A185438 a(n) = 8*n^2 - 2*n + 1.

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

Offset: 0

Author

Paul Curtz, Feb 03 2011

Keywords

• nonn
• easy

Comments

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

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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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

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

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

Programs

Magma

[1-2*n+8*n^2: n in [0..50]]; // Vincenzo Librandi, Feb 03 2011

Mathematica

Table[1 - 2n + 8n^2, {n, 0, 39}] (* Alonso del Arte, Feb 03 2011 *)
CoefficientList[Series[(-1 - 4 x - 11 x^2)/(x - 1)^3, {x, 0, 47}], x] (* Michael De Vlieger, Aug 01 2016 *)

PARI

a(n)=8*n^2-2*n+1 \\ Charles R Greathouse IV, Jun 17 2017

Formula

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

a(n) = A014635(n) + 1. - Bruno Berselli, Apr 09 2011

E.g.f.: exp(x)*(1 + 6*x + 8*x^2). - Elmo R. Oliveira, Nov 17 2024