Sequences

Flavius Josephus's sieve: Start with the natural numbers; at the k-th sieving step, remove every (k+1)-st term of the sequence remaining after the (k-1)-st sieving step; iterate.

Powers of primes. Alternatively, 1 and the prime powers (p^k, p prime, k >= 1).

The convergent sequence A_n for the ternary continued fraction (3,1;2,2) of period 2.

The convergent sequence B_n for the ternary continued fraction (3,1;2,2) of period 2.

The convergent sequence C_n for the ternary continued fraction (3,1;2,2) of period 2.

Numerators of expansion of e.g.f. sinh(x) / sin(x) (even powers only).

n! never ends in this many 0's.

Sum of Fermat coefficients.

Sum of odd Fermat coefficients rounded to nearest integer.

Expansion of g.f. (1 + x + 2*x^2)/((1 - x)^2*(1 - x^3)).

Fermat coefficients.

Fermat coefficients.

Fermat coefficients.

Fermat coefficients.

Conjecturally the number of even integers the sum of two primes in exactly n ways.

a(2n) = 2*a(2n-1), a(2n+1) = 2*a(2n)+1 (also a(n) is the n-th number without consecutive equal binary digits).

Period of 1/n! in base 10.

Numbers that are divisible by at least three different primes.

Wagstaff numbers: numbers k such that (2^k + 1)/3 is prime.

Wagstaff primes: primes of form (2^p + 1)/3.

Number of ways of writing 0 as Sum_{k=-n..n} e(k)*k, where e(k) is 0 or 1.

Numbers beginning with letter 'n' in English.

a(n) = ceiling(n^2/2).

Size of minimal binary covering code of length n and covering radius 1.

Central binomial coefficients: binomial(2*n,n) = (2*n)!/(n!)^2.

Number of n X n symmetric matrices with nonnegative entries and all row sums 2.

Number of n X n symmetric matrices with (0,1) entries and all row sums 2.

Number of stochastic matrices of integers.

Number of one-sided polyominoes with n cells.

3-adic valuation of binomial(2*n, n): largest k such that 3^k divides binomial(2*n, n).

Number of plane partitions of n with at most two rows.

Number of 3-line partitions of n.

"Half-Catalan numbers": a(n) = Sum_{k=1..floor(n/2)} a(k)*a(n-k) with a(1) = 1.

Number of distinct quadratic residues mod 10^n; also number of distinct n-digit endings of base-10 squares.

Shifts 2 places left under binomial transform.

Shifts left two terms under the binomial transform.

Shifts 3 places left under binomial transform.

From a differential equation.

From a differential equation.

5-adic valuation of binomial(2*n,n): largest k such that 5^k divides binomial(2*n, n).

a(n) = least m such that if a/b < c/d where a,b,c,d are integers in [0,n], then a/b < k/m < c/d for some integer k.

Number of sublattices of index n in generic 3-dimensional lattice.

Number of dissections of a convex (n+2)-gon into triangles and quadrilaterals by nonintersecting diagonals.

Schroeder's second problem (generalized parentheses); also called super-Catalan numbers or little Schroeder numbers.

Number of nonequivalent dissections of an (n+2)-gon by nonintersecting diagonals up to rotation and reflection.

Number of ways of partitioning n points on a circle into subsets only of sizes 2 and 3.

Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle.

a(n) = ( Sum C(p,i); i=1,...,floor(2p/3) ) / p^2, where p = prime(n).

a(n) = numerator of harmonic number H(n) = Sum_{i=1..n} 1/i.

Triangle giving number L(n,k) of normalized k X n Latin rectangles.