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A001571 a(0) = 0, a(1) = 2, a(n) = 4a(n-1) - a(n-2) + 1.
(Formerly M1928 N0762)
16
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).

V. Thebault, Consecutive cubes with difference a square, Amer. Math. Monthly, 56 (1949), 174-175.

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.

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, Jul 31 2008]

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 *)

CROSSREFS

Sequence in context: A200050 A032601 A083141 * A092431 A147762 A077837

Adjacent sequences:  A001568 A001569 A001570 * A001572 A001573 A001574

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

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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Last modified November 25 18:32 EST 2014. Contains 250000 sequences.