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A374084
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a(n) = (1 + (n+1)^2 + (n-2)*(-1)^n)/2.
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1
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3, 5, 8, 14, 17, 27, 30, 44, 47, 65, 68, 90, 93, 119, 122, 152, 155, 189, 192, 230, 233, 275, 278, 324, 327, 377, 380, 434, 437, 495, 498, 560, 563, 629, 632, 702, 705, 779, 782, 860, 863, 945, 948, 1034, 1037, 1127, 1130, 1224, 1227, 1325, 1328, 1430, 1433, 1539
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OFFSET
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1,1
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COMMENTS
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Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is column 2 of the boustrophedon-style array (see example).
In general, column k is given by (1+(t-1)^2+(n-k)*(-1)^t)/2, t = n+k. Here, k=2.
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LINKS
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FORMULA
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a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: -x*(2*x^3-3*x^2+2*x+3)/((x+1)^2*(x-1)^3).
E.g.f.: (2 + x)*(exp(x)*x + 2*sinh(x))/2. - Stefano Spezia, Jun 29 2024
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EXAMPLE
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[ 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 ...
...
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MATHEMATICA
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CoefficientList[Series[-(2*x^3 - 3*x^2 + 2*x + 3)/((x + 1)^2*(x - 1)^3), {x, 0, 50}], x]
k := 2; Table[(1 + (n + k - 1)^2 + (n - k) (-1)^(n + k))/2, {n, 60}]
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PROG
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(Magma) [(1 + (n+1)^2 + (n-2)*(-1)^n)/2: n in [1..80]];
(Python)
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CROSSREFS
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Cf. A128918 (Column 1) n>0, this sequence (Column 2).
Row 2 of the rectangular array in A056023.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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