0,2

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+3x^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 (ck6(AT)evansville.edu), Aug 27 2008

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

E. K. Lloyd "The standard deviation of 1, 2, .., n, Pell's equation and rational triangles", preprint.

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.

Index entries for sequences related to linear recurrences with constant coefficients

C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.

Tanya Khovanova, Recursive Sequences

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.

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]

a(n) = 5[a(n-1)-a(n-2)] + a(n-3); a(0) = 0, a(1) = 3, a(2) = 12; n > 3; a(n) = (sqrt(3)/2)*(2+sqrt(3))^n-(sqrt(3)/2)*(2-sqrt(3))^n. - Antonio Alberto Olivares (tonioolivares(AT)todito.com), Jan 17 2004

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 (heinz(AT)hs-heilbronn.de), Aug 14 2008]

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

Cf. A082841.

Sequence in context: A109437 A005656 A064017 this_sequence A062561 A128593 A085481

Adjacent sequences: A005317 A005318 A005319 this_sequence A005321 A005322 A005323

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 14 2008

Typo in definition corrected by Johannes Boot, Feb 05 2009