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A042948 Numbers congruent to 0 or 1 mod 4. 58
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 (list; graph; refs; listen; history; text; internal format)
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

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

Partial sums of a(n) in A035608(n). A035608(n) = Expansion of x*(1 + 3x)/((1 + x)(1 - x)^3). - Jaroslav Krizek, Dec 18 2009

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 n for which the binomial coefficient n choose 2 is even.- Tanya Khovanova, Oct 20,2018.

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. 44 (2006), no. 4, 302-315. See p. 311.

T. Skolem, On certain distributions of integers in pairs with given differences, Math. Scand., 1957, vol. 5, 57-68.

Harry Tamvakis and O. P. Lossers, Amenable Numbers: 10454, The American Mathematical Monthly, Vol. 105, No. 4 (Apr., 1998), p. 368.

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]

G.f.: x*(1 + 3*x)/((1 + x)*(1 - x)^2). a(n) = a(n-1) + 2 + (-1)^n. - Michael Somos, Jan 12 2000

a(n) = -1/2 + 1/2*(-1)^n + 2*n. - Paolo P. Lava, Oct 03 2008

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

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

seq(add(irem(3^k, 4), k=4..n), n=3..57); # Zerinvary Lajos, Apr 20 2008

a:=n->add(2+(-1)^j, j=1..n):seq(a(n), n=0..52); # Zerinvary Lajos, Dec 13 2008

MATHEMATICA

bb = {}; Do[Do[Do[d = b^2 - 4 a c; If[d < 0, [null], AppendTo[bb, d]], {a, 0, 50}], {b, 0, 50}], {c, 0, 50}]; Union[bb] (* Artur Jasinski, Apr 28 2008 *)

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 *)

PROG

(PARI) a(n)=2*n-n%2

(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

(PARI) concat(0, Vec(x*(1+3*x)/((1+x)*(1-x)^2) + O(x^100))) \\ Altug Alkan, Dec 09 2015

CROSSREFS

Sequence in context: A284906 A285260 A190671 * A126001 A269984 A188085

Adjacent sequences:  A042945 A042946 A042947 * A042949 A042950 A042951

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 21 07:20 EDT 2018. Contains 316405 sequences. (Running on oeis4.)