cp's OEIS Frontend

Odd numbers (A005408) written clockwise as a square spiral:

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41--43--45--47--49--51

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39 13--15--17--19 53

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37 11 1---3 21 55

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35 9---7---5 23 57

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33--31--29--27--25 59

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71--69--67--65--63--61

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Walking in straight lines away from the center:

1, 17, 49, ... = A069129(n+1) = 1 - 8*n + 8*n^2,

1, 3, 21, ... = A033567(n) = 1 - 6*n + 8*n^2,

1, 15, 45, ... = A014634(n) = 1 + 6*n + 8*n^2,

1, 5, 25, ... = A080856(n) = 1 - 4*n + 8*n^2,

1, 13, 41, ... = A102083(n) = 1 + 4*n + 8*n^2,

1, 7, 29, ... = a(n) = 1 - 2*n + 8*n^2,

1, 11, 37, ... = A188135(n) = 1 + 2*n + 8*n^2,

1, 9, 33, ... = A081585(n) = 1 + 8*n^2,

5, 29, 69, ... = A108928(n+1) = -3 + 8*n^2,

7, 31, 71, ... = A157914(n+1) = -1 + 8*n^2,

9, 35, 77, ... = A033566(n+1) = -1 + 2*n + 8*n^2.

All are quadrisections of sequences in A181407(n) (example: A014634(n) and A033567(n) in A064038(n+1)) or of this family (?): a(n) is a quadrisection of f(n) = 1,1,1,1,2,7,11,8,11,29,37,23,28,67,79,46,... f(n) is just before A064038(n+1) (fifth vertical) in A181407(n). The companion to a(n) is A188135(n), another quadrisection of f(n). Two last quadrisections of f(n) are A054552(n) and A033951(n).

For n >= 1, bisection of A193867. - Omar E. Pol, Aug 16 2011

Also the sequence may be obtained by starting with the segment (1, 7) followed by the line from 7 in the direction 7, 29, ... in the square spiral whose vertices are the generalized hexagonal numbers (A000217). - Omar E. Pol, Aug 01 2016

First quadrisection of A176126(n). Take clockwise (square) spiral from A023443(n)=n-1: a(n) is on the negative x-axis. Fourth quadrisection (-1-n+4*n^2) is on the negative y-axis.

Conjecture: the 4 quadrisections of (the family) A064038, A160050, A176126, A178242 (see A181407) come from square spiral.

a(n) mod 9 has period 9: 8,4,8,2,4,5,5,4,2. a(n) mod 10 has period 10: 9,4,7,8,7,4,9,2,3,2. Each polynomial modulo some constant c has a period of length c (and perhaps shorter ones). - Paul Curtz and Bruno Berselli, Feb 05 2011