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Home » From Child to Adult: These Happy Birthday Song Lyrics Hit Different—Learn Why Now! - Simpleprint

From Child to Adult: These Happy Birthday Song Lyrics Hit Different—Learn Why Now! - Simpleprint

📅 March 16, 2026 👤 scraface

Mar 16, 2026

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📌 00000, 00001, 00010, 00100, 00101, 01000, 01001, 01010, 10000, 10001, 10010, 10100, 10101 → 13

📌 So \(a_5 = 13\)

📌 But this counts **classifications** — each position gets H or S, no two adjacent H’s.

📌 Since the substations are in a fixed line (distinct positions), and each is assigned one mode, the total number is exactly the number of such binary strings: 13.

📌 Wait — but earlier we computed \(\binom{n-k+1}{k} = \binom{5-4+1}{4} = \binom{2}{4} = 0\) — invalid.

📌 Mistake: formula \(\binom{n-k+1}{k}\) is for placing \(k\) non-adjacent indistinct items? No — standard formula for number of binary strings of length \(n\) with no two consecutive 1’s is \(F_{n+2}\), where \(F_1=1, F_2=1, F_3=2, F_4=3, F_5=5, F_6=8, F_7=13, F_8=21\)

📌 So \(F_{5+2} = F_7 = 13\) — matches.

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Expand the numerator:
Adding them:
The denominator is:
\frac{2a^2 + 2b^2}{a^2 - b^2} = 3
Factor out 2 in the numerator:
\frac{2(a^2 + b^2)}{a^2 - b^2} = 3
\frac{a^2 + b^2}{a^2 - b^2} = \frac{3}{2}
Question: Let $ g(x^2 + 1) = x^4 + 2x^2 + 2 $. Find $ g(x^2 - 1) $.
g(x^2 + 1) = x^4 + 2x^2 + 2
Let’s express the right-hand side in terms of $ x^2 + 1 $.

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