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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.

%I A357946 #57 Sep 01 2023 03:58:23
%S A357946 1,6,8,20,21,40,40,66,65,98,96,136,133,180,176,230,225,286,280,348,
%T A357946 341,416,408,490,481,570,560,656,645,748,736,846,833,950,936,1060,
%U A357946 1045,1176,1160,1298,1281,1426,1408,1560,1541,1700,1680,1846,1825,1998,1976
%N 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.
%C A357946 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.
%C A357946 The even-indexed terms are the positive octagonal numbers (cf. A000567) and are lined up in a straight line.
%H A357946 Nicolay Avilov, <a href="/A357946/a357946.jpg">Drawing with the beginning of the route</a>,
%H A357946 Nicolay Avilov, <a href="https://www.diofant.ru/problem/4198/">Problem 2403. Sequence in the Pythagorean table</a> (in Russian).
%H A357946 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A357946 a(n) = (3*n^2 + 8*n + 4)/4 if n is an even number,
%F A357946 a(n) = (3*n^2 + 16*n + 5)/4 if n is an odd number.
%F A357946 a(n) = (6*n + 3 + (-1)^n)*(2*n + 7 - 3*(-1)^n)/16, where n is any natural number.
%F A357946 a(n) = A001651(n+1)*A052938(n).
%e A357946 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:
%e A357946 a(0) = 1*1 =  1,
%e A357946 a(1) = 2*3 =  6,
%e A357946 a(2) = 4*2 =  8,
%e A357946 a(3) = 5*4 = 20.
%Y A357946 Cf. A001651 (route abscissas), A052938 (route ordinates).
%Y A357946 Cf. A000567, A003991 (multiplication table)
%K A357946 nonn,easy
%O A357946 0,2
%A A357946 _Nicolay Avilov_, Oct 21 2022