Sequences

Add 1, multiply by 1, add 2, multiply by 2, etc.; start with 0.

Add 1, multiply by 1, add 2, multiply by 2, etc., start with 3.

Add 1, multiply by 1, add 2, multiply by 2, etc., start with 1.

Multiply by 1, add 1, multiply by 2, add 2, etc., start with 1.

Multiply by 1, add 1, multiply by 2, add 2, etc., start with 2.

Multiply by 1, add 1, multiply by 2, add 2, etc.; start with 3.

(n-2)nd Catalan number is congruent to n/3 mod n.

(n-2)-th Catalan number is congruent to 2n/3 mod n.

Numbers k such that k does not divide binomial(2*k-4, k-2).

Numbers k that divide binomial(2*k-4, k-2).

Maximum number of states needed to accept an n-th order approximation to a language over (0+1)^*, n=1..infinity.

Weak preference orderings of n alternatives, i.e., orderings that have indifference between at least two alternatives.

Number of stable n-celled patterns ("still lifes") in Conway's Game of Life, up to rotation and reflection.

Continued fraction expansion of W(1), where W(x) is the Lambert W function (the root of w*exp(w) = x).

Define the sequence S(a(0),a(1)) by a(n+2) is the least integer such that a(n+2)/a(n+1) > a(n+1)/a(n) for n >= 0. This is S(2,10).

a(n) = 5*a(n-1) + a(n-2) - 2*a(n-3).

Define the sequence S(a(0),a(1)) by a(n+2) is the least integer such that a(n+2)/a(n+1) > a(n+1)/a(n) for n >= 0. This is S(3,15) (agrees with A019478 only for n <= 23).

a(n) = 5*a(n-1) + a(n-2) - 3*a(n-3).

Define the sequence S(a_0,a_1) by a_{n+2} is the least integer such that a_{n+2}/a_{n+1}>a_{n+1}/a_n for n >= 0. This is S(4,8).

Define the sequence S(a(0),a(1)) by a(n+2) is the least integer such that a(n+2)/a(n+1) > a(n+1)/a(n) for n >= 0. This is S(4,12) (agrees with A019481 for n <= 19 only).

a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3) (agrees with A019480 for n <= 19 only).

Define the sequence S(a(0),a(1)) by a(n+2) is the least integer such that a(n+2)/a(n+1) > a(n+1)/a(n) for n >= 0. This is S(4,28).

Expansion of 1/((1-4x)(1-6x)(1-10x)).

Expansion of (8 + 7 x - 7 x^2 - 7 x^3)/(1 - 6 x - 7 x^2 + 5 x^3 + 6 x^4).

a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3), with a(0) = 2, a(1) = 5, a(2) = 12.

a(n) = 2*a(n-1) + a(n-2) - a(n-4) - a(n-5) - a(n-6) - a(n-7).

a(n) = 3*a(n-1) - a(n-2) + 2*a(n-3) - 2*a(n-4).

Expansion of 1/((1-4*x)*(1-6*x)*(1-11*x)).

Define the sequence T(a(0),a(1)) by a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n) for n >= 0. This is T(3,7).

Expansion of 1/((1-4*x)*(1-6*x)*(1-12*x)).

Numbers n for which number of distinct prime divisors of binomial(n,k) has local minimum at k = n/2.

Pisot sequence T(4,9), a(n) = floor(a(n-1)^2/a(n-2)).

a(n) = 3*a(n-1) - 4*a(n-3) + a(n-6).

Define the sequence T(a(0),a(1)) by a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n) for n >= 0. This is T(4,10).

Define the sequence T(a(0),a(1)) by a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n) for n >= 0. This is T(4,11).

a(n) = 3*a(n-1) - 3*a(n-3) + 2*a(n-4), with a(0)=4, a(1)=11.

Number of ternary search trees on n keys.

Number of 4-ary search trees on n keys.

Number of 5-ary search trees on n keys.

Number of 6-ary search trees on n keys.

Number of 7-ary search trees on n keys.

Number of simplices in minimal decomposition of an n-cube.

Simplexity of the n-cube: minimal cardinality of triangulation of n-cube using n-simplices whose vertices are vertices of the n-cube.

Number of simplices in minimal corner-slicing triangulation of n-cube.

a(1)=1; for n > 1, a(n) is the smallest number with the same number of divisors as 2*a(n-1).

Hoax numbers: composite numbers whose digit-sum equals the sum of the digit-sums of its distinct prime factors.

Droll numbers: numbers > 1 whose sum of even prime factors equals the sum of odd prime factors.

X^m=X rings without normal forms: integers m > 1 for which there exist a prime p and integers a,b > 0 such that both p^a-1 and p^b-1 divide m-1 but p^lcm(a,b)-1 does not divide m-1.

Nim-values for the impartial game Take-a-Triangle.

a(n) = gcd( binomial(n+3, n) + binomial(n+4, n+1), binomial(n+5, n+2) ).