A188647 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.

1, 5, 1, 29, 17, 1, 169, 305, 37, 1, 985, 5473, 1405, 65, 1, 5741, 98209, 53353, 4289, 101, 1, 33461, 1762289, 2026009, 283009, 10301, 145, 1, 195025, 31622993, 76934989, 18674305, 1050601, 21169, 197, 1, 1136689, 567451585, 2921503573, 1232221121, 107151001, 3090529, 39005, 257, 1

Offset: 0

Author

Charles L. Hohn, Apr 06 2011

Keywords

• nonn
• tabl

Comments

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.

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

Examples

			Square array begins:
     | 0    1       2          3             4
-----+---------------------------------------------
   1 | 1,   5,     29,       169,          985, ...
   2 | 1,  17,    305,      5473,        98209, ...
   3 | 1,  37,   1405,     53353,      2026009, ...
   4 | 1,  65,   4289,    283009,     18674305, ...
   5 | 1, 101,  10301,   1050601,    107151001, ...
   6 | 1, 145,  21169,   3090529,    451196065, ...
   7 | 1, 197,  39005,   7722793,   1529074009, ...
   8 | 1, 257,  66305,  17106433,   4413393409, ...
   9 | 1, 325, 105949,  34539049,  11259624025, ...
  10 | 1, 401, 161201,  64802401,  26050404001, ...
  11 | 1, 485, 235709, 114554089,  55673051545, ...
  12 | 1, 577, 333505, 192765313, 111418017409, ...
  13 | 1, 677, 459005, 311204713, 210996336409, ...
  14 | 1, 785, 617009, 484968289, 381184458145, ...
  15 | 1, 901, 812701, 733055401, 661215159001, ...
  ...
		

Links

• Wikipedia, Chebyshev polynomials.
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

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.

Column 1 is A053755.

A(n,n) gives A323012.

Cf. A188645, A188646 (f(x, y) as above with y=-1).

Formula

A(n,k) = 2 * A188645(n,k) - A(n,k-1).

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

Extensions

Edited and extended by Seiichi Manyama, Jan 02 2019