OFFSET
0,3
COMMENTS
Maximum number of squares attacked by a bishop on an (n + 1) X (n + 1) chessboard. - Stewart Gordon, Mar 23 2001
Maximum vertex degree of the (n + 1) X (n + 1) bishop graph and black bishop graph. - Eric W. Weisstein, Jun 26 2017
Also number of squares attacked by a bishop on a toroidal chessboard. - Diego Torres (torresvillarroel(AT)hotmail.com), May 30 2001
Numbers n such that {1, 2, 3, ..., n-1, n} is a perfect Skolem set. - Emeric Deutsch, Nov 24 2006
The number of terms which lie on the principal diagonals of an n X n square spiral. - William A. Tedeschi, Mar 02 2008
Possible nonnegative discriminants of quadratic equation a*x^2 + b*x + c or discriminants of binary quadratic forms a*x^2 + b*x*y + c^y^2. - Artur Jasinski, Apr 28 2008
Nonnegative m for which floor(k*m/4) = k*floor(m/4), where k = 2 or 3. Example: 13 is in the sequence because floor(2*13/4) = 2*floor(13/4), and also floor(3*13/4) = 3*floor(13/4). - Bruno Berselli, Dec 09 2015
Also number of maximal cliques in the n X n white bishop graph. - Eric W. Weisstein, Dec 01 2017
The offset should have been 1. - Jianing Song, Oct 06 2018
Numbers k for which the binomial coefficient C(k,2) is even. - Tanya Khovanova, Oct 20 2018
Numbers m such that there exists a permutation (x(1), x(2), ..., x(m)) with all absolute differences |x(k) - k| distinct. - Jukka Kohonen, Oct 02 2021
Numbers m such that there exists a multiset of integers whose size is m, and sum and product are both -m. - Yifan Xie, Mar 25 2024
LINKS
James Spahlinger, Table of n, a(n) for n = 0..10000
H. W. Gould, The inverse of a finite series and a third-order recurrent sequence, Fibonacci Quart., Vol. 44, No. 4 (2006), pp. 302-315. See p. 311.
M. J. Pelling and J. H. Steelman, E3269. Permutations with distinct displacements, (problem by Pelling and solution by Steelman), The American Mathematical Monthly, 96 (1989), 843-844.
T. Skolem, On certain distributions of integers in pairs with given differences, Math. Scand., Vol. 5 (1957), pp. 57-68.
Harry Tamvakis and O. P. Lossers, Amenable Numbers: 10454, The American Mathematical Monthly, Vol. 105, No. 4 (Apr., 1998), p. 368.
Leo Tavares, Illustration: Diamond Crosses
Eric Weisstein's World of Mathematics, Bishop Graph.
Eric Weisstein's World of Mathematics, Black Bishop Graph.
Eric Weisstein's World of Mathematics, Maximal Clique.
Eric Weisstein's World of Mathematics, Maximum Vertex Degree.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
FORMULA
a(n) = A042963(n+1) - 1. [Corrected by Jianing Song, Oct 06 2018]
From Michael Somos, Jan 12 2000: (Start)
G.f.: x*(1 + 3*x)/((1 + x)*(1 - x)^2).
a(n) = a(n-1) + 2 + (-1)^n. (End)
a(n) = 4*n - a(n-1) - 3 with a(0) = 0. - Vincenzo Librandi, Nov 17 2010
a(n) = floor((4/3)*floor(3*n/2)). - Clark Kimberling, Jul 04 2012
a(n) = n + 2*floor(n/2) = 2*n - (n mod 2). - Bruno Berselli, Apr 30 2016
E.g.f.: 2*exp(x)*x - sinh(x). - Stefano Spezia, Sep 09 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + 3*log(2)/4. - Amiram Eldar, Dec 05 2021
MAPLE
a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-2]+4 od: seq(a[n], n=0..54); # Zerinvary Lajos, Mar 16 2008
MATHEMATICA
Select[Range[0, 150], Or[Mod[#, 4] == 0, Mod[#, 4] == 1] &] (* Vincenzo Librandi, Dec 09 2015 *)
Table[(4 n - 5 - (-1)^n)/2, {n, 20}] (* Eric W. Weisstein, Dec 01 2017 *)
LinearRecurrence[{1, 1, -1}, {1, 4, 5}, {0, 20}] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[x (1 + 3 x)/((-1 + x)^2 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)
{#, # + 1} & /@ (4 Range[0, 40]) // Flatten (* Harvey P. Dale, Jan 15 2024 *)
PROG
(PARI) a(n)=2*n-n%2;
(PARI) concat(0, Vec(x*(1+3*x)/((1+x)*(1-x)^2) + O(x^100))) \\ Altug Alkan, Dec 09 2015
(Maxima) makelist(-1/2+1/2*(-1)^n+2*n, n, 0, 60); /* Martin Ettl, Nov 05 2012 */
(Magma) [n: n in [0..150]|n mod 4 in {0, 1}]; // Vincenzo Librandi, Dec 09 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved