|
%I M1245 #102 Jan 23 2023 05:00:01
%S 1,1,2,4,11,23,64,134,373,781,2174,4552,12671,26531,73852,154634,
%T 430441,901273,2508794,5253004,14622323,30616751,85225144,178447502,
%U 496728541,1040068261,2895146102,6061962064,16874148071,35331704123
%N a(n) = 6*a(n-2) - a(n-4).
%C Solution to a Diophantine equation.
%C Integers k such that k^2-1 is a triangular number. - _Benoit Cloitre_, Apr 05 2002
%C For all elements "x" of the sequence, 8*x^2 - 7 is a square. - _Gregory V. Richardson_, Oct 07 2002
%C a(n) mod 10 is a sequence of period 12: repeat (1, 1, 2, 4, 1, 3, 4, 4, 3, 1, 4, 2). - _Paul Curtz_, Dec 07 2012
%C a(n)^2 - 1 = A006454(n - 1) is a Sophie Germain triangular number of the second kind as defined in A217278. - _Raphie Frank_, Feb 08 2013
%C Except for the first term, positive values of x (or y) satisfying x^2 - 6xy + y^2 + 7 = 0. - _Colin Barker_, Feb 04 2014
%C Except for the first term, positive values of x (or y) satisfying x^2 - 34xy + y^2 + 252 = 0. - _Colin Barker_, Mar 04 2014
%C From _Wolfdieter Lang_, Feb 26 2015: (Start)
%C a(n+1), for n >= 0, gives one half of all positive y solutions of the Pell equation x^2 - 2*y^2 = -7. The corresponding x-solutions are x(n) = A077446(n+1).
%C See a comment on A077446 for the first and second class solutions separately, and the connection to the Pell equation X^2 - 2*Y^2 = 14. (End)
%C For n > 0, a(n) is the n-th almost balancing number of second type (see Tekcan and Erdem). - _Stefano Spezia_, Nov 26 2022
%D A. J. Gottlieb, How four dogs meet in a field, etc., Technology Review, Jul/Aug 1973 pp. 73-74.
%D Jeffrey Shallit, personal communication.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A006452/b006452.txt">Table of n, a(n) for n = 0..1000</a>
%H A. J. Gottlieb, <a href="/A006451/a006451.pdf">How four dogs meet in a field, etc.</a> (scanned copy)
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H J. Shallit, <a href="/A006449/a006449.pdf">Letter to N. J. A. Sloane, Oct. 1975</a>
%H Ahmet Tekcan and Alper Erdem, <a href="https://arxiv.org/abs/2211.08907">General Terms of All Almost Balancing Numbers of First and Second Type</a>, arXiv:2211.08907 [math.NT], 2022.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-1).
%F Bisection: a(2n) = A006452(n). a(2n+1) = A038723(n).
%F G.f.: ( 1+x-4*x^2-2*x^3 ) / ( (1-2*x-x^2)*(1+2*x-x^2) ).
%F From _Gregory V. Richardson_, Oct 07 2002: (Start)
%F For n (even), a(n) = ( ((3 + sqrt(8))^((n/2)+1) - (3 - sqrt(8))^((n/2)+1)) - 2*((3 + sqrt(8))^((n/2)-1) - (3 - sqrt(8))^((n/2)-1)) ) / (6*sqrt(8)).
%F For n (odd), a(n) = ( ((3 + sqrt(8))^((n+1)/2) - (3 - sqrt(8))^((n+1)/2)) - 2*((3 + sqrt(8))^((n-1)/2) - (3 - sqrt(8))^((n-1)/2)) ) / (2*sqrt(8)).
%F Limit_{n->oo} a(n)/a(n-2) = 3 + sqrt(8).
%F If n is odd, lim_{n->oo} a(n)/a(n-1) = (9 + 2*sqrt(8))/7.
%F If n is even, lim_{n->oo} a(n)/a(n-1) = (11 + 3*sqrt(8))/7. (End)
%F a(n+2) = (A001333(n+3) + (-1)^n *A001333(n))/4. - _Paul Curtz_, Dec 06 2012
%F a(n+2) = sqrt(17*a(n)^2 + 6*(sqrt(8*a(n)^2 - 7))*a(n)*sgn(2*n - 1) - 7) with a(0) = 1, a(1) = 1. - _Raphie Frank_, Feb 08 2013
%F a(n+2) = (A216134(n+2) - A216134(n))/2. - _Raphie Frank_, Feb 11 2013
%F E.g.f.: (2*cosh(sqrt(2)*x)*(2*cosh(x) - sinh(x)) + sqrt(2)*(3*cosh(x) - sinh(x))*sinh(sqrt(2)*x))/4. - _Stefano Spezia_, Nov 26 2022
%e n = 3: 11^2 - 2*(2*4)^2 = -7 (see the Pell comment above);
%e (4*4)^2 - 2*11^2 = +14. - _Wolfdieter Lang_, Feb 26 2015
%p A006452:=-(z-1)*(z**2+3*z+1)/(z**2+2*z-1)/(z**2-2*z-1); # conjectured by _Simon Plouffe_ in his 1992 dissertation; gives sequence except for one of the leading 1's
%t s=0;lst={1}; Do[s+=n;If[Sqrt[s+1]==Floor[Sqrt[s+1]],AppendTo[lst, Sqrt[s+1]]], {n,0,8!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Apr 02 2009 *)
%t a[0]=a[1]= 1; a[2]=2; a[3]=4; a[n_]:= 6*a[n-2] -a[n-4]; Array[a, 30, 0] (* _Robert G. Wilson v_, Jun 11 2010 *)
%t CoefficientList[Series[(1+x-4x^2-2x^3)/((1-2x-x^2)(1+2x-x^2)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 09 2013 *)
%o (Magma) I:=[1,1,2,4]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Jun 09 2013
%o (PARI) a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,6,0]^n*[1;1;2;4])[1,1] \\ _Charles R Greathouse IV_, May 10 2016
%o (SageMath)
%o def A001333(n): return lucas_number2(n, 2, -1)/2
%o def A006452(n): return (A001333(n+1) + (-1)^n *A001333(n-2))/4
%o [A006452(n) for n in range(41)] # _G. C. Greubel_, Jan 22 2023
%Y Cf. A001333, A006451, A006452, A006454, A038723, A077446, A216134, A217278.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, _Jeffrey Shallit_
%E More terms from _James A. Sellers_, May 03 2000
| 