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

%I A374004 #7 Jun 24 2024 19:27:13
%S A374004 7,14,18,25,33,40,52,59,75,82,102,109,133,140,168,175,207,214,250,257,
%T A374004 297,304,348,355,403,410,462,469,525,532,592,599,663,670,738,745,817,
%U A374004 824,900,907,987,994,1078,1085,1173,1180,1272,1279,1375,1382,1482,1489,1593
%N A374004 a(n) = (1 + (n+3)^2 - (n-4)*(-1)^n)/2.
%C A374004 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).
%C A374004 In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=4.
%H A374004 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A374004 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
%F A374004 G.f.: -x*(7*x^4-7*x^3-10*x^2+7x+7)/((x+1)^2*(x-1)^3).
%F A374004 a(n) = A373663(n+1) + (-1)^n.
%e A374004        [ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9] [10] [11] [12]
%e A374004   [ 1]   1    3    4   10   11   21   22   36   37   55   56   78   ...
%e A374004   [ 2]   2    5    9   12   20   23   35   38   54   57   77   ...
%e A374004   [ 3]   6    8   13   19   24   34   39   53   58   76   ...
%e A374004   [ 4]   7   14   18   25   33   40   52   59   75   ...
%e A374004   [ 5]  15   17   26   32   41   51   60   74   ...
%e A374004   [ 6]  16   27   31   42   50   61   73   ...
%e A374004   [ 7]  28   30   43   49   62   72   ...
%e A374004   [ 8]  29   44   48   63   71   ...
%e A374004   [ 9]  45   47   64   70   ...
%e A374004   [10]  46   65   69   ...
%e A374004   [11]  66   68   ...
%e A374004   [12]  67   ...
%e A374004         ...
%t A374004 CoefficientList[Series[-(7*x^4 - 7*x^3 - 10*x^2 + 7 x + 7)/((x + 1)^2*(x - 1)^3), {x, 0, 50}], x]
%t A374004 k := 4; Table[(1 + (n+k-1)^2 + (n-k) (-1)^(n+k-1))/2, {n, 80}]
%o A374004 (Magma) [(1 + (n+3)^2 - (n-4)*(-1)^n)/2: n in [1..80]];
%Y A374004 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).
%Y A374004 Row 4 of the table in A056011.
%Y A374004 Column 4 of the rectangular array in A056023.
%K A374004 nonn,easy
%O A374004 1,1
%A A374004 _Wesley Ivan Hurt_, Jun 24 2024