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 A159678 The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2-equation problem 7*n(j) + 1 = a(j)*a(j) and 9*n(j) + 1 = b(j)*b(j) with positive integer numbers. 6
 1, 17, 271, 4319, 68833, 1097009, 17483311, 278635967, 4440692161, 70772438609, 1127918325583, 17975920770719, 286486814005921, 4565813103324017, 72766522839178351, 1159698552323529599, 18482410314337295233, 294558866477073194129, 4694459453318833810831 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The sequence a(j) is A157456, the sequence n(j) is A159679, the sequence b(j) the sequence given here. Numbers n such that 7*n^2 + 2 is a square. - Colin Barker, Mar 17 2014 LINKS Colin Barker, Table of n, a(n) for n = 1..800 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 (16,-1). FORMULA The b(j) recurrence (this sequence here) is b(1)=1; b(2)=17; b(t+2) = 16*b(t+1) - b(t). G.f. x*(1+x) / ( 1-16*x+x^2 ). a(n) = A077412(n-1) + A077412(n-2). - R. J. Mathar, Oct 31 2011 a(1)=1, a(2)=17, a(n) = 16*a(n-1) - a(n-2). - Harvey P. Dale, Dec 25 2011 a(n) = (-(8-3*sqrt(7))^n*(3+sqrt(7))-(-3+sqrt(7))*(8+3*sqrt(7))^n)/(2*sqrt(7)). - Colin Barker, Jul 25 2016 MAPLE for a from 1 by 2 to 100000 do b:=sqrt((9*a*a-2)/7): if (trunc(b)=b) then n:=(a*a-1)/7: La:=[op(La), a]:Lb:=[op(Lb), b]:Ln:=[op(Ln), n]: end if: end do: MATHEMATICA Rest[CoefficientList[Series[x (1+x)/(1-16x+x^2), {x, 0, 30}], x]] (* or *) LinearRecurrence[{16, -1}, {1, 17}, 30] (* Harvey P. Dale, Dec 25 2011 *) PROG (Sage) [(lucas_number2(n, 16, 1)-lucas_number2(n-1, 16, 1))/14 for n in xrange(1, 20)] # Zerinvary Lajos, Nov 10 2009 (PARI) Vec(x*(1+x)/(1-16*x+x^2) + O(x^30)) \\ Michel Marcus, Jan 03 2016 (PARI) a(n) = round((-(8-3*sqrt(7))^n*(3+sqrt(7))-(-3+sqrt(7))*(8+3*sqrt(7))^n)/(2*sqrt(7))) \\ Colin Barker, Jul 25 2016 (MAGMA) I:=[1, 17]; [n le 2 select I[n] else 16*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jun 03 2018 CROSSREFS Cf. A077412, A157456, A159679, A266698. Sequence in context: A259849 A090380 A142898 * A162803 A097830 A163093 Adjacent sequences:  A159675 A159676 A159677 * A159679 A159680 A159681 KEYWORD nonn,easy AUTHOR Paul Weisenhorn, Apr 19 2009 EXTENSIONS More terms from Zerinvary Lajos, Nov 10 2009 STATUS approved

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Last modified June 18 14:52 EDT 2019. Contains 324213 sequences. (Running on oeis4.)