cp's OEIS Frontend

Partial sums are A097063, whose pairwise sums are A002061.

Binomial transform is A097064.

Permutation of the natural numbers. a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.

Enumeration table T(n,k) by diagonals. The order of the list

if n is odd - T(n-1,2),T(n-3,4),...,T(2,n-1),T(1,n),T(3,n-2),...T(n,1).

if n is even - T(n-1,2),T(n-3,4),...,T(3,n-2),T(1,n),T(2,n-1),...T(n,1).

Table T(n,k) contains:

Column number 1 A000217,

column number 2 A000124,

column number 3 A000096,

column number 4 A152948,

column number 5 A034856,

column number 6 A152950,

column number 7 A055998.

Row number 1 A000982,

row number 2 A097063.

T(n,k) gives the maximum number of inversions in a permutation on n symbols containing a single k-cycle and (n-k) other fixed points.

T(n,n) = A000982(n).

T(n,n-1) = A097063(n).

Permutation of the natural numbers.

a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.

Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). The order of the list:

T(1,1)=1;

T(1,2), T(2,2), T(2,1);

. . .

T(1,n), T(2,n), ... T(n-1,n), T(n,n), T(n,n-1), ... T(n,1);

. . .

Permutation of the natural numbers.

a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.

Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Table read by boustrophedonic ("ox-plowing") method. Let m be natural number. The order of the list:

T(1,1)=1;

T(2,1), T(2,2), T(1,2);

. . .

T(1,2*m+1), T(2,2*m+1), ... T(2*m,2*m+1), T(2*m+1,2*m+1), T(2*m+1,2*m), ... T(2*m+1,1);

T(2*m,1), T(2*m,2), ... T(2*m,2*m-1), T(2*m,2*m), T(2*m-1,2*m), ... T(1,2*m);

. . .

The first row is layer read clockwise, the second row is layer counterclockwise.

Let G(n) be the graph whose vertices are sequences of length n with entries in {1,2,3} for which adjacent terms differ by +-1 and {s,t} is an edge if the i-th term of sequence s differs from the i-th term of sequence t by +-1. Let A be the adjacency matrix of this graph. Then a(n) is the sum of the entries in A^(n-1). That is, a(n) is the number of paths of length n-1 in the graph G(n). Conjecture: a(n) = 2^A097063(n) for n even and 2*2^A097063(n) if n is odd.

Proof that a(n) = 2^A097063(n) for n even and 2*2^A097063(n) if n is odd, by Max Alekseyev: Suppose that elements of the n X n matrix are colored in black and white like a chessboard. Then either all black or all white elements must equal 2. Each element of the other color can be 1 or 3 independently. For even n, the number of black and white elements is the same and equal n/2. For odd n, the number of black/white squares differs by 1. Therefore the number of n X n matrices defined in A146161 is 2*2^(n^2/2)=2^((n^2+2)/2) if n is even, and 2^((n^2+1)/2) + 2^((n^2-1)/2) = 3*2^((n^2-1)/2) if n is odd. The explicit formula for A097063 gives A097063(n) = (n^2+2)/2 for even n, and A097063(n) = (n^2-1)/2 for odd n. So the conjecture is true.

Same definition for k:

k+b(n)+n is a square for each term b(n) of A097063 except the first;

k+b(n)+n+1 is a square for each term b(n) of A007590 except the first;

k+b(n)+n is a cube for each term b(n) of the sequence 0, 7, 18, 43, 78, 133, 204, 301, 420, 571, 750, 967, 1218, 1513, 1848, 2233, 2664, 3151, 3690, 4291, 4950, 5677, 6468, 7333, ... (last digit repeats with period 10);

k+b(n)+n is a triangular number for each term b(n) of A002378 (oblong numbers);

k+b(n)+n is an oblong number for each term b(n) of A000217 (triangular numbers);

k+b(n)+n is a prime for each term b(n) of the sequence 0, 1, 2, 6, 7, 11, 12, 18, 21, 23, 26, 30, 31, 35, 40, 42, 43, 47, 48, 60, 69, 73, 78, 80, 87, 99, 102, 104, 107, 115, 118, 120, 125, 135, ...