A042948 Numbers congruent to 0 or 1 (mod 4).

0, 1, 4, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29, 32, 33, 36, 37, 40, 41, 44, 45, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93, 96, 97, 100, 101, 104, 105, 108

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

A133872(a(n)) = 1; complement of A042964. - Reinhard Zumkeller, Oct 03 2008

Partial sums are A035608. - Jaroslav Krizek, Dec 18 2009 [corrected by Werner Schulte, Dec 04 2023]

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).

Crossrefs

Cf. A000290, A002620.

Programs

Magma

[n: n in [0..150]|n mod 4 in {0, 1}]; // Vincenzo Librandi, Dec 09 2015

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 *)

Maxima

makelist(-1/2+1/2*(-1)^n+2*n, n, 0, 60); /* Martin Ettl, Nov 05 2012 */

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

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) = Sum_{k>=0} A030308(n,k)*A151821(k+1). - Philippe Deléham, Oct 17 2011

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

a(n) = A000290(n) - 4*A002620(n-1). - Leo Tavares, Oct 04 2022