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A251599
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Centers of rows of the triangular array formed by the natural numbers.
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5
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1, 2, 3, 5, 8, 9, 13, 18, 19, 25, 32, 33, 41, 50, 51, 61, 72, 73, 85, 98, 99, 113, 128, 129, 145, 162, 163, 181, 200, 201, 221, 242, 243, 265, 288, 289, 313, 338, 339, 365, 392, 393, 421, 450, 451, 481, 512, 513, 545, 578, 579, 613, 648, 649, 685, 722, 723
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OFFSET
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1,2
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COMMENTS
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Forms a cascade of 3-number triangles down the center of the triangle array. Related to A000124 (left/west bank of same triangular array), A000217 (right/east bank) and A001844 (center column).
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LINKS
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FORMULA
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Terms for n=1 (mod 3): 2m^2+2m+1, for n=2 (mod 3): 2m^2+4m+2, for n=0 (mod 3): 2m^2+4m+3, where m = floor((n-1)/3).
G.f.: -x*(x^2+1)*(x^4-x^3+x+1)/((x^2+x+1)^2*(x-1)^3). - Alois P. Heinz, Dec 10 2014
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EXAMPLE
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First ten terms (1,2,3,5,8,9,13,18,19,25) may be read down the center of the triangular formation:
1
2 3
4 5 6
7 8 9 10
11 12 13 14 15
16 17 18 19 20 21
22 23 24 25 26 27 28
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MAPLE
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a:= n-> (m-> 2*(m+1)^2-[2*m+1, 0, -1][1+r])(iquo(n-1, 3, 'r')):
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MATHEMATICA
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LinearRecurrence[{1, 0, 2, -2, 0, -1, 1}, {1, 2, 3, 5, 8, 9, 13}, 60] (* Jean-François Alcover, Jan 09 2016 *)
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PROG
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(Haskell)
a251599 n = a251599_list !! (n-1)
a251599_list = f 0 $ g 1 [1..] where
f i (us:vs:wss) = [head $ drop i us] ++ (take 2 $ drop i vs) ++
f (i + 1) wss
g k zs = ys : g (k + 1) xs where (ys, xs) = splitAt k zs
(PARI) Vec(-x*(x^2+1)*(x^4-x^3+x+1)/((x^2+x+1)^2*(x-1)^3) + O(x^80)) \\ Michel Marcus, Jan 09 2016
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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