(The Mathematics of poetry5) The Indian writer Acharya Hemachandra

(c. 1150 AD) studied the rhythms of Sanskrit poetry. Syllables in Sanskrit are either long (L)

or short (S). Long syllables have twice the length of short syllables. The question he asked is how many rhythm patterns with a given total length can be formed by a sequence of short and long syllables?

For instance, the number of patterns that have the length of four short syllables is 4: SSSS, SSL, SLS, LSS.
(i) For 1 ≤ k ≤ 7, find the number rk of patterns with total length k.
(ii) Prove that, for k = 2, 3,…, the following identity holds:

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Tk=rk-1+rk-2