A373663 a(n) = (1 + (n+2)^2 + (n-3)*(-1)^n)/2.

6, 8, 13, 19, 24, 34, 39, 53, 58, 76, 81, 103, 108, 134, 139, 169, 174, 208, 213, 251, 256, 298, 303, 349, 354, 404, 409, 463, 468, 526, 531, 593, 598, 664, 669, 739, 744, 818, 823, 901, 906, 988, 993, 1079, 1084, 1174, 1179, 1273, 1278, 1376, 1381, 1483, 1488, 1594

Offset: 1

Author

Wesley Ivan Hurt, Jun 12 2024

Keywords

• nonn
• easy

Comments

Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 3 of the boustrophedon-style array (see example).

In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=3.

Examples

			       [ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9] [10] [11] [12]
  [ 1]   1    3    4   10   11   21   22   36   37   55   56   78   ...
  [ 2]   2    5    9   12   20   23   35   38   54   57   77   ...
  [ 3]   6    8   13   19   24   34   39   53   58   76   ...
  [ 4]   7   14   18   25   33   40   52   59   75   ...
  [ 5]  15   17   26   32   41   51   60   74   ...
  [ 6]  16   27   31   42   50   61   73   ...
  [ 7]  28   30   43   49   62   72   ...
  [ 8]  29   44   48   63   71   ...
  [ 9]  45   47   64   70   ...
  [10]  46   65   69   ...
  [11]  66   68   ...
  [12]  67   ...
        ...
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

For rows k = 1..10: A131179 (k=1) n>0, A373662 (k=2), this sequence (k=3), A374004 (k=4), A374005 (k=5), A374007 (k=6), A374008 (k=7), A374009 (k=8), A374010 (k=9), A374011 (k=10).

Row 3 of the example in A056011, Column 3 of the rectangular array in A056023.

Programs

Magma

[(1 + (n+2)^2 + (n-3)*(-1)^n)/2: n in [1..80]];

Mathematica

k := 3; Table[(1 + (n+k-1)^2 + (n-k) (-1)^(n+k-1))/2, {n, 80}]

Python

def A373663(n): return ((n+1)*(n+2)+6 if n&1 else (n+2)*(n+3)-4)>>1 # Chai Wah Wu, Jun 23 2024

Formula

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).

a(n) = A373662(n+1) - (-1)^n.

G.f.: -x*(x^4+2*x^3-7*x^2+2*x+6)/((x+1)^2*(x-1)^3).