A188647 - OEIS

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A188647

Array read by anti-diagonals of a(n)=a(n-1)*k-((k-1)/(k^n)) where a(0)=1 and k=((x^2+1)^(1/2)+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 (list; table; graph; refs; listen; history; text; internal format)

	OFFSET	`0,2`
	COMMENTS	`Conjecture by C Hohn: Given function f(x, y)=((x^2+y)^(1/2)+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`
	LINKS	`Table of n, a(n) for n=0..34.`
	FORMULA	`a(n)=A188645(n)*2-a(n-1)`
	EXAMPLE	`1, 5, 29, 169, 985, 5741, 33461, 195025, ...` `1, 17, 305, 5473, 98209, 1762289, ...` `1, 37, 1405, 53353, 2026009, 76934989, ...` `1, 65, 4289, 283009, 18674305, ...` `1, 101, 10301, 1050601, 107151001, ...` `1, 145, 21169, 3090529, 451196065, ...` `1, 197, 39005, 7722793, 1529074009, ...` `1, 257, 66305, 17106433, 4413393409, ...` `1, 325, 105949, 34539049, 11259624025, ...` `1, 401, 161201, 64802401, 26050404001, ...` `1, 485, 235709, 114554089, 55673051545, ...` `1, 577, 333505, 192765313, 111418017409, ...` `1, 677, 459005, 311204713, ...` `1, 785, 617009, 484968289, ...` `1, 901, 812701, 733055401, ...` `...`
	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 2 is A053755.` `Cf. A188646 (f(x, y) as above with y=-1).` `Sequence in context: A125906 A146414 A146374 * A214882 A144890 A144891` `Adjacent sequences: A188644 A188645 A188646 * A188648 A188649 A188650`
	KEYWORD	`nonn,tabl`
	AUTHOR	`Charles L. Hohn, Apr 06 2011`
	STATUS	`approved`

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Last modified May 23 03:06 EDT 2013. Contains 225585 sequences.