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

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

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.

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.]

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.

Adjacent sequences: A005317 A005318 A005319 this_sequence A005321 A005322 A005323

Sequence in context: A109437 A005656 A064017 this_sequence A062561 A128593 A085481

nonn,easy,more

njas