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A008590 Multiples of 8. 41
0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 408, 416, 424, 432 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

For n>3, the number of squares on the infinite 4-column half-strip chessboard at <=n knight moves from any fixed point on the short edge.

First differences of odd squares: a(n)=A016754(n)-A016754(n-1) for n>0. - Reinhard Zumkeller, Nov 08 2009

Complement of A047592; A168181(a(n)) = 0. - Reinhard Zumkeller, Nov 30 2009

For n>=1, number of pairs (x, y) of Z^2, such that max(abs(x), abs(y)) = n. - Michel Marcus, Nov 28 2014

These terms are the area of square frames (using integer lengths), with specific instances where the area equals the sum of inner and outer perimeters (see example and formula below).  The thickness of the frames are always 2, which is of further significance when considering that all regular polygons have an area that is equal to perimeter when apothem is 2.

LINKS

Ivan Panchenko, Table of n, a(n) for n = 0..200

Tanya Khovanova, Recursive Sequences

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 320

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

Index entries for linear recurrences with constant coefficients, signature (2,-1)

FORMULA

a(n) = (2*n+1)^2-(2*n-1)^2. - Xavier Acloque Oct 22 2003

a(n) = 8*n = 2*a(n-1)-a(n-2). G.f.: 8*x/(x-1)^2. - Vincenzo Librandi, Dec 24 2010

a(n) = sum((i^k+1)*(i^(4n-k)+1), k=1..4n), where i=sqrt(-1). - Bruno Berselli, Mar 19 2012

a(n) = (n+2)^2-(n-2)^2 = 4*(n+2)+4*(n-2), as exemplified below. Peter M. Chema, Apr 03 2016

EXAMPLE

Beginning with n = 2, illustration of the terms as the area of square frames, where area equals the sum of inner and outer perimeters:

                                                                _ _ _ _ _ _ _ _

                                              _ _ _ _ _ _ _    |               |

                              _ _ _ _ _ _    |             |   |    _ _ _ _    |

                _ _ _ _ _    |           |   |    _ _ _    |   |   |       |   |

   _ _ _ _     |         |   |    _ _    |   |   |     |   |   |   |       |   |

  |       |    |    _    |   |   |   |   |   |   |     |   |   |   |       |   |

  |       |    |   |_|   |   |   |_ _|   |   |   |_ _ _|   |   |   |_ _ _ _|   |

  |       |    |         |   |           |   |             |   |               |

  |_ _ _ _|    |_ _ _ _ _|   |_ _ _ _ _ _|   |_ _ _ _ _ _ _|   |_ _ _ _ _ _ _ _|

  a(2) = 16      a(3) = 24     a(4) = 32        a(5) = 40          a(6) = 48

The inner square has side n-2 and outer square side n+2, pursuant to the above and related formula.  Note that a(2) is simply square 4x4, with the inner square having side 0; considering the inner square as a center point, this frame also has thickness of 2.

E.g., for a(4), the square frame is formed by a 6x6 outer square and a 2x2 inner square, with the area (6x6 minus 2x2) equal to the perimeter (4*6+4*2) at 32.Peter M. Chema, Apr 03 2016

MATHEMATICA

Table[8*n, {n, 0, 5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2010 *)

PROG

(Haskell)

a008590 = (* 8)

a008590_list = [0, 8..]  -- Reinhard Zumkeller, Apr 02 2012

(PARI) a(n) = 8*n; \\ Altug Alkan, Apr 08 2016

CROSSREFS

Cf. A010014.

Essentially the same as A022144.

Subsequence of A185359, apart initial 0.

Sequence in context: A185359 A022144 A181390 * A186544 A061824 A085131

Adjacent sequences:  A008587 A008588 A008589 * A008591 A008592 A008593

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 26 03:45 EDT 2017. Contains 285426 sequences.