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Home / $ b_n = c_{n-1} $ (can only append 3 if previous digit is not 3)

$ b_n = c_{n-1} $ (can only append 3 if previous digit is not 3)

February 22, 2026

$$ b_n = c_{n-1} $ (can only append 3 if previous digit is not 3)$

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$ k = 3 $: $ -\binom{3}{3} \cdot 0^7 = -1 \cdot 0 = 0 $
$ b_n $: sequences ending in 3
$ c_n $: sequences ending in 7
$ c_n = a_{n-1} $ (append 7 to any valid sequence)
$ a_n = a_{n-1} + a_{n-2} $ — standard Fibonacci.
$ n = 1 $: sequences: "3", "7" → both valid → $ a_1 = 2 $

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