\frac\ln\left(\frac\sqrt\right)\\
&=\frac13\left(\frac+\frac52\ln(5) -\sqrt5 \ln(\phi)\right)\\
&=1.3227792531223888567\dots
\end

with

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...

\phi = \tfrac2

Heptagonal roots

In analogy to the

square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...

of ''x, ''one can calculate the heptagonal root of ''x'', meaning the number of terms in the sequence up to and including ''x''. The heptagonal root of ''x '' is given by the formula :

n = \frac,

which is obtained by using the

quadratic formula In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations, such as completing the square, yield the same solutions. Given a general quadr ...

to solve

x = \frac

for its unique positive root ''n''.

References

{{series (mathematics) Figurate numbers