A081348 -id:A081348 - cp's OEIS frontend

A080335 Diagonal in square spiral or maze arrangement of natural numbers.

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

Offset: 0

Paul Barry, Mar 19 2003

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

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).

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.

[(3+4*n+2*n^2-(-1)^n)/2: n in [0..50]]; // Vincenzo Librandi, Sep 06 2011

A080335:=n->(n mod 2) + (n+1)^2; seq(A080335(k),k=0..49); # Wesley Ivan Hurt, Oct 10 2013

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 *)

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

A081347 First column in maze arrangement of natural numbers.

Original entry on oeis.org

1, 2, 3, 12, 13, 30, 31, 56, 57, 90, 91, 132, 133, 182, 183, 240, 241, 306, 307, 380, 381, 462, 463, 552, 553, 650, 651, 756, 757, 870, 871, 992, 993, 1122, 1123, 1260, 1261, 1406, 1407, 1560, 1561, 1722, 1723, 1892, 1893, 2070, 2071, 2256, 2257, 2450, 2451

Offset: 0

Paul Barry, Mar 19 2003

Interleaves two times the hexagonal numbers A000384 with A054554.

			Starting with 1,2,3, turn (LL) and then repeat (RRR)(LLL) to get
1 6 7 20
2 5 8 19
3 4 9 18
12 11 10 17

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A081348, A080335.

[((1+2*n^2)+(1-2*n)*(-1)^n)/2: n in [0..50]]; // Vincenzo Librandi, Aug 08 2013

CoefficientList[Series[(1 + x - x^2 + 7 x^3) / ((1 - x)^3 (1 + x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 08 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{1,2,3,12,13},60] (* Harvey P. Dale, Aug 13 2025 *)

a(n) = ((1+2*n^2)+(1-2*n)*(-1)^n)/2.
a(2n) = A054554(n).
a(2n+1) = 2*A000384(n).
G.f.: (1+x-x^2+7*x^3)/((1-x)^3*(1+x)^2). [Colin Barker, Apr 17 2012]

A093650 Natural numbers arranged in a square maze beginning 1, 2, 3, then moving right, then up, right, down, left, down, right, etc., and read by antidiagonals upwards.

Original entry on oeis.org

1, 2, 6, 3, 5, 7, 12, 4, 8, 20, 13, 11, 9, 19, 21, 30, 14, 10, 18, 22, 42, 31, 29, 15, 17, 23, 41, 43, 56, 32, 28, 16, 24, 40, 44, 72, 57, 55, 33, 27, 25, 39, 45, 71, 73, 90, 58, 54, 34, 26, 38, 46, 70, 74, 110, 91, 89, 59, 53, 35, 37, 47, 69, 75, 109, 111

Offset: 1

Michael Joseph Halm, May 15 2004

			a(3) = 6 because the maze begins 2 under 1, 3 under 2, 4 right of 3, 5 right of 2 and 6 right of 1.
Array begins:
   1   6---7  20 ...
   |   |   |   |
   2   5   8  19 ...
   |   |   |   |
   3---4   9  18 ...
           |   |
  12--11--10  17 ...
   |           |
  13--14--15--16 ...
  ...

Cf. A080335, A081347, A081348.

Other square mazes: A081344, A081349.

More terms from Jinyuan Wang, Jun 15 2022

A226940 a(0)=0; if a(n-1) is odd, a(n) = n + a(n-1), otherwise a(n) = n - a(n-1).

Original entry on oeis.org

0, 1, 3, 6, -2, 7, 13, 20, -12, 21, 31, 42, -30, 43, 57, 72, -56, 73, 91, 110, -90, 111, 133, 156, -132, 157, 183, 210, -182, 211, 241, 272, -240, 273, 307, 342, -306, 343, 381, 420, -380, 421, 463, 506, -462, 507, 553, 600, -552, 601, 651, 702, -650, 703, 757

Offset: 0

Enrico Santilli, Jun 23 2013

Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A081348 (second bisection); A002939, A054554, A054569, A068377.

[IsZero(n) select 0 else IsOdd(Self(n)) select n+Self(n) else n-Self(n): n in [0..60]]; // Bruno Berselli, Jul 01 2013

LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {0, 1, 3, 6, -2, 7, 13, 20, -12, 21, 31, 42}, 60] (* Bruno Berselli, Jul 01 2013 *)

makelist(coeff(taylor(x*(1+3*x+6*x^2-2*x^3+4*x^4+4*x^5+2*x^6-6*x^7+3*x^8+x^9)/((1-x)^3*(1+x)^3*(1+x^2)^3), x, 0, n), x, n), n, 0, 60); /* Bruno Berselli, Jul 01 2013 */

G.f.: x*(1 +3*x +6*x^2 -2*x^3 +4*x^4 +4*x^5 +2*x^6 -6*x^7 +3*x^8 +x^9)/((1-x)^3*(1+x)^3*(1+x^2)^3). [Bruno Berselli, Jul 01 2013]
a(n) = 3*a(n-4) -3*a(n-8) +a(n-12). [Bruno Berselli, Jul 01 2013]
a(4n) = -A002939(n), a(4n+1) = A054569(n+1), a(4n+2) = A054554(n+2), a(4n+3) = A068377(n+2). [Bruno Berselli, Jul 02 2013]

More terms from Bruno Berselli, Jul 01 2013

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A080335 Diagonal in square spiral or maze arrangement of natural numbers.

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A081347 First column in maze arrangement of natural numbers.

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A093650 Natural numbers arranged in a square maze beginning 1, 2, 3, then moving right, then up, right, down, left, down, right, etc., and read by antidiagonals upwards.

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A226940 a(0)=0; if a(n-1) is odd, a(n) = n + a(n-1), otherwise a(n) = n - a(n-1).

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