A374004 a(n) = (1 + (n+3)^2 - (n-4)*(-1)^n)/2.

7, 14, 18, 25, 33, 40, 52, 59, 75, 82, 102, 109, 133, 140, 168, 175, 207, 214, 250, 257, 297, 304, 348, 355, 403, 410, 462, 469, 525, 532, 592, 599, 663, 670, 738, 745, 817, 824, 900, 907, 987, 994, 1078, 1085, 1173, 1180, 1272, 1279, 1375, 1382, 1482, 1489, 1593

Offset: 1

Author

Wesley Ivan Hurt, Jun 24 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 4 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=4.

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), A373663 (k=3), this sequence (k=4), A374005 (k=5), A374007 (k=6), A374008 (k=7), A374009 (k=8), A374010 (k=9), A374011 (k=10).

Row 4 of the table in A056011.

Column 4 of the rectangular array in A056023.

Programs

Magma

[(1 + (n+3)^2 - (n-4)*(-1)^n)/2: n in [1..80]];

Mathematica

CoefficientList[Series[-(7*x^4 - 7*x^3 - 10*x^2 + 7 x + 7)/((x + 1)^2*(x - 1)^3), {x, 0, 50}], x]
k := 4; Table[(1 + (n+k-1)^2 + (n-k) (-1)^(n+k-1))/2, {n, 80}]

Formula

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

G.f.: -x*(7*x^4-7*x^3-10*x^2+7x+7)/((x+1)^2*(x-1)^3).

a(n) = A373663(n+1) + (-1)^n.