Some formulas listed in Mathworld

A golden example (formula (34) here)

A BBPP (for BBP+Pochhammer) formula for Pi (formula (57) here)

A double series for zeta(2) (formula (8) here)

A limit formula for log(2) involving all zeta numbers (formula (129) here)

A formula for zeta(3) rediscovered (formula (10) here)

A series for log(2) (formula (45) here)

An identity for generalised harmonic numbers H(n,2) (formula (33) here)

A limit formula for the gamma constant (formula (36) here)

A false Lehmer cotengent expansion for Phi (formula (4) here)

Some formulas listed in the OEIS

1's counting sequence

Asymptotic formula for Motzkin numbers

Bisection of Fibonacci numbers

Bisection of Lucas numbers

Fibonacci(2n+1) - 1

Fibonacci(6n+3)/2

(Fibonacci(3n+2)-1)/4

Generalized Fibonacci numbers

Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-4).

Binomial(n,4)

Binomial(n,5)

Binomial(3n,n)

A001519

Asymptotic formula for 3-smooth numbers

Central trinomial coefficients

Narayana's cows sequence

Little Schroeder numbers

Large Schroeder numbers

Numbers n such that 2*n^2 - 1 is a square

Kolakoski sequence

Fermat numbers

a(n)^2 is a triangular number

n-th number without consecutive equal binary digits

Central Delannoy numbers

Number of graphs on n unlabeled nodes

Number of partitions of n into at most 3 parts

NSW numbers

The multiplicative partition function

Fredholm-Rueppel sequence

a(n) = n*a(n-1) + (n-1)*a(n-2), a(0) = 1, a(1) = 1

Asymptotic for number of necklaces of partitions of n+1 labeled beads

Conjectured formula for the least k such that H(k)>n

Genocchi numbers of first kind

Pillai's arithmetical function: sum of gcd(k, n) for 1 = k = n

Number of ways of writing n as a sum of 4 squares

Number of elements in the set {(x,y): 1=x,y=n, gcd(x,y)=1}

Number of (positive) squarefree numbers n

Numbers whose ternary expansion contains no 1's

a(n) = least k such that n divides Fibonacci number F_k

Denominators of Farey tree fractions

Regular paper-folding sequence

Number of ternary (0,1,2) sequences without a consecutive '012'

Number of combinations of n things from 1 to n at a time, with repeats allowed

Number of cyclic subgroups of the group C_n X C_n (where C_n is the cyclic group of order n)

Minimal number of steps to get from 0 to n by (a) adding 1 or (b) multiplying by 2

Numerator of tan(2n arctan(1/2))

Denominator of tan(2n arctan(1/2))

Number of n X n matrices with nonnegative entries and every row and column sum 2.

Unique monotonic sequence of nonnegative integers satisfying a(a(n)) = 3n

Number of paths with n turns when light is reflected from 5 glass plates

Glaisher's G numbers

De Bruijn's S(4,n)

Number of minimax trees on n nodes