Properties of Generalized (r,s,t,u)-Numbers

Abstract

In this paper, we investigate the generalized (r,s,t,u) sequence and we deal with, in detail, three special cases which we call them (r,s,t,u), Lucas (r,s,t,u) and modified (r,s,t,u) sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.

On the Recurrence Properties of Generalized Tribonacci Sequence

“…where W 0 ; W 1 ; W 2 are arbitrary complex (or real) numbers and r; s; t are real numbers. This sequence has been studied by many authors, see for example [3] and references therein. The sequence fW n g n 0 can be extended to negative subscripts by de…ning…”

“…In literature, for example, the following names and notations (see Table 2) are used for the special case of r; s; t and initial values. (3) n g = fWn(0; 1; 2; 2; 1; 1)g A077939, A077978…”

“…8 third order Pell-Lucas fQ (3) n g = fWn(3; 2; 6; 2; 1; 1)g A276225, A276228 9 third order m odi…ed Pell fE (3) n g = fWn(0; 1; 1; 2; 1; 1)g A077997, A078049 third order Pell-Perrin fR (3) n g = fWn(3; 0; 2; 2; 1; 1)g Padovan (Cordonnier) fPng = fWn(1; 1; 1; 0; 1; 1)g A000931 Perrin (Padovan-Lucas) fEng = fWn(3; 0; 2; 0; 1; 1)g A001608, A078712 Padovan-Perrin fSng = fWn(0; 0; 1; 0; 1; 1)g A000931, A176971 m odi…ed Padovan fAng = fWn(3; 1; 3; 0; 1; 1)g adjusted Padovan fUng = fWn(0; 1; 0; 0; 1; 1)g Pell-Padovan fRng = fWn(1; 1; 1; 0; 2; 1)g A066983, A128587 Pell-Perrin fCng = fWn(3; 0; 2; 0; 2; 1)g third order Fib onacci-Pell fGng = fWn(1; 0; 2; 0; 2; 1)g third order Lucas-Pell fBng = fWn(3; 0; 4; 0; 2; 1)g adjusted Pell-Padovan fMng = fWn(0; 1; 0; 0; 2; 1)g Jacobsthal-Padovan fQng = fWn(1; 1; 1; 0; 1; 2)g A159284 Jacobsthal-Perrin (-Lucas) fLng = fWn(3; 0; 2; 0; 1; 2)g A072328 adjusted Jacobsthal-Padovan fKng = fWn(0; 1; 0; 0; 1; 2)g m odi…ed Jacobsthal-Padovan fMng = fWn(3; 1; 3; 0; 1; 2)g Narayana fNng = fWn(0; 1; 1; 1; 0; 1)g A078012 Narayana-Lucas fUng = fWn(3; 1; 1; 1; 0; 1)g A001609 Narayana-Perrin fHng = fWn(3; 0; 2; 1; 0; 1)g third order Jacobsthal fJ (3) n g = fWn(0; 1; 1; 1; 1; 2)g A077947 third order Jacobsthal-Lucas fj (3) n g = fWn(2; 1; 5; 1; 1; 2)g A226308 m o di…ed third order Jacobsthal-Lucas fK (3) n g = fWn(3; 1; 3; 1; 1; 2)g third order Jacobsthal-Perrin fQ (3) n g = fWn(3; 0; 2; 1; 1; 2)g 3-prim es fGng = fWn(0; 1; 2; 2; 3; 5)g Lucas 3-prim es fHng = fWn(3; 2; 10; 2; 3; 5)g m odi…ed 3-prim es fEng = fWn(0; 1; 1; 2; 3; 5)g reverse 3-prim es fNng = fWn(0; 1; 5; 5; 3; 2)g reverse Lucas 3-prim es fSng = fWn(3; 5; 31; 5; 3; 2)g reverse m odi…ed 3-prim es fUng = fWn(0; 1; 4; 5; 3; 2)g…”

On the Recurrence Properties of Generalized Tribonacci Sequence

“…where W 0 , W 1 , W 2 are arbitrary complex (or real) numbers and r, s, t are real numbers. This sequence has been studied by many authors, see for example [1,2,3,4,5,7,9,10,11,13,22,24,25].…”

Recurrence Properties of Generalized Hexanacci Sequence

“…for n = 1, 2, 3, .... Therefore, recurrence (1.1) holds for all integers n. Hexanacci sequence has been studied by many authors, see for example [1,2,3] and references therein.…”