Integer sequence

A divisibility sequence associated with multiples of a point on an elliptic curve, using suitably normalized division polynomials. With $W_0=0$ and $W_1=1$, its terms satisfy the Ward identity $W_{m+n}{W}_{m-n}=W_{m+1}{W}_{m-1}{W}_n^2-W_{n+1}{W}_{n-1}{W}_m^2$. Setting nearby indices gives doubling identities, enabling logarithmic-depth evaluation.

Starting with two distinct positive integers $a<b$, repeatedly append the smallest integer larger than the previous term that is expressible in exactly one way as the sum of two distinct earlier terms. Requiring distinct summands and counting unordered pairs are both essential parts of the definition.

The nondecreasing sequence of positive integers beginning with $G(1)=1$ in which the integer $n$ occurs exactly $G(n)$ times. Its first terms are $1,2,2,3,3,4,4,4,5,5,5,\dots$. The prefix sum ${\sum }_{j=1}^{k}G(j)$ gives the last position occupied by $k$.

Terms of a sequence satisfying the third-order linear recurrence $T_{n+3}=T_n+T_{n+1}+T_{n+2}$. Initial values vary by convention; some counting problems specify $T_1=T_2=T_3=1$. To study divisibility by $m$, retain three consecutive remainders modulo $m$, a deterministic state with at most $m^3$ possibilities.