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 A161591 The list of the B values in the common solutions to the 2 equations 13*k + 1 = A^2, 17*k + 1 = B^2. 3
 1, 16, 239, 3569, 53296, 795871, 11884769, 177475664, 2650250191, 39576277201, 590993907824, 8825332340159, 131788991194561, 1968009535578256, 29388354042479279, 438857301101610929, 6553471162481684656, 97863210136123658911, 1461394680879373199009 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The 2 equations are equivalent to the Pell equation x^2 - 221*y^2 = 1, with x = (221*k+15)/2 and y = A*B/2, case C=13 in A160682. LINKS Andersen, K., Carbone, L. and Penta, D., Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9. Index entries for linear recurrences with constant coefficients, signature (15,-1). FORMULA B(t+2) = 15*B(t+1) - B(t). B(t) = ((221+17*w)*((15+w)/2)^(t-1) + (221-17*w)*((15-w)/2)^(t-1))/442 where w=sqrt(221). B(t) = floor of ((221+17*w)*((15+w)/2)^(t-1))/442 = A078364(t-2) + A078364(t-1). G.f.: x*(1+x)/(1-15*x+x^2). MAPLE t:=0: for b from 1 to 1000000 do a:=sqrt((13*b^2+4)/17): if (trunc(a)=a) then t:=t+1: n:=(b^2-1)/17: print(t, a, b, n): end if: end do: MATHEMATICA LinearRecurrence[{15, -1}, {1, 16}, 30] (* Harvey P. Dale, Dec 04 2015 *) PROG (Sage) [(lucas_number2(n, 15, 1)-lucas_number2(n-1, 15, 1))/13 for n in range(1, 20)] # Zerinvary Lajos, Nov 10 2009 CROSSREFS Cf. A160682 (sequence of A), A161584 (sequence of k). Sequence in context: A266099 A264343 A283411 * A227440 A103975 A162791 Adjacent sequences: A161588 A161589 A161590 * A161592 A161593 A161594 KEYWORD nonn AUTHOR Paul Weisenhorn, Jun 14 2009 EXTENSIONS Edited, extended by R. J. Mathar, Sep 02 2009 STATUS approved

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Last modified December 10 02:09 EST 2022. Contains 358712 sequences. (Running on oeis4.)