cp's OEIS Frontend

This is the second of four polka dot arrays; see A185868.

row 1: A130883;

row 2: A100037;

row 3: A100038;

row 4: A100039;

col 1: A014107;

col 2: A033537;

col 3: A100040;

col 4: A100041;

diag (2,18,...): A077591;

diag (7,31,...): A157914;

diag (16,48,...): A035008;

diag (29,69,...): A108928;

antidiagonal sums: A033431;

antidiagonal sums: 2*(1^3, 2^3, 3^3, 4^3,...) = 2*A000578.

A060432(n) + n is odd if and only if n is in this sequence. - Peter Kagey, Feb 03 2016

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

.

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

| |

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

| | | |

37 11 1---3 21 55

| | | | |

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

| | |

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

|

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

.

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