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 A001571 a(0) = 0, a(1) = 2, a(n) = 4a(n-1) - a(n-2) + 1. (Formerly M1928 N0762) 17
 0, 2, 9, 35, 132, 494, 1845, 6887, 25704, 95930, 358017, 1336139, 4986540, 18610022, 69453549, 259204175, 967363152, 3610248434, 13473630585, 50284273907, 187663465044, 700369586270, 2613814880037, 9754889933879, 36405744855480, 135868089488042 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Second member of the Diophantine pair (m,k) that satisfies 3(m^2+m)=k^2+k: a(n)=k. - Bruce Corrigan (scentman(AT)myfamily.com), Nov 04 2002 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..200 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. V. Thebault, Consecutive cubes with difference a square, Amer. Math. Monthly, 56 (1949), 174-175. FORMULA a(n) = (A001834(n)-1)/2. a(n) = -(1/2)-(1/4)*sqrt(3)*[2-sqrt(3)]^n+(1/4)*sqrt(3)*[2+sqrt(3)]^n+(1/4)*[2-sqrt(3)]^n+(1/4) *[2+sqrt(3)]^n, with n>=0. [Paolo P. Lava, Jul 31 2008] a(n) = sqrt((-2+(2-sqrt(3))^n+(2+sqrt(3))^n)*(2+(2-sqrt(3))^(1+n)+(2+sqrt(3))^(1+n)))/(2*sqrt(2)). - Gerry Martens, Jun 05 2015 MAPLE A001571:=z*(-2+z)/(-1+z)/(z**2-4*z+1); # Conjectured by Simon Plouffe in his 1992 dissertation. MATHEMATICA a[0] = 0; a[1] = 2; a[n_] := a[n] = 4a[n - 1] - a[n - 2] + 1; Table[ a[n], {n, 0, 24}] (* Robert G. Wilson v, Apr 24 2004 *) PROG (MAGMA) I:=[0, 2]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2)+1: n in [1..30]]; // Vincenzo Librandi, Jun 07 2015 CROSSREFS Sequence in context: A032601 A255410 A083141 * A092431 A147762 A077837 Adjacent sequences:  A001568 A001569 A001570 * A001572 A001573 A001574 KEYWORD nonn AUTHOR EXTENSIONS Better description from Bruce Corrigan (scentman(AT)myfamily.com), Nov 04 2002 More terms and new description from Robert G. Wilson v, Apr 24 2004 STATUS approved

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