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A006452 a(n) = 6*a(n-2) - a(n-4).
(Formerly M1245)
24
1, 1, 2, 4, 11, 23, 64, 134, 373, 781, 2174, 4552, 12671, 26531, 73852, 154634, 430441, 901273, 2508794, 5253004, 14622323, 30616751, 85225144, 178447502, 496728541, 1040068261, 2895146102, 6061962064, 16874148071, 35331704123 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Solution to a Diophantine equation.

n such that n^2-1 is a triangular number. - Benoit Cloitre, Apr 05 2002

For all elements "x" of the sequence, 8*x^2 - 7 is a square. Lim n-> inf. a(n)/a(n-2) = 3 + sqrt(8). If n is odd, lim n -> inf. a(n)/a(n-1) = (9 + 2*sqrt(8))/7. If n is even, lim n -> inf. a(n)/a(n-1) = (11 + 3*sqrt(8))/7. - Gregory V. Richardson, Oct 07 2002

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

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

Except for the first term, positive values of x (or y) satisfying x^2 - 6xy + y^2 + 7 = 0. - Colin Barker, Feb 04 2014

Except for the first term, positive values of x (or y) satisfying x^2 - 34xy + y^2 + 252 = 0. - Colin Barker, Mar 04 2014

From Wolfdieter Lang, Feb 26 2015: (Start)

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).

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)

REFERENCES

A. J. Gottlieb, How four dogs meet in a field, etc., Technology Review, Jul/Aug 1973 pp. 73-74.

Jeffrey Shallit, personal communication.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

A. J. Gottlieb, How four dogs meet in a field, etc. (scanned copy)

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

FORMULA

Bisection: a(2n) = A006452(n). a(2n+1) = A038723(n).

G.f.: ( 1+x-4*x^2-2*x^3 ) / ( (x^2+2*x-1)*(x^2-2*x-1) ).

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)) 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)). - Gregory V. Richardson, Oct 07 2002

a(n+2) = (A001333(n+3) + (-1)^n *A001333(n))/4. - Paul Curtz, Dec 06 2012

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

a(n+2) = (A216134(n+2) - A216134(n))/2. - Raphie Frank, Feb 11 2013

EXAMPLE

n = 3: 11^2 - 2*(2*4)^2 = -7 (see the Pell comment above);

(4*4)^2 - 2*11^2 = +14. - Wolfdieter Lang, Feb 26 2015

MAPLE

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

MATHEMATICA

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 *)

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 *)

CoefficientList[Series[(1 + x - 4 x^2 - 2 x^3) / ((x^2 + 2 x - 1) (x^2 - 2 x - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)

PROG

(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

(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

CROSSREFS

Cf. A006451, A006454, A006452, A038723, A077446.

Sequence in context: A026531 A038047 A061152 * A104430 A256801 A103669

Adjacent sequences:  A006449 A006450 A006451 * A006453 A006454 A006455

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Jeffrey Shallit

EXTENSIONS

More terms from James A. Sellers, May 03 2000

STATUS

approved

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Last modified August 19 06:42 EDT 2017. Contains 290794 sequences.