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%I #47 Jan 31 2024 10:08:19
%S 1,5,9,17,25,37,49,65,81,101,121,145,169,197,225,257,289,325,361,401,
%T 441,485,529,577,625,677,729,785,841,901,961,1025,1089,1157,1225,1297,
%U 1369,1445,1521,1601,1681,1765,1849,1937,2025,2117,2209,2305,2401,2501
%N Diagonal in square spiral or maze arrangement of natural numbers.
%C Interleaves the odd squares A016754 with (1+4n^2), A053755.
%C Squares of positive integers (plus 1 if n is odd). - _Wesley Ivan Hurt_, Oct 10 2013
%C 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
%C 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
%H Vincenzo Librandi, <a href="/A080335/b080335.txt">Table of n, a(n) for n = 0..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EdgeChromaticNumber.html">Edge Chromatic Number</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookComplementGraph.html">Rook Complement Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F a(n) = (3 + 4*n + 2*n^2 - (-1)^n)/2.
%F a(2*n) = A016754(n), a(2*n+1) = A053755(n+1).
%F 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
%F 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
%F a(n)+(-1)^n = A137928(n+1). - _Philippe Deléham_, Feb 17 2012
%F G.f.: (1 + 3*x - x^2 + x^3)/((1-x)^3*(1+x)). - _Colin Barker_, Mar 18 2012
%F a(n) = A000035(n) + A000290(n+1). - _Wesley Ivan Hurt_, Oct 10 2013
%F From _Bob Selcoe_, Feb 12 2015: (Start)
%F a(n) = A137932(n+2) + 1.
%F a(n) = (n+1)^2 when n is even; a(n) = (n+1)^2 + 1 when n is odd.
%F a(n) = A002378(n+2) - A047238(n+3) + 1.
%F (End)
%F Sum_{n>=0} 1/a(n) = Pi*coth(Pi/2)/4 + Pi^2/8 - 1/2. - _Amiram Eldar_, Jul 07 2022
%p A080335:=n->(n mod 2) + (n+1)^2; seq(A080335(k),k=0..49); # _Wesley Ivan Hurt_, Oct 10 2013
%t With[{nn = 60}, Riffle[Range[1, nn, 2]^2, 4 Range[nn]^2 + 1]] (* _Harvey P. Dale_, Jan 29 2012 *)
%t LinearRecurrence[{2, 0, -2, 1}, {1, 5, 9, 17}, 60] (* _Harvey P. Dale_, Jan 29 2012 *)
%t Table[(3 + 4 n + 2 n^2 - (-1)^n)/2, {n, 0, 50}] (* _Wesley Ivan Hurt_, Oct 10 2013 *)
%t Table[Mod[n, 2] + (n + 1)^2, {n, 0, 20}] (* _Eric W. Weisstein_, Jan 31 2024 *)
%o (Magma) [(3+4*n+2*n^2-(-1)^n)/2: n in [0..50]]; // _Vincenzo Librandi_, Sep 06 2011
%Y Cf. A081347, A081348.
%Y Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
%Y 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.
%Y 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.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Mar 19 2003
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