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A005320
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a(n) = 4*a(n-1) - a(n-2), with a(0) = 0, a(1) = 3.
(Formerly M2919)
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5
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0, 3, 12, 45, 168, 627, 2340, 8733, 32592, 121635, 453948, 1694157, 6322680, 23596563, 88063572, 328657725, 1226567328, 4577611587, 17083879020, 63757904493, 237947738952, 888033051315, 3314184466308, 12368704813917
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| For n > 1, a(n-1) is the determinant of the n-by-n band matrix which has {2,4,4,...,4,4,2} on the diagonal and a 1 on the entire super- and subdiagonal. This matrix appears when constructing a natural cubic spline interpolating n equally spaced data points. - g.degroot(AT)phys.uu.nl, Feb 14 2007
Integer values of x that make sqrt(9+3*x^2) a perfect square. - Lorenz H. Menke, Jr. (lnz2004(AT)mindspring.com), Mar 26 2008
The intermediate convergents to 3^(1/2), beginning with 3/2, 12/7, 45/26, 168/97, comprise a strictly increasing sequence; numerators=A005320, denominators=A001075. - Clark Kimberling, Aug 27 2008
a(n) also give the altitude to the middle side of a Super-Heronian Triangle. [From Johannes Boot, Oct 14 2010]
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REFERENCES
| Clark Kimberling, "Best lower and upper approximates to irrational numbers," Elemente der Mathematik, 52 (1997) 122-126.
Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
E. K. Lloyd "The standard deviation of 1, 2, .., n, Pell's equation and rational triangles", preprint.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
Tanya Khovanova, Recursive Sequences
E. Keith Lloyd, The Standard Deviation of 1, 2,..., n: Pell's Equation and Rational Triangles, Math. Gaz. vol 81 (1997), 231-243. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 21 2009]
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.
William H. Richardson, Super-Heronian Triangles from Johannes Boot, Oct 14 2010.
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| a(n) = (sqrt(3)/2)*(2+sqrt(3))^n-(sqrt(3)/2)*(2-sqrt(3))^n. - Antonio Alberto Olivares, Jan 17 2004
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MAPLE
| A005320:=3*z/(1-4*z+z**2); [S. Plouffe in his 1992 dissertation.]
a := n -> (Matrix([[3, 0]]). Matrix([[4, 1], [ -1, 0]])^n)[1, 2]; seq (a(n), n=0..50); [From Alois P. Heinz, Aug 14 2008]
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MATHEMATICA
| Det[SparseArray[{{i_, i_} -> If[i == 1 || i == n, 2, 4], {i_, j_} -> If[Abs[i - j] == 1, 1, 0]}, {n, n}]] (* the recurrence relation is faster! *) - g.degroot(AT)phys.uu.nl, Feb 14 2007
Do[If[IntegerQ[Sqrt[(9 + 3 x^2)]], Print[{x, Sqrt[(9 + 3 x^2)]}]], {x, 0, 2000000}] - Lorenz H. Menke, Jr. (lnz2004(AT)mindspring.com), Mar 26 2008
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CROSSREFS
| Cf. A082841.
Sequence in context: A109437 A005656 A064017 * A062561 A128593 A085481
Adjacent sequences: A005317 A005318 A005319 * A005321 A005322 A005323
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 14 2008
Typo in definition corrected by Johannes Boot, Feb 05 2009
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