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

Author

Nicolay Avilov, Oct 21 2022

Keywords

• nonn
• easy

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.

Examples

			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.
		

Links

• 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).

Crossrefs

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

Cf. A000567, A003991 (multiplication table)

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).