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A113249
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Corresponds to m = 3 in a family of 4th order linear recurrence sequences given by a(m,n) = m^4*a(n-4) + (2*m)^2*a(n-3) - 4*a(n-1), a(m,0) = -1, a(m,1) = 4, a(m,2) = -13 + 6*(m-1) + 3*(m-1)^2, a(m,3) = (-8+m^2)^2.
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7
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-1, 4, 11, 1, 59, 484, -1009, 6241, -2761, 13924, 87251, 57121, 49139, 4072324, -7165609, 35058241, 10350959, 30492484, 559712411, 973502401, -1957852501, 30450948004, -41421000289, 174055005601, 241428053159, 9658565284, 2872244917091, 11300885699041, -25300162140061
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OFFSET
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0,2
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COMMENTS
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Conjecture: a(m, 2*n+1) is a perfect square for all m, n. Disregarding signs and/or initial term, we have: m = 0 (A000302), m = 1 (A097948), m = 2 (A056450), m = 3 (a(n)), m = 4 (A113250), m = 5 (A113251), m = 6 (A113252), m = 7 (A113253), m = 8 (A113254), m = 9 (A113255), m = 10 (A113256)
In this case, a(2n+1) = b(n)^2 where b(n) = Re((2+sqrt(-5))^(n+1)) satisfies the recurrence b(n) = 4 b(n-1) - 9 b(n-2) with b(0)=2, b(1)=-1. - Robert Israel, Oct 23 2017
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REFERENCES
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C. Dement, Floretion Integer Sequences (work in progress)
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LINKS
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Table of n, a(n) for n=0..28.
Robert Munafo, Sequences Related to Floretions
Index entries for linear recurrences with constant coefficients, signature (-4, 0, 36, 81).
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FORMULA
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G.f. (-1+27*x^2+81*x^3)/((-3*x+1)*(3*x+1)*(9*x^2+4*x+1)).
a(2k+1) = (2*A176333(k)-3*A190967(k))^2. - Robert Israel, Oct 23 2017
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EXAMPLE
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a(3, 13) = 93161710957356599364/((-2+I*sqrt(5))^14*(2+I*sqrt(5))^14) = 4072324 = (2)^2*(1009)^2
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MAPLE
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Floretion Algebra Multiplication Program, FAMP Code: tesseq[A(3)*B]; A(m) = + 'i - 'k + i' - k' - m'jj' - 'ij' - 'ji' - 'jk' - 'kj' B = + .5'kk' + .5'ij' + .5'ji' + .5e or a(n) = f(3, n) with (Maple): f := (m, n) -> -1/4*m^2*(-m^2/(2-sqrt(4-m^2)))^n/(2-sqrt(4-m^2))-1/4*m^2*(-m^2/(2+sqrt(4-m^2)))^n/(2+sqrt(4-m^2))+1/4*m*m^n-1/4*m*(-m)^n;
# Alternative:
f:= gfun:-rectoproc({a(n) = 81*a(n-4)+36*a(n-3)-4*a(n-1), a(0) = -1, a(1) = 4, a(2) = 11, a(3) = 1}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Oct 23 2017
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MATHEMATICA
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LinearRecurrence[{-4, 0, 36, 81}, {-1, 4, 11, 1}, 29] (* Jean-François Alcover, Sep 25 2017 *)
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CROSSREFS
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Cf. A000302, A097948, A056450, A113250, A113251, A113252, A113253, A113254, A113255, A113256.
Cf. A176333, A190967.
Sequence in context: A132150 A091389 A175668 * A087171 A282026 A066333
Adjacent sequences: A113246 A113247 A113248 * A113250 A113251 A113252
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KEYWORD
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easy,sign
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AUTHOR
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Creighton Dement, Oct 20 2005
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STATUS
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approved
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