A357946 a(n) is the number in the infinite multiplication table that the chess knight reaches in n moves, starting from the number 1, the angle between adjacent segments being 90 degrees alternately changing direction to the left and to the right.

1, 6, 8, 20, 21, 40, 40, 66, 65, 98, 96, 136, 133, 180, 176, 230, 225, 286, 280, 348, 341, 416, 408, 490, 481, 570, 560, 656, 645, 748, 736, 846, 833, 950, 936, 1060, 1045, 1176, 1160, 1298, 1281, 1426, 1408, 1560, 1541, 1700, 1680, 1846, 1825, 1998, 1976

Offset

0,2

Comments

The route of the chess knight is an endless zigzag broken line starting from (1,1) and taking steps alternately (+1,+2) and (+2,-1). Successive steps are 90-degree turns left and right.

The even-indexed terms are the positive octagonal numbers (cf. A000567) and are lined up in a straight line.

Links

Table of n, a(n) for n=0..50.

Nicolay Avilov, Drawing with the beginning of the route,

Nicolay Avilov, Problem 2403. Sequence in the Pythagorean table (in Russian).

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

a(n) = (3*n^2 + 8*n + 4)/4 if n is an even number,

a(n) = (3*n^2 + 16*n + 5)/4 if n is an odd number.

a(n) = (6*n + 3 + (-1)^n)*(2*n + 7 - 3*(-1)^n)/16, where n is any natural number.

a(n) = A001651(n+1)*A052938(n).

Example

The route of the chess knight (1,1)-(2,3)-(4,2)-(5,4)-(7,3)-(8,5)-(10,4)-(11,6)- ... by the cells of the multiplication table generates the beginning of this sequence, therefore:
a(0) = 1*1 =  1,
a(1) = 2*3 =  6,
a(2) = 4*2 =  8,
a(3) = 5*4 = 20.

Crossrefs

Cf. A001651 (route abscissas), A052938 (route ordinates).

Cf. A000567, A003991 (multiplication table)

Sequence in context: A199884 A028331 A279729 * A309653 A113806 A105775

Adjacent sequences: A357943 A357944 A357945 * A357947 A357948 A357949

Keyword

nonn,easy

Author

Nicolay Avilov, Oct 21 2022

Status

approved