The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A080335 Diagonal in square spiral or maze arrangement of natural numbers. 32
 1, 5, 9, 17, 25, 37, 49, 65, 81, 101, 121, 145, 169, 197, 225, 257, 289, 325, 361, 401, 441, 485, 529, 577, 625, 677, 729, 785, 841, 901, 961, 1025, 1089, 1157, 1225, 1297, 1369, 1445, 1521, 1601, 1681, 1765, 1849, 1937, 2025, 2117, 2209, 2305, 2401, 2501 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Interleaves the odd squares A016754 with (1+4n^2), A053755. Squares of positive integers (plus 1 if n is odd). - Wesley Ivan Hurt, Oct 10 2013 a(n) is the maximum total number of queens that can coexist without attacking each other on an [n+3] X [n+3] chessboard, when the lone queen is in the most vulnerable position on the board. Specifically, the lone queen will placed in any center position, facing an opponent's "army" of size a(n)-1 == A137932(n+2). - Bob Selcoe, Feb 12 2015 a(n) is also the edge chromatic number of the complement of the (n+2) X (n+2) rook graph. - Eric W. Weisstein, Jan 31 2024 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..10000 Eric Weisstein's World of Mathematics, Edge Chromatic Number Eric Weisstein's World of Mathematics, Rook Complement Graph Eric Weisstein's World of Mathematics, Rook Graph Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1). FORMULA a(n) = (3 + 4*n + 2*n^2 - (-1)^n)/2. a(2*n) = A016754(n), a(2*n+1) = A053755(n+1). E.g.f.: exp(x)*(2 + 3*x + x^2) - cosh(x). The sequence 1,1,5,9,... is given by n^2+(1+(-1)^n)/2 with e.g.f. exp(1+x+x^2)*exp(x)-sinh(x). - Paul Barry, Sep 02 2003 and Sep 19 2003 a(0)=1, a(1)=5, a(2)=9, a(3)=17, a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Harvey P. Dale, Jan 29 2012 a(n)+(-1)^n = A137928(n+1). - Philippe Deléham, Feb 17 2012 G.f.: (1 + 3*x - x^2 + x^3)/((1-x)^3*(1+x)). - Colin Barker, Mar 18 2012 a(n) = A000035(n) + A000290(n+1). - Wesley Ivan Hurt, Oct 10 2013 From Bob Selcoe, Feb 12 2015: (Start) a(n) = A137932(n+2) + 1. a(n) = (n+1)^2 when n is even; a(n) = (n+1)^2 + 1 when n is odd. a(n) = A002378(n+2) - A047238(n+3) + 1. (End) Sum_{n>=0} 1/a(n) = Pi*coth(Pi/2)/4 + Pi^2/8 - 1/2. - Amiram Eldar, Jul 07 2022 MAPLE A080335:=n->(n mod 2) + (n+1)^2; seq(A080335(k), k=0..49); # Wesley Ivan Hurt, Oct 10 2013 MATHEMATICA With[{nn = 60}, Riffle[Range[1, nn, 2]^2, 4 Range[nn]^2 + 1]] (* Harvey P. Dale, Jan 29 2012 *) LinearRecurrence[{2, 0, -2, 1}, {1, 5, 9, 17}, 60] (* Harvey P. Dale, Jan 29 2012 *) Table[(3 + 4 n + 2 n^2 - (-1)^n)/2, {n, 0, 50}] (* Wesley Ivan Hurt, Oct 10 2013 *) Table[Mod[n, 2] + (n + 1)^2, {n, 0, 20}] (* Eric W. Weisstein, Jan 31 2024 *) PROG (Magma) [(3+4*n+2*n^2-(-1)^n)/2: n in [0..50]]; // Vincenzo Librandi, Sep 06 2011 CROSSREFS Cf. A081347, A081348. Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951. Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754. Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335. Sequence in context: A210978 A211434 A182388 * A351837 A089109 A100449 Adjacent sequences: A080332 A080333 A080334 * A080336 A080337 A080338 KEYWORD nonn,easy AUTHOR Paul Barry, Mar 19 2003 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 22 01:34 EDT 2024. Contains 371887 sequences. (Running on oeis4.)