A374005 - OEIS

%I #8 Jun 24 2024 19:44:22

%S 15,17,26,32,41,51,60,74,83,101,110,132,141,167,176,206,215,249,258,

%T 296,305,347,356,402,411,461,470,524,533,591,600,662,671,737,746,816,

%U 825,899,908,986,995,1077,1086,1172,1181,1271,1280,1374,1383,1481,1490,1592,1601

%N a(n) = (1 + (n+4)^2 + (n-5)*(-1)^n)/2.

%C Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 5 of the boustrophedon-style array (see example).

%C In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=5.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

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

%F G.f.: -x*(6*x^4+2*x^3-21*x^2+2*x+15)/((x+1)^2*(x-1)^3).

%F a(n) = A374004(n+1) - (-1)^n.

%e        [ 1] [ 2] [ 3] [ 4] [ 5] [ 6] [ 7] [ 8] [ 9] [10] [11] [12]

%e   [ 1]   1    3    4   10   11   21   22   36   37   55   56   78   ...

%e   [ 2]   2    5    9   12   20   23   35   38   54   57   77   ...

%e   [ 3]   6    8   13   19   24   34   39   53   58   76   ...

%e   [ 4]   7   14   18   25   33   40   52   59   75   ...

%e   [ 5]  15   17   26   32   41   51   60   74   ...

%e   [ 6]  16   27   31   42   50   61   73   ...

%e   [ 7]  28   30   43   49   62   72   ...

%e   [ 8]  29   44   48   63   71   ...

%e   [ 9]  45   47   64   70   ...

%e   [10]  46   65   69   ...

%e   [11]  66   68   ...

%e   [12]  67   ...

%e         ...

%t CoefficientList[Series[-(6*x^4 + 2*x^3 - 21*x^2 + 2*x + 15)/((x + 1)^2*(x - 1)^3), {x, 0, 50}], x]

%t k := 5; Table[(1 + (n + k - 1)^2 + (n - k) (-1)^(n + k - 1))/2, {n, 80}]

%o (Magma) [(1 + (n+4)^2 + (n-5)*(-1)^n)/2: n in [1..80]];

%Y For rows k = 1..10: A131179 (k=1) n>0, A373662 (k=2), A373663 (k=3), A374004 (k=4), this sequence (k=5), A374007 (k=6), A374008 (k=7), A374009 (k=8), A374010 (k=9), A374011 (k=10).

%Y Row 5 of the table in A056011.

%Y Column 5 of the rectangular array in A056023.

%K nonn,easy

%O 1,1

%A _Wesley Ivan Hurt_, Jun 24 2024