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

%I A373663 #27 Jun 24 2024 19:43:31
%S A373663 6,8,13,19,24,34,39,53,58,76,81,103,108,134,139,169,174,208,213,251,
%T A373663 256,298,303,349,354,404,409,463,468,526,531,593,598,664,669,739,744,
%U A373663 818,823,901,906,988,993,1079,1084,1174,1179,1273,1278,1376,1381,1483,1488,1594
%N A373663 a(n) = (1 + (n+2)^2 + (n-3)*(-1)^n)/2.
%C A373663 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).
%C A373663 In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=3.
%H A373663 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A373663 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
%F A373663 a(n) = A373662(n+1) - (-1)^n.
%F A373663 G.f.: -x*(x^4+2*x^3-7*x^2+2*x+6)/((x+1)^2*(x-1)^3).
%e A373663        [ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9] [10] [11] [12]
%e A373663   [ 1]   1    3    4   10   11   21   22   36   37   55   56   78   ...
%e A373663   [ 2]   2    5    9   12   20   23   35   38   54   57   77   ...
%e A373663   [ 3]   6    8   13   19   24   34   39   53   58   76   ...
%e A373663   [ 4]   7   14   18   25   33   40   52   59   75   ...
%e A373663   [ 5]  15   17   26   32   41   51   60   74   ...
%e A373663   [ 6]  16   27   31   42   50   61   73   ...
%e A373663   [ 7]  28   30   43   49   62   72   ...
%e A373663   [ 8]  29   44   48   63   71   ...
%e A373663   [ 9]  45   47   64   70   ...
%e A373663   [10]  46   65   69   ...
%e A373663   [11]  66   68   ...
%e A373663   [12]  67   ...
%e A373663         ...
%t A373663 k := 3; Table[(1 + (n+k-1)^2 + (n-k) (-1)^(n+k-1))/2, {n, 80}]
%o A373663 (Magma) [(1 + (n+2)^2 + (n-3)*(-1)^n)/2: n in [1..80]];
%o A373663 (Python)
%o A373663 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
%Y A373663 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).
%Y A373663 Row 3 of the example in A056011, Column 3 of the rectangular array in A056023.
%K A373663 nonn,easy
%O A373663 1,1
%A A373663 _Wesley Ivan Hurt_, Jun 12 2024