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 A093700 Number of 9's immediately following the decimal point in the expansion of (3+sqrt(8))^n. 0
 0, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 13, 13, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 22, 22, 23, 24, 25, 26, 26, 27, 28, 29, 29, 30, 31, 32, 32, 33, 34, 35, 35, 36, 37, 38, 39, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 48, 49, 50, 51, 52, 52, 53, 54, 55, 55, 56, 57 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Number of 0's immediately following the decimal point in the expansion of (3-sqrt(8))^n. LINKS Math Forum, Triangular Numbers That are Perfect Squares Math Pages, On m = sqrt(sqrt(n) + sqrt(kn+1)) [Wrong link] Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Convergence Acceleration of Alternating Series, Experiment. Math. Volume 9, Issue 1 (2000), 3-12, Project Euclid - Cornell Univ (see Proposition 1). Robert Simms, Using counting numbers to generate Pythagorean triples FORMULA Roughly, floor(3*n/4) EXAMPLE Let n=10, (3+sqrt(8))^10= 45239073.9999999778... (the fractional part starts with seven 9's), so the 10th element in this sequence is 7. The 132nd element is 100. The 1000th element is 765. The 1307th element is 1000. The arrangement of repeating elements are like A074184 (Index of the smallest power of n >= n!) and A076539 (Numerators a(n) of fractions slowly converging to pi) and A080686 (Number of 19-smooth numbers <= n). MATHEMATICA For[n = 1, n < 999, n++, Block[{\$MaxExtraPrecision = 50*n}, Print[ -Floor[Log[10, 1 - N[FractionalPart[(3 + 2Sqrt)^n], n]]] - 1]]] f[n_] := Block[{}, -MantissaExponent[(3 - Sqrt)^n][]]; Table[ f[n], {n, 75}] (* Robert G. Wilson v, Apr 10 2004 *) CROSSREFS Cf. A003499, A050608, A080686, A076539, A074184. Sequence in context: A076539 A074184 A187329 * A213854 A214449 A118168 Adjacent sequences:  A093697 A093698 A093699 * A093701 A093702 A093703 KEYWORD nonn,base AUTHOR Marvin Ray Burns, Apr 10 2004 STATUS approved

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Last modified June 20 02:34 EDT 2021. Contains 345154 sequences. (Running on oeis4.)