cp's OEIS Frontend

See A093883 for a discussion and guide to related sequences.

From Clark Kimberling, Jan 02 2013: (Start)

Each term divides its successor, as in A006963, and by the corresponding superfactorial, A000178(n), as in A203469.

Abbreviate "Vandermonde" as V. The V permanent of a set S={s(1),s(2),...,s(n)} is a product of sums s(j)+s(k) in analogy to the V determinant as a product of differences s(k)-s(j). Let D(n) and P(n) denote the V determinant and V permanent of S, and E(n) the V determinant of the numbers s(1)^2, s(2)^2, ..., s(n)^2; then P(n) = E(n)/D(n). This is one of many divisibility properties associated with V determinants and permanents. Another is that if S consists of distinct positive integers, then D(n) divides D(n+1) and P(n) divides P(n+1).

Guide to related sequences:

...

s(n).............. D(n)....... P(n)

n................. A000178.... (this)

n+1............... A000178.... A203470

n+2............... A000178.... A203472

n^2............... A202768.... A203475

2^(n-1)........... A203303.... A203477

2^n-1............. A203305.... A203479

n!................ A203306.... A203482

n(n+1)/2.......... A203309.... A203511

Fibonacci(n+1).... A203311.... A203518

prime(n).......... A080358.... A203521

odd prime(n)...... A203315.... A203524

nonprime(n)....... A203415.... A203527

composite(n)...... A203418.... A203530

2n-1.............. A108400.... A203516

n+floor(n/2)...... A203430

n+floor[(n+1)/2].. A203433

1/n............... A203421

1/(n+1)........... A203422

1/(2n)............ A203424

1/(2n+2).......... A203426

1/(3n)............ A203428

Generalizing, suppose that f(x,y) is a function of two variables and S=(s(1),s(2),...s(n)). The phrase, "Vandermonde sequence using f(x,y) applied to S" means the sequence a(n) whose n-th term is the product f(s(j,k)) : 1<=j

...

If f(x,y) is a (bivariate) cyclotomic polynomial and S is a strictly increasing sequence of positive integers, then a(n) consists of integers, each of which divides its successor. Guide to sequences for which f(x,y) is x^2+xy+y^2 or x^2-xy+y^2 or x^2+y^2:

...

s(n) ............ x^2+xy+y^2.. x^2-xy+y^2.. x^2+y^2

n ............... A203012..... A203312..... A203475

n+1 ............. A203581..... A203583..... A203585

2n-1 ............ A203514..... A203587..... A203589

n^2 ............. A203673..... A203675..... A203677

2^(n-1) ......... A203679..... A203681..... A203683

n! .............. A203685..... A203687..... A203689

n(n+1)/2 ........ A203691..... A203693..... A203695

Fibonacci(n) .... A203742..... A203744..... A203746

Fibonacci(n+1) .. A203697..... A203699..... A203701

prime(n) ........ A203703..... A203705..... A203707

floor(n/2) ...... A203748..... A203752..... A203773

floor((n+1)/2) .. A203759..... A203763..... A203766

For f(x,y)=x^4+y^4, see A203755 and A203770. (End)