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A185438
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a(n) = 8*n^2 - 2*n + 1.
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4
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1, 7, 29, 67, 121, 191, 277, 379, 497, 631, 781, 947, 1129, 1327, 1541, 1771, 2017, 2279, 2557, 2851, 3161, 3487, 3829, 4187, 4561, 4951, 5357, 5779, 6217, 6671, 7141, 7627, 8129, 8647, 9181, 9731, 10297, 10879, 11477, 12091, 12721, 13367, 14029, 14707, 15401, 16111, 16837, 17579
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OFFSET
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0,2
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COMMENTS
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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).
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
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LINKS
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FORMULA
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a(n) = a(n-1) + 16*n - 10 (n > 0).
a(n) = 2*a(n-1) - a(n-2) + 16 (n > 1).
a(n) = 3*(n-1) - 3*a(n-2) + a(n-3) (n > 2).
G.f.: ( -1-4*x-11*x^2 )/(x-1)^3. - R. J. Mathar, Feb 03 2011
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MATHEMATICA
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CoefficientList[Series[(-1 - 4 x - 11 x^2)/(x - 1)^3, {x, 0, 47}], x] (* Michael De Vlieger, Aug 01 2016 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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