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A374009 a(n) = (1 + (n+7)^2 - (n-8)*(-1)^n)/2.

%I A374009 #6 Jun 24 2024 19:23:35
%S A374009 29,44,48,63,71,86,98,113,129,144,164,179,203,218,246,261,293,308,344,
%T A374009 359,399,414,458,473,521,536,588,603,659,674,734,749,813,828,896,911,
%U A374009 983,998,1074,1089,1169,1184,1268,1283,1371,1386,1478,1493,1589,1604,1704,1719
%N A374009 a(n) = (1 + (n+7)^2 - (n-8)*(-1)^n)/2.
%C A374009 Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 8 of the boustrophedon-style array (see example).
%C A374009 In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=8.
%H A374009 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A374009 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
%F A374009 G.f.: -x*(29*x^4-15*x^3-54*x^2+15*x+29)/((x+1)^2*(x-1)^3).
%F A374009 a(n) = A374008(n+1) + (-1)^n.
%e A374009        [ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9] [10] [11] [12]
%e A374009   [ 1]   1    3    4   10   11   21   22   36   37   55   56   78   ...
%e A374009   [ 2]   2    5    9   12   20   23   35   38   54   57   77   ...
%e A374009   [ 3]   6    8   13   19   24   34   39   53   58   76   ...
%e A374009   [ 4]   7   14   18   25   33   40   52   59   75   ...
%e A374009   [ 5]  15   17   26   32   41   51   60   74   ...
%e A374009   [ 6]  16   27   31   42   50   61   73   ...
%e A374009   [ 7]  28   30   43   49   62   72   ...
%e A374009   [ 8]  29   44   48   63   71   ...
%e A374009   [ 9]  45   47   64   70   ...
%e A374009   [10]  46   65   69   ...
%e A374009   [11]  66   68   ...
%e A374009   [12]  67   ...
%e A374009         ...
%t A374009 CoefficientList[Series[-(29*x^4 - 15*x^3 - 54*x^2 + 15*x + 29)/((x + 1)^2*(x - 1)^3), {x, 0, 50}], x]
%t A374009 k := 8; Table[(1 + (n + k - 1)^2 + (n - k) (-1)^(n + k - 1))/2, {n, 80}]
%o A374009 (Magma) [(1 + (n+7)^2 - (n-8)*(-1)^n)/2: n in [1..80]];
%Y A374009 For rows k = 1..10: A131179 (k=1) n>0, A373662 (k=2), A373663 (k=3), A374004 (k=4), A374005 (k=5), A374007 (k=6), A374008 (k=7), this sequence (k=8), A374010 (k=9), A374011 (k=10).
%Y A374009 Row 8 of the table in A056011.
%Y A374009 Column 8 of the rectangular array in A056023.
%K A374009 nonn,easy
%O A374009 1,1
%A A374009 _Wesley Ivan Hurt_, Jun 24 2024