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%I #32 Jan 01 2019 12:02:03
%S 1,3,1,17,9,1,99,161,19,1,577,2889,721,33,1,3363,51841,27379,2177,51,
%T 1,19601,930249,1039681,143649,5201,73,1,114243,16692641,39480499,
%U 9478657,530451,10657,99,1,665857,299537289,1499219281,625447713,54100801,1555849,19601,129,1
%N Array of ((k^n)+(k^(-n)))/2 where k=(sqrt(x^2+1)+x)^2 for integers x>=1.
%C 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.
%C 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
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A(n,k) = (A188647(n,k-1) + A188647(n,k))/2.
%F 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
%e Square array begins:
%e | 0 1 2 3 4
%e -----+---------------------------------------------
%e 1 | 1, 3, 17, 99, 577, ...
%e 2 | 1, 9, 161, 2889, 51841, ...
%e 3 | 1, 19, 721, 27379, 1039681, ...
%e 4 | 1, 33, 2177, 143649, 9478657, ...
%e 5 | 1, 51, 5201, 530451, 54100801, ...
%e 6 | 1, 73, 10657, 1555849, 227143297, ...
%e 7 | 1, 99, 19601, 3880899, 768398401, ...
%e 8 | 1, 129, 33281, 8586369, 2215249921, ...
%e 9 | 1, 163, 53137, 17322499, 5647081537, ...
%e 10 | 1, 201, 80801, 32481801, 13057603201, ...
%e 11 | 1, 243, 118097, 57394899, 27893802817, ...
%e 12 | 1, 289, 167041, 96549409, 55805391361, ...
%e 13 | 1, 339, 229841, 155831859, 105653770561, ...
%e 14 | 1, 393, 308897, 242792649, 190834713217, ...
%e 15 | 1, 451, 406801, 366934051, 330974107201, ...
%e ...
%t 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 *)
%Y Row 1 is A001541, row 2 is A023039, row 3 is A078986, row 4 is A099370, row 5 is A099397, row 6 is A174747, row 8 is A176368, (row 1)*2 is A003499, (row 2)*2 is A087215.
%Y Column 1 is A058331, (column 1)*2 is A005899.
%Y A188644 (f(x, y) as above with y=-1).
%Y Diagonal gives A173128.
%Y Cf. A188647.
%K nonn,tabl
%O 0,2
%A _Charles L. Hohn_, Apr 06 2011
%E Edited and extended by _Seiichi Manyama_, Jan 01 2019
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