A Novel Formula in Solving Tribonacci-like Sequence

In mathematical terms, sequence is a list of numbers arranged in a specific order. Tribonacci sequence is a sequence of whole numbers with equations T0 = 0, T1 = 0 and T2 = 1 in which Tn = Tn-1 + Tn-2 + Tn-3. When the first three terms of the tribonacci sequence become arbitrary, it is known as tribonacci-like sequence. Tribonacci-like sequence can start at any desired number. This study aimed to derive and validate a formula in solving tribonacci-like sequence. Basic equations were derived on finding nth term (Sn) of Tribonacci-like sequence given its arbitrary first three terms. Formulas of Sn for four terms (n = 4), five terms (n = 5), six terms (n = 6) and seven terms (n = 7) were solved. From these formulas, numerical coefficients of the first three terms were tabulated and patterns were recognized. In these patterns, a formula was obtained in solving nth term of Tribonacci-like sequence using its first three terms and Tribonacci numbers. The novel formula is Sn= Tn-2S1 + (Tn-2+Tn-3)S2+Tn-1S3 Where Sn = nth term of Tribonacci-like sequence S1 = first term S2 = second term S3 = third term Tn-1, Tn-2, Tn-3 = corresponding Tribonacci numbers.