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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

%I

%S 1,17,271,4319,68833,1097009,17483311,278635967,4440692161,

%T 70772438609,1127918325583,17975920770719,286486814005921,

%U 4565813103324017,72766522839178351,1159698552323529599,18482410314337295233,294558866477073194129,4694459453318833810831

%N 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.

%C The sequence a(j) is A157456, the sequence n(j) is A159679, the sequence b(j) the sequence given here.

%C Numbers n such that 7*n^2 + 2 is a square. - _Colin Barker_, Mar 17 2014

%H Colin Barker, <a href="/A159678/b159678.txt">Table of n, a(n) for n = 1..800</a>

%H Andersen, K., Carbone, L. and Penta, D., <a href="https://pdfs.semanticscholar.org/8f0c/c3e68d388185129a56ed73b5d21224659300.pdf">Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields</a>, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,-1).

%F The b(j) recurrence (this sequence here) is b(1)=1; b(2)=17; b(t+2) = 16*b(t+1) - b(t).

%F G.f. x*(1+x) / ( 1-16*x+x^2 ). a(n) = A077412(n-1) + A077412(n-2). - _R. J. Mathar_, Oct 31 2011

%F a(1)=1, a(2)=17, a(n) = 16*a(n-1) - a(n-2). - _Harvey P. Dale_, Dec 25 2011

%F 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

%p for a from 1 by 2 to 100000 do b:=sqrt((9*a*a-2)/7): if (trunc(b)=b) then

%p n:=(a*a-1)/7: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: end if: end do:

%t 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 *)

%o (Sage) [(lucas_number2(n,16,1)-lucas_number2(n-1,16,1))/14 for n in xrange(1, 20)] # _Zerinvary Lajos_, Nov 10 2009

%o (PARI) Vec(x*(1+x)/(1-16*x+x^2) + O(x^30)) \\ _Michel Marcus_, Jan 03 2016

%o (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

%o (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

%Y Cf. A077412, A157456, A159679, A266698.

%K nonn,easy

%O 1,2

%A _Paul Weisenhorn_, Apr 19 2009

%E More terms from _Zerinvary Lajos_, Nov 10 2009

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Last modified July 17 16:38 EDT 2019. Contains 325107 sequences. (Running on oeis4.)