Sequences

Smallest natural number k such that periodic part of 1/sqrt(k) is n (or 0 if no such number exists).

Theta series of lattice A_2 tensor A_15+ (dimension 30, det 3^15, min. norm 4).

Expansion of Sum_{n>=0} n*q^(n^2).

Expansion of ( Sum_{k>=0} k*q^(k^2) )^2.

Expansion of ( Sum_{k>=0} k*q^(k^2) )^3.

Expansion of ( Sum_{k>=0} k*q^(k^2) )^4.

Expansion of ( Sum_{k>=0} k*q^(k^2) )^8.

Smallest natural number k such that periodic part of 1/k is the prime p(n), or 0 if no such k exists.

Theta series for 10-dimensional 4-modular lattice Q10 with minimal norm 4.

Summarize the previous term!.

Number of near-rings (or nearrings) definable on cyclic group of order n.

Decimal expansion of Pi*e^2.

Number of solutions to non-attacking rooks problem on n X n board that are invariant under 180-degree rotation.

Number of permutations p of {1,2,3...,n} that are fixed points under the operation of first reversing p, then taking the inverse.

a(n) = phi(2n+1).

a(n) = phi(2n+1) / multiplicative order of 2 mod 2n+1.

If n = 2^m*k, k odd, then a(n) = 2*m+1.

a(n) = phi(n) - pi(n).

Numbers k such that pi(k) >= phi(k).

Numbers k such that phi(k) < pi(k).

Primes which set a new record for length of Pratt certificate.

Number of matchings in graph C_{3} X C_{3} X P_{n}.

Order of (4,n) cage, i.e., minimal order of 4-regular graph of girth n.

a(n) = number of rings with n elements.

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

Expansion of (3+2*x^2)/(1-x)^4.

Expansion of (3 + x^2) / (1 - x)^4.

5x - 1 sequence starting at 19 (a(n+1) = a(n)/2 if a(n) is even, or 5*a(n)-1 if a(n) is odd).

Numerator of Pi^(2n)/(Gamma(2n)*(1-2^(-2n))*Zeta(2n)); = 8*(highest power of 2 dividing n).

Molien series for 3-D group X1.

Expansion of (1+3*x^2+7*x^3+15*x^4+13*x^5+15*x^6+8*x^7+4*x^8)/((1-x)*(1-x^2)^3*(1-x^3)^2).

Molien series for 3-D group R2+R3.

Molien series for 3-D group R4.

Base 100 expansion of Pi.

Number of unrooted self-avoiding walks of n steps on square lattice.

Total number of fixed points in free homeomorphically irreducible trees with n nodes.

Reversion of g.f. (beginning with x term) for number of trees with n nodes.

a(n) = n^6*(n^2 - 1)*(n^6 - 1).

a(n) = n^24*(n^2-1)*(n^6-1)*(n^8-1)*(n^12-1).

a(n) = n^2*(n^2 + 1)*(n-1).

a(n) = n^3*(n^3 + 1)*(n-1).

a(n) = n^12*(n^6 + 1)*(n^4 - 1)*(n^3 + 1)*(n-1).

a(n) = n^12*(n^8+n^4+1)*(n^6-1)*(n^2-1).

Triangle read by rows: T(n,k) (n >= 1, 1 <= k< = n) gives number of non-distorting tie-avoiding integer vote weights.

For n weights, number of combinations when limited to two weights per pan.

a(n) = n!*Sum_{i=0..n-1} (n-i)*(-2)^i/(i+1)!.

a() = 1,3,... [ A037257 ], differences = 2,... [ A037258 ] and 2nd differences [ A037259 ] are disjoint and monotonic; adjoin next free number to 2nd differences unless it would produce a duplicate in which case ignore.

First differences of A037257.

Second differences of A037257.

a()=A037260 and its first [ A037261 ], 2nd [ A037262 ] and 3rd [ A037263 ] differences together include every number at most once and are monotonic and minimal.