Ramus's Identity/Examples/k = 1, m = 3
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Example of Ramus's Identity
| \(\ds \sum_{j \mathop \ge 0} \binom n {3 j + 1}\) | \(=\) | \(\ds \binom n 1 + \binom n 4 + \binom n 7 + \cdots\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 3 \paren {2^n + 2 \cos \frac {\paren {n - 2} \pi} 3}\) |
Proof
From :
| \(\ds \sum_{j \mathop \ge 0} \binom n {3 j + 1}\) | \(=\) | \(\ds \frac 1 3 \sum_{0 \mathop \le j \mathop < 3} \paren {2 \cos \frac {j \pi} 3}^n \cos \frac {j \paren {n - 2} \pi} 3\) | Ramus's Identity: $k = 1, m = 3$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 3 \paren {\paren {2 \cos 0}^n \cos 0 + \paren {2 \cos \frac \pi 3}^n \cos \frac {\paren {n - 2} \pi} 3 + \paren {2 \cos \frac {2 \pi} 3}^n \cos \frac {2 \paren {n - 2} \pi} 3}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 3 \paren {2^n + \paren {2 \cos \frac \pi 3}^n \cos \frac {\paren {n - 2} \pi} 3 + \paren {2 \cos \frac {2 \pi} 3}^n \cos \frac {2 \paren {n - 2} \pi} 3}\) | Cosine of Zero is One | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 3 \paren {2^n + \paren {2 \paren {\frac 1 2} }^n \cos \frac {\paren {n - 2} \pi} 3 + \paren {2 \cos \frac {2 \pi} 3}^n \cos \frac {2 \paren {n - 2} \pi} 3}\) | Cosine of $60^\circ$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 3 \paren {2^n + \cos \frac {\paren {n - 2} \pi} 3 + \paren {2 \paren {-\frac 1 2} }^n \cos \frac {2 \paren {n - 2} \pi} 3}\) | Cosine of $120^\circ$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 3 \paren {2^n + \cos \frac {\paren {n - 2} \pi} 3 + \paren {-1}^n \cos \frac {2 \paren {n - 2} \pi} 3}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 3 \paren {2^n + \cos \frac {\paren {n - 2} \pi} 3 + \paren {-1}^{2 - n} \cos \frac {2 \paren {n - 2} \pi} 3}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 3 \paren {2^n + \cos \frac {\paren {n - 2} \pi} 3 + \cos \frac {2 \paren {n - 2} \pi + \paren {2 - n} \pi} 3}\) | Cosine of Angle plus Integer Multiple of Pi | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 3 \paren {2^n + \cos \frac {\paren {n - 2} \pi} 3 + \cos \frac {-\paren {n - 2} \pi} 3}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 3 \paren {2^n + 2 \cos \frac {\paren {n - 2} \pi} 3}\) | Cosine Function is Even |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $38$