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Search: id:A005320
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| A005320 |
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a(n) = 4a(n-1) - a(n-2), with a(0) = 0, a(1) = 1. (Formerly M2919)
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+0 2
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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
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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
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REFERENCES
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E. K. Lloyd "The standard deviation of 1, 2, .., n, Pell's equation and rational triangles", preprint.
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LINKS
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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.
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FORMULA
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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
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MAPLE
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A005320:=3*z/(1-4*z+z**2); [S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
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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
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Cf. A082841.
Adjacent sequences: A005317 A005318 A005319 this_sequence A005321 A005322 A005323
Sequence in context: A109437 A005656 A064017 this_sequence A062561 A128593 A085481
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KEYWORD
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nonn,easy,more
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AUTHOR
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njas
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