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A001571
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a(0) = 0, a(1) = 2, a(n) = 4a(n-1) - a(n-2) + 1.
(Formerly M1928 N0762)
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15
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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
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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).
V. Thebault, Consecutive cubes with difference a square, Amer. Math. Monthly, 56 (1949), 174-175.
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LINKS
| S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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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 [From Paolo P. Lava (paoloplava(AT)gmail.com), Jul 31 2008]
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MAPLE
| A001571:=z*(-2+z)/(-1+z)/(z**2-4*z+1); [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| a[0] = 0; a[1] = 2; a[n_] := a[n] = 4a[n - 1] - a[n - 2] + 1; Table[ a[n], {n, 0, 24}] (from Robert G. Wilson v Apr 24 2004)
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CROSSREFS
| Sequence in context: A140217 A032601 A083141 * A092431 A147762 A077837
Adjacent sequences: A001568 A001569 A001570 * A001572 A001573 A001574
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Better description from Bruce Corrigan (scentman(AT)myfamily.com), Nov 04 2002
More terms and new description from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 24 2004
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