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A188645
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Array of ((k^n)+(k^(-n)))/2 where k=(sqrt(x^2+1)+x)^2 for integers x>=1.
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13
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1, 3, 1, 17, 9, 1, 99, 161, 19, 1, 577, 2889, 721, 33, 1, 3363, 51841, 27379, 2177, 51, 1, 19601, 930249, 1039681, 143649, 5201, 73, 1, 114243, 16692641, 39480499, 9478657, 530451, 10657, 99, 1, 665857, 299537289, 1499219281, 625447713, 54100801, 1555849, 19601, 129, 1
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OFFSET
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0,2
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COMMENTS
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Conjecture: Given function f(x, y)=(sqrt(x^2+y)+x)^2; and constant k=f(x, y); then for all integers x>=1 and y=[+-]1, k may be irrational, but ((k^n)+(k^(-n)))/2 always produces integer sequences; y=1 results shown here; y=-1 results are A188644.
Also square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{k}(x), evaluated at x=2*n^2+1. - Seiichi Manyama, Jan 01 2019
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LINKS
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FORMULA
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A(n,k) = Sum_{j=0..k} binomial(2*k,2*j)*(n^2+1)^(k-j)*n^(2*j). - Seiichi Manyama, Jan 01 2019
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EXAMPLE
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Square array begins:
| 0 1 2 3 4
-----+---------------------------------------------
1 | 1, 3, 17, 99, 577, ...
2 | 1, 9, 161, 2889, 51841, ...
3 | 1, 19, 721, 27379, 1039681, ...
4 | 1, 33, 2177, 143649, 9478657, ...
5 | 1, 51, 5201, 530451, 54100801, ...
6 | 1, 73, 10657, 1555849, 227143297, ...
7 | 1, 99, 19601, 3880899, 768398401, ...
8 | 1, 129, 33281, 8586369, 2215249921, ...
9 | 1, 163, 53137, 17322499, 5647081537, ...
10 | 1, 201, 80801, 32481801, 13057603201, ...
11 | 1, 243, 118097, 57394899, 27893802817, ...
12 | 1, 289, 167041, 96549409, 55805391361, ...
13 | 1, 339, 229841, 155831859, 105653770561, ...
14 | 1, 393, 308897, 242792649, 190834713217, ...
15 | 1, 451, 406801, 366934051, 330974107201, ...
...
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MATHEMATICA
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max = 9; y = 1; t = Table[k = ((x^2 + y)^(1/2) + x)^2; ((k^n) + (k^(-n)))/2 // FullSimplify, {n, 0, max - 1}, {x, 1, max}]; Table[ t[[n - k + 1, k]], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 17 2013 *)
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CROSSREFS
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A188644 (f(x, y) as above with y=-1).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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