%I #53 Nov 17 2024 16:42:54

%S 1,7,29,67,121,191,277,379,497,631,781,947,1129,1327,1541,1771,2017,

%T 2279,2557,2851,3161,3487,3829,4187,4561,4951,5357,5779,6217,6671,

%U 7141,7627,8129,8647,9181,9731,10297,10879,11477,12091,12721,13367,14029,14707,15401,16111,16837,17579

%N a(n) = 8*n^2 - 2*n + 1.

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

%C .

%C 41--43--45--47--49--51

%C | |

%C 39 13--15--17--19 53

%C | | | |

%C 37 11 1---3 21 55

%C | | | | |

%C 35 9---7---5 23 57

%C | | |

%C 33--31--29--27--25 59

%C |

%C 71--69--67--65--63--61

%C .

%C Walking in straight lines away from the center:

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

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

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

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

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

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

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

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

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

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

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

%C 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).

%C For n >= 1, bisection of A193867. - _Omar E. Pol_, Aug 16 2011

%C 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

%H G. C. Greubel, <a href="/A185438/b185438.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = a(n-1) + 16*n - 10 (n > 0).

%F a(n) = 2*a(n-1) - a(n-2) + 16 (n > 1).

%F a(n) = 3*(n-1) - 3*a(n-2) + a(n-3) (n > 2).

%F G.f.: (-1 - 4*x - 11*x^2)/(x-1)^3. - _R. J. Mathar_, Feb 03 2011

%F a(n) = A014635(n) + 1. - _Bruno Berselli_, Apr 09 2011

%F E.g.f.: exp(x)*(1 + 6*x + 8*x^2). - _Elmo R. Oliveira_, Nov 17 2024

%t Table[1 - 2n + 8n^2, {n, 0, 39}] (* _Alonso del Arte_, Feb 03 2011 *)

%t CoefficientList[Series[(-1 - 4 x - 11 x^2)/(x - 1)^3, {x, 0, 47}], x] (* _Michael De Vlieger_, Aug 01 2016 *)

%o (Magma) [1-2*n+8*n^2: n in [0..50]]; // _Vincenzo Librandi_, Feb 03 2011

%o (PARI) a(n)=8*n^2-2*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A000217, A005408, A014634, A014635, A033566, A033567, A033951, A054552, A064038.

%Y Cf. A069129, A080856, A081585, A102083, A108928, A157914, A181407, A188135, A193867.

%K nonn,easy

%O 0,2

%A _Paul Curtz_, Feb 03 2011