

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



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
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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 90degree turns left and right.
The evenindexed terms are the positive octagonal numbers (cf. A000567) and are lined up in a straight line.


LINKS



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.


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



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



