cp's OEIS Frontend

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.

%I A159678 #49 Jun 24 2025 06:38:46
%S A159678 1,17,271,4319,68833,1097009,17483311,278635967,4440692161,
%T A159678 70772438609,1127918325583,17975920770719,286486814005921,
%U A159678 4565813103324017,72766522839178351,1159698552323529599,18482410314337295233,294558866477073194129,4694459453318833810831
%N 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.
%C A159678 The sequence a(j) is A157456, the sequence n(j) is A159679, the sequence b(j) the sequence given here.
%C A159678 Numbers k such that 7*k^2 + 2 is a square. - _Colin Barker_, Mar 17 2014
%H A159678 Colin Barker, <a href="/A159678/b159678.txt">Table of n, a(n) for n = 1..800</a>
%H A159678 K. Andersen, L. Carbone, and D. Penta, <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 A159678 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,-1).
%F A159678 The b(j) recurrence (this sequence) is b(1)=1, b(2)=17, b(t+2) = 16*b(t+1) - b(t).
%F A159678 From _R. J. Mathar_, Oct 31 2011: (Start)
%F A159678 G.f.: x*(1+x) / ( 1-16*x+x^2 ).
%F A159678 a(n) = A077412(n-1) + A077412(n-2). (End)
%F A159678 a(n) = 16*a(n-1) - a(n-2), with a(1)=1, a(2)=17. - _Harvey P. Dale_, Dec 25 2011
%F A159678 a(n) = ( (3-sqrt(7))*(8+3*sqrt(7))^n - (3+sqrt(7))*(8-3*sqrt(7))^n )/(2*sqrt(7)). - _Colin Barker_, Jul 25 2016
%p A159678 for a from 1 by 2 to 100000 do b:=sqrt((9*a*a-2)/7): if (trunc(b)=b) then
%p A159678 n:=(a*a-1)/7: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: end if: end do:
%p A159678 # Second program
%p A159678 seq(simplify(ChebyshevU(n-1,8) + ChebyshevU(n-2,8)), n=1..30); # _G. C. Greubel_, Sep 27 2022
%t A159678 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 A159678 (Sage) [(lucas_number2(n,16,1)-lucas_number2(n-1,16,1))/14 for n in range(1, 20)] # _Zerinvary Lajos_, Nov 10 2009
%o A159678 (PARI) Vec(x*(1+x)/(1-16*x+x^2) + O(x^30)) \\ _Michel Marcus_, Jan 03 2016
%o A159678 (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 A159678 (Magma) [n le 2 select 17^(n-1) else 16*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jun 03 2018
%Y A159678 Cf. A077412, A157456, A159679, A266698.
%K A159678 nonn,easy
%O A159678 1,2
%A A159678 _Paul Weisenhorn_, Apr 19 2009
%E A159678 More terms from _Zerinvary Lajos_, Nov 10 2009