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A001571
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a(n) = 4*a(n-1) - a(n-2) + 1, with a(0) = 0, a(1) = 2.
(Formerly M1928 N0762)
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24
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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G.f.: x*(2-x)/( (1-x)*(1-4*x+x^2) ). - Simon Plouffe in his 1992 dissertation.
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
a(n) = ((1+sqrt(3))*(2+sqrt(3))^n + (1-sqrt(3))*(2-sqrt(3))^n)/4 - (1/2). - Vladimir Pletser, Jan 15 2021
a(n) = (ChebyshevU(n, 2) + ChebyshevU(n-1, 2) - 1)/2. - G. C. Greubel, Feb 02 2022
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MAPLE
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f := gfun:-rectoproc({a(0) = 0, a(1) = 2, a(n) = 4*a(n - 1) - a(n - 2) + 1}, a(n), remember): map(f, [$ (0 .. 40)])[]; # Vladimir Pletser, Jul 25 2020
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MATHEMATICA
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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 *)
Table[(ChebyshevU[n, 2] +ChebyshevU[n-1, 2] -1)/2, {n, 0, 30}] (* G. C. Greubel, Feb 02 2022 *)
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PROG
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(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
(Magma) [(Evaluate(ChebyshevU(n+1), 2) + Evaluate(ChebyshevU(n), 2) - 1)/2 : n in [0..30]]; // G. C. Greubel, Feb 02 2022
(Sage) [(chebyshev_U(n, 2) + chebyshev_U(n-1, 2) - 1)/2 for n in (0..30)] # G. C. Greubel, Feb 02 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Better description from Bruce Corrigan (scentman(AT)myfamily.com), Nov 04 2002
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STATUS
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approved
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