A008590 Multiples of 8.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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. - Peter M. Chema, Apr 03 2016

From Lechoslaw Ratajczak, Sep 03 2017: (Start)

Conjecture: let gcd_2(b,c) be the second greatest common divisor and lcd_2(b,c) be the second least common divisor of not coprime integers b and c. Consecutive elements of this sequence (for a(n) > 0) are consecutive integers m for which both Sum_{k=1..m, gcd(k,m)<>1} gcd_2(k,m) and Sum_{k=1..m, gcd(k,m) <>1} lcd_2(k,m) are even numbers.

a(1) = 8 because 1+2+1+4 = 8 (8 is even) and 2+2+2+2 = 8 (8 is even).

a(2) = 16 because 1+2+1+4+1+2+1+8 = 20 (20 is even) and 2+2+2+2+2+2+2+2 = 16 (16 is even).

a(3) = 24 because 1+1+2+3+4+1+1+6+1+1+4+3+2+1+1+12 = 44 (44 is even) and 2+3+2+2+2+3+2+2+2+3+2+2+2+3+2+2 = 36 (36 is even).

The conjecture was checked for 5*10^4 consecutive integers. (End)

Examples

			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 the square 4*4, 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 6 X 6 outer square and a 2 X 2 inner square, with the area (6 X 6 minus 2 X 2) equal to the perimeter (4*6 + 4*2) at 32. - _Peter M. Chema_, Apr 03 2016
		

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..200
• Ch. Berdellé, Démonstration élémentaire d’un théorème énoncé par M. E. Catalan, Bulletin de la S. M. F., tome 17 (1889), p. 102. [Every positive multiple of 8 is the sum of 8 odd squares.]
• E. Catalan, Extrait d’une lettre, Bulletin de la S. M. F., tome 17 (1889), pp. 205-206.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 320.
• Tanya Khovanova, Recursive Sequences.
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
• Leo Tavares, Illustration: Square Ray Frames
• Index entries for linear recurrences with constant coefficients, signature (2,-1)

Crossrefs

Cf. A010014.

Essentially the same as A022144.

Subsequence of A185359, apart initial 0.

Cf. A000567, A016754, A045944, A047592, A168181.

Programs

Haskell

a008590 = (* 8)
a008590_list = [0,8..]  -- Reinhard Zumkeller, Apr 02 2012

Mathematica

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

PARI

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

Python

def A008590(n): return n<<3 # Chai Wah Wu, Mar 11 2025

Formula

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

From Vincenzo Librandi, Dec 24 2010: (Start)

a(n) = 8*n = 2*a(n-1) - a(n-2).

G.f.: 8*x/(x-1)^2. (End)

a(n) = Sum_{k=1..4n} (i^k + 1)*(i^(4n-k) + 1), 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

a(n) = A000567(n+1) - A045944(n-1). - Leo Tavares, Mar 25 2022

E.g.f.: 8*x*exp(x). - Stefano Spezia, Apr 03 2023