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We derive, using matrices, new classes of linear recurrence identities linking the general term of the so called Horadam sequence with those of an associated 'cohort' sequence that differs only in its initial values. [ABSTRACT FROM AUTHOR]

Time spent surveying work on the so called Horadam sequence {wn}n=0∞ = {wn}0∞ = {wn(a, b; p, q)}0∞--defined by the linear recurrence wn = pwn-1 - qwn-2 in its most general form (n ≥ 2; arbitrary initial values w0 = a, w1 = b)--has identified considerable variation in the nature of results generated but also, more importantly, in the quality of papers containing them. It urges the following short opinion piece which discusses this observation and touches on some wider issues relating to research. [ABSTRACT FROM AUTHOR]

We state and prove a non-linear recurrence identity for terms of the so called Ho- radam sequence, and then offer its generalisation which is available from the same methodology. We illustrate how the overarching idea may be used to sequentially produce extended versions, each in turn with an extra level of non-linearity and term index complexity. These identities can all be captured in matrix determinant form. [ABSTRACT FROM AUTHOR]

We state, and prove using matrices, two (related) linear recurrence identities for terms of the so called Horadam sequence; each result expresses the general term of the sequence as a linear combination of terms with particular initial values. First offered by A.F. Horadam himself in the 1960s, our approach to their formulation is quite different and, we believe, new. [ABSTRACT FROM AUTHOR]

We state, and prove by a succinct matrix method, a non-linear recurrence identity class for terms of the so called Horadam sequence. A particular instance was established (in equivalent form) over half a century ago by A.F. Horadam, which provides a starting point for the discussion and an introduction to our formulation technique. [ABSTRACT FROM AUTHOR]