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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. 17
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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
LINKS
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
EXAMPLE
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, ...
...
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).
Sequence in context: A125906 A146414 A146374 * A232015 A214882 A144890
KEYWORD
nonn,tabl
AUTHOR
Charles L. Hohn, Apr 06 2011
EXTENSIONS
Edited and extended by Seiichi Manyama, Jan 02 2019
STATUS
approved

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Last modified March 19 06:56 EDT 2024. Contains 370953 sequences. (Running on oeis4.)