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%I #45 Jan 03 2019 05:40:07
%S 1,5,1,29,17,1,169,305,37,1,985,5473,1405,65,1,5741,98209,53353,4289,
%T 101,1,33461,1762289,2026009,283009,10301,145,1,195025,31622993,
%U 76934989,18674305,1050601,21169,197,1,1136689,567451585,2921503573,1232221121,107151001,3090529,39005,257,1
%N Array read by antidiagonals of a(n) = a(n-1)*k-((k-1)/(k^n)) where a(0)=1 and k=(sqrt(x^2+1)+x)^2 for integers x>=1.
%C Conjecture: Given function f(x, y)=(sqrt(x^2+y)+x)^2; constant k=f(x, y); and initial term a(0)=1; then for all integers x>=1 and y=[+-]1, k may be irrational, but sequence a(n)=a(n-1)*k-((k-1)/(k^n)) always produces integer sequences; y=1 results shown here; y=-1 results are A188646.
%C Also square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is (1/sqrt(n^2+1)) * T_{2*k+1}(sqrt(n^2+1)), with T the Chebyshev polynomial of the first kind. - _Seiichi Manyama_, Jan 02 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) = 2 * A188645(n,k) - A(n,k-1).
%F A(n,k) = Sum_{j=0..k} binomial(2*k+1,2*j)*(n^2+1)^(k-j)*n^(2*j). - _Seiichi Manyama_, Jan 02 2019
%e Square array begins:
%e | 0 1 2 3 4
%e -----+---------------------------------------------
%e 1 | 1, 5, 29, 169, 985, ...
%e 2 | 1, 17, 305, 5473, 98209, ...
%e 3 | 1, 37, 1405, 53353, 2026009, ...
%e 4 | 1, 65, 4289, 283009, 18674305, ...
%e 5 | 1, 101, 10301, 1050601, 107151001, ...
%e 6 | 1, 145, 21169, 3090529, 451196065, ...
%e 7 | 1, 197, 39005, 7722793, 1529074009, ...
%e 8 | 1, 257, 66305, 17106433, 4413393409, ...
%e 9 | 1, 325, 105949, 34539049, 11259624025, ...
%e 10 | 1, 401, 161201, 64802401, 26050404001, ...
%e 11 | 1, 485, 235709, 114554089, 55673051545, ...
%e 12 | 1, 577, 333505, 192765313, 111418017409, ...
%e 13 | 1, 677, 459005, 311204713, 210996336409, ...
%e 14 | 1, 785, 617009, 484968289, 381184458145, ...
%e 15 | 1, 901, 812701, 733055401, 661215159001, ...
%e ...
%Y Row 1 is A001653, row 2 is A007805, row 3 is A097315, row 4 is A078988, row 5 is A097727, row 6 is A097730, row 7 is A097733, row 8 is A097736, row 9 is A097739, row 10 is A097742, row 11 is A097767, row 12 is A097770, row 13 is A097773.
%Y Column 1 is A053755.
%Y A(n,n) gives A323012.
%Y Cf. A188645, A188646 (f(x, y) as above with y=-1).
%K nonn,tabl
%O 0,2
%A _Charles L. Hohn_, Apr 06 2011
%E Edited and extended by _Seiichi Manyama_, Jan 02 2019
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