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For questions about sequences of integers. References are often made to the online resource oeis.org.

8 votes
0 answers
87 views

Generalization of Lucas sequences to order 3 (and above)

For fixed integer parameters $(P,Q)$, Lucas sequences represent a pair of complimentary integer sequences satisfying the same recurrence with the characteristic polynomial $f(x):=x^2 - Px + Q$. The tw …
69 votes
1 answer
4k views

Iterations of $2^{n-1}+5$: the strong law of small numbers, or something bigger?

I've discovered what I believe is a quite remarkable sequence (A318970), defined by $$n_1 = 3,\qquad n_{k+1} = 2^{n_k-1}+5\quad(k\geq 1).$$ Here are the first four terms with their prime factorization …
4 votes
0 answers
299 views

Identities for powers of functions based on generalization of Lagrange interpolation

Lagrange polynomial can be used to obtain an identity: $$(k+t)^n = \sum_{i=0}^n (k+d_i)^n \prod_{\substack{j=0\\ j\not=i}}^n \frac{t-d_j}{d_i-d_j},$$ which holds for any integer $n>0$, any real number …