cp's OEIS Frontend

a(n) = largest odd divisor of A059100(n+1). Proof: Observe that a(2n) = A059100(2n+1) and a(2n+1) = (A059100(2n+2))/2 and note that (A059100(m))/2 is odd for even m. - Jeremy Gardiner, Aug 25 2013

a(n) is also the denominator of the (n+1)-st largest circle in a special case of the Pappus chain inspired by the Yin-Yang symbol. See illustration in the links. - Kival Ngaokrajang, Jun 20 2015

A 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). Enumeration table T(n,k) is layer by layer. The order of the list:

T(1,1)=1;

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

. . .

T(1,n), T(3,n), ... T(n,3), T(n,1), T(2,n), T(4,n), ... T(n,4), T(n,2);

...

The previous definition was "a(n) = normalized inverse radius of the inscribed circle that is tangent to the left circle of the symmetric arbelos and the n-th and (n-1)-st circles in the Pappus chain".

See links section for image of these circles, via Wolfram MathWorld (there an asymmetric arbelos is shown).

The Rothman-Fukagawa article has another picture of the circles, based on a Japanese 1788 sangaku problem. - N. J. A. Sloane, Jan 02 2020

Conjecture: a(n) exists for any n > 1.

Note that k*n - prime(k) < 0 if k > e^(n + 1).

This triangle is obtained from the array A(m, k) = m^2 + 2*k^2, for k and m >= 1, read by upwards antidiagonals. This array A is of interest for representing numbers as a sum of three non-vanishing squares with two squares coinciding.

For the numbers represented this way, see A154777. To find the actual values for m and k (taken positive), given a representable number from A154777, one can also use the number triangle T(n, k) = A(n-k+1, k).

To find the number of representations of value N (from A154777), it is sufficient to consider the rows n >= 1 not exceeding n_{max} = Floor(N, Min), where the sequence Min gives the minima of the numbers in each row: Min = {min(n)}_{n>=1} with min(n) = min(T(n, 1), T(n, 2), ..., T(n, n)) and Floor(N, Min) is the greatest member of Min not exceeding N.

Conjecture: min(n) = T(n, ceiling(n/3)), n >= 1. This is the sequence (n+1)^2 - ceiling(n/3)*(2*(n+1) - 3*ceiling(n/3)) = A071619(n+1) = ceiling((2/3)*(n+1)^2) = (n+1)^2 - floor((1/3)*(n+1)^2) = 3, 6, 11, 17, 24, 33, 43, .... (Proof of these identities by considering the three n (mod 3) cases.)

For the multiplicities of the representable values A154777(n), see A339047.

The author met this representation problem in connection with special triples of integer curvatures in the Descartes-Steiner five circle problem.

Iterate A028392, starting with n: a(n) is the number of steps until a square will be reached. - Reinhard Zumkeller, Feb 23 2012

Shifted variant of A102305. - R. J. Mathar, Oct 20 2011

a(n) is not A173642, a compact Bohr-Stoner model (1924), modified by Charles Janet in 1930. The good distribution is A168208.

Only sequences N16(n) in A234398 are used:

N16(1)= 1 followed by 2's = A040000,

N16(2)= 1, 2, 3, 4, 5, followed by 6's = A101272,

N16(3)= 1 to 9, followed by 10's,

N16(4)= 1 to 13, followed by 14's, etc.

The distribution by rows are in the example.

The N16(n)'s are respectively on columns (hence triangle T)

1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, A002620(n+2)

3, 5, 8, 11, 15, 19, 24, 29, 35, A024206(n+2)

7, 10, 14, 18, 23, 28, 34, A014616(n+3)

13, 17, 22, 27, 33, A004116(n+4)

21, 26, 32,

31, etc.

See A163255.

Antidiagonals give the natural numbers A000027, like rows sums in the example.

A033638=1, 1, 2, 3, 5, 7,... is upon the triangle T.

Refer to sequential curvatures from Wikipedia. For any integer k > 0, there exists an Apollonian gasket defined by the following curvatures:

(-k, k+1, k*(k+1), k*(k+1)+1).

For example, the gaskets defined by (-1, 2, 2, 3), (-2, 3, 6, 7), (-3, 4, 12, 13), ..., all follow this pattern (all curvatures are integral). Because every interior circle that is defined by k+1 can become the bounding circle (defined by -k) in another gasket, these gaskets can be nested. When one considers only circles that contact both circles -k and k+1, the pattern will be nested Pappus chains. T(n,k) is the curvature when n = 0 is the circle at the center and n > 0 is in the clockwise direction, k >= 1 for each nested iteration. See illustration in links.