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A001571
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a(0) = 0, a(1) = 2, a(n) = 4*a(n-1) - a(n-2) + 1.
(Formerly M1928 N0762)
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21
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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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T. D. Noe, Table of n, a(n) for n = 0..200
Niccolò Castronuovo, On the number of fixed points of the map gamma, arXiv:2102.02739 [math.NT], 2021. Mentions this sequence.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
V. Thebault, Consecutive cubes with difference a square, Amer. Math. Monthly, 56 (1949), 174-175.
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).
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FORMULA
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a(n) = (A001834(n) - 1)/2.
G.f.: x*(-2+x) / ( (x-1)*(x^2-4*x+1) ). - Simon Plouffe in his 1992 dissertation.
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
E.g.f.: (exp(2*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) - exp(x))/2. - Franck Maminirina Ramaharo, Nov 12 2018
a(n) = 2*A061278(n) - A061278(n-1). - R. J. Mathar, Feb 06 2020
a(n) = ((1+sqrt(3))*(2+sqrt(3))^n + (1-sqrt(3))*(2-sqrt(3))^n)/4 - (1/2). - Vladimir Pletser, Jan 15 2021
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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 *)
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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
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CROSSREFS
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Cf. A001834, A061278, A071954, A082840.
Sequence in context: A032601 A255410 A083141 * A092431 A147762 A077837
Adjacent sequences: A001568 A001569 A001570 * A001572 A001573 A001574
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KEYWORD
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nonn,easy,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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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
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STATUS
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approved
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