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A374010
a(n) = (1 + (n+8)^2 + (n-9)*(-1)^n)/2.
9
45, 47, 64, 70, 87, 97, 114, 128, 145, 163, 180, 202, 219, 245, 262, 292, 309, 343, 360, 398, 415, 457, 474, 520, 537, 587, 604, 658, 675, 733, 750, 812, 829, 895, 912, 982, 999, 1073, 1090, 1168, 1185, 1267, 1284, 1370, 1387, 1477, 1494, 1588, 1605, 1703, 1720, 1822
OFFSET
1,1
COMMENTS
Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 9 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=9.
FORMULA
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: -x*(28*x^4+2*x^3-73*x^2+2*x+45)/((x+1)^2*(x-1)^3).
a(n) = A374009(n+1) - (-1)^n.
EXAMPLE
[ 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 ...
...
MATHEMATICA
CoefficientList[Series[-(28*x^4 + 2*x^3 - 73*x^2 + 2*x + 45)/((x + 1)^2*(x - 1)^3), {x, 0, 50}], x]
k := 9; Table[(1 + (n + k - 1)^2 + (n - k) (-1)^(n + k - 1))/2, {n, 80}]
PROG
(Magma) [(1 + (n+8)^2 + (n-9)*(-1)^n)/2: n in [1..80]];
CROSSREFS
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), A374009 (k=8), this sequence (k=9), A374011 (k=10).
Row 9 of the table in A056011.
Column 9 of the rectangular array in A056023.
Sequence in context: A042011 A031064 A183983 * A116334 A291788 A291787
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Jun 24 2024
STATUS
approved