This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A008590 #115 Mar 11 2025 13:24:07
%S A008590 0,8,16,24,32,40,48,56,64,72,80,88,96,104,112,120,128,136,144,152,160,
%T A008590 168,176,184,192,200,208,216,224,232,240,248,256,264,272,280,288,296,
%U A008590 304,312,320,328,336,344,352,360,368,376,384,392,400,408,416,424,432
%N A008590 Multiples of 8.
%C A008590 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.
%C A008590 First differences of odd squares: a(n) = A016754(n) - A016754(n-1) for n > 0. - _Reinhard Zumkeller_, Nov 08 2009
%C A008590 Complement of A047592; A168181(a(n)) = 0. - _Reinhard Zumkeller_, Nov 30 2009
%C A008590 For n >= 1, number of pairs (x, y) of Z^2, such that max(abs(x), abs(y)) = n. - _Michel Marcus_, Nov 28 2014
%C A008590 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
%C A008590 From _Lechoslaw Ratajczak_, Sep 03 2017: (Start)
%C A008590 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.
%C A008590 a(1) = 8 because 1+2+1+4 = 8 (8 is even) and 2+2+2+2 = 8 (8 is even).
%C A008590 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).
%C A008590 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).
%C A008590 The conjecture was checked for 5*10^4 consecutive integers. (End)
%H A008590 Ivan Panchenko, <a href="/A008590/b008590.txt">Table of n, a(n) for n = 0..200</a>
%H A008590 Ch. Berdellé, <a href="https://doi.org/10.24033/bsmf.383">Démonstration élémentaire d’un théorème énoncé par M. E. Catalan</a>, Bulletin de la S. M. F., tome 17 (1889), p. 102. [Every positive multiple of 8 is the sum of 8 odd squares.]
%H A008590 E. Catalan, <a href="https://doi.org/10.24033/bsmf.401">Extrait d’une lettre</a>, Bulletin de la S. M. F., tome 17 (1889), pp. 205-206.
%H A008590 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=320">Encyclopedia of Combinatorial Structures 320</a>.
%H A008590 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A008590 Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
%H A008590 Leo Tavares, <a href="/A008590/a008590.jpg">Illustration: Square Ray Frames</a>
%H A008590 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1)
%F A008590 a(n) = (2*n+1)^2 - (2*n-1)^2. - Xavier Acloque, Oct 22 2003
%F A008590 From _Vincenzo Librandi_, Dec 24 2010: (Start)
%F A008590 a(n) = 8*n = 2*a(n-1) - a(n-2).
%F A008590 G.f.: 8*x/(x-1)^2. (End)
%F A008590 a(n) = Sum_{k=1..4n} (i^k + 1)*(i^(4n-k) + 1), where i=sqrt(-1). - _Bruno Berselli_, Mar 19 2012
%F A008590 a(n) = (n+2)^2 - (n-2)^2 = 4*(n+2) + 4*(n-2), as exemplified below. - _Peter M. Chema_, Apr 03 2016
%F A008590 a(n) = A000567(n+1) - A045944(n-1). - _Leo Tavares_, Mar 25 2022
%F A008590 E.g.f.: 8*x*exp(x). - _Stefano Spezia_, Apr 03 2023
%e A008590 Beginning with n = 2, illustration of the terms as the area of square frames, where area equals the sum of inner and outer perimeters:
%e A008590 _ _ _ _ _ _ _ _
%e A008590 _ _ _ _ _ _ _ | |
%e A008590 _ _ _ _ _ _ | | | _ _ _ _ |
%e A008590 _ _ _ _ _ | | | _ _ _ | | | | |
%e A008590 _ _ _ _ | | | _ _ | | | | | | | | |
%e A008590 | | | _ | | | | | | | | | | | | |
%e A008590 | | | |_| | | |_ _| | | |_ _ _| | | |_ _ _ _| |
%e A008590 | | | | | | | | | |
%e A008590 |_ _ _ _| |_ _ _ _ _| |_ _ _ _ _ _| |_ _ _ _ _ _ _| |_ _ _ _ _ _ _ _|
%e A008590 a(2) = 16 a(3) = 24 a(4) = 32 a(5) = 40 a(6) = 48
%e A008590 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 A008590 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
%t A008590 Table[8*n,{n,0,5!}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 03 2010 *)
%o A008590 (Haskell)
%o A008590 a008590 = (* 8)
%o A008590 a008590_list = [0,8..] -- _Reinhard Zumkeller_, Apr 02 2012
%o A008590 (PARI) a(n) = 8*n; \\ _Altug Alkan_, Apr 08 2016
%o A008590 (Python)
%o A008590 def A008590(n): return n<<3 # _Chai Wah Wu_, Mar 11 2025
%Y A008590 Cf. A010014.
%Y A008590 Essentially the same as A022144.
%Y A008590 Subsequence of A185359, apart initial 0.
%Y A008590 Cf. A000567, A016754, A045944, A047592, A168181.
%K A008590 nonn,easy
%O A008590 0,2
%A A008590 _N. J. A. Sloane_