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In his Vttajātisamuccaya (c. 600 AD), Virahāka discussed the prastāra or enumeration of mātrā-vttas or moric metres, metres of value n consisting of long and short syllables, laghu and guru, assigning value 1 to laghu and 2 to guru, the value of the metre being equal to the sum of the values of its constituent syllables. Virahāka noted that the sakhyākas or number of rows in the prastāra is given by a sequence of numbers, which were rediscovered much later in the 13th century by Fibonacci in a different context. Similar problems of enumeration were considered in the context of tāla-prastāra in music by Śārgadeva in his Sagītaratnākara (c. 1250 AD). A general mathematical treatment of most of the combinatorial problems considered in the earlier literature was given by Nārāyaa Paita in his work Gaitakaumudī (c.1356 AD). One of the problems discussed by Nārāyaa is the prastāra of a general class of mātrā-vttas where, apart from the syllabic units laghu and guru (of values 1, 2), there could be other syllabic units (such as pluta, etc.) of values 3,4,... , q. This is also a general form of tāla-prastāra, but it does not subsume the specific tāla-prastāra considered by Śārgadeva in Sagītaratnākara, where the tāla-units have values 1, 2, 4 and 6. The questions of naa and uddia, which consist of finding the row number associated with a generalised mātrā-vtta and its converse, in relation to this prastāra, have been dealt with by Nārāyaa Paita by means of a tabular form called unmeru. In this paper, we shall show that these naa and uddia processes are indeed based upon a certain representation of each number uniquely as a sum of the generalised Virahāka-Fibonacci numbers, i.e., sakhyākas associated with the generalised mātrā-vttaprastāra of Nārāyaa.

Nārāyaṇā’s Generalisation of Mātrā-Vṛtta-Prastāra and the Generalised Virahāṅka-Fibonacci Representation of Numbers