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Search: id:A097947
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| A097947 |
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G.f.: -(2+7*x+2*x^2)/(1+4*x-4*x^3-x^4). |
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+0
3
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| -2, 1, -6, 16, -62, 225, -842, 3136, -11706, 43681, -163022, 608400, -2270582, 8473921, -31625106, 118026496, -440480882, 1643897025, -6135107222, 22896531856, -85451020206, 318907548961, -1190179175642, 4441809153600, -16577057438762, 61866420601441, -230888624967006
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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One of 4 related sequences. This is the sequence "les(n)". "jes(n)" = [1, -4, 15, -56, ...] is (-1)^(n+1)*A001353(n+1), "tes(n)" is A097948 and "ves(n)" is A099949.
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FORMULA
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Properties (from Creighton Dement, Sep 06 2004):
I: jes(n) + les(n) + tes(n) = ves(n)
II: All of the following are perfect squares: {les(2n+1); tes(2n+1); ves(2n+1); ves(2n+1) - jes(2n+1) - 1 = les(2n+1) + tes(2n+1) - 1; 3*les(2n+1) + 1 = 3*jes(n)^2 + 1}.
III: les(2n+1) divides ves(2n+1) - jes(2n+1) - 1 = les(2n+1) + tes(2n+1) - 1
IV: (jes(n))^2 = les(2n+1)
V: tes(2n) = A001570(n), sqrt( tes(2n+1) ) = A001075(n)
VI: sqrt( ves(2n+1) ) = A001835(n)
VII: sqrt( les(2n+1) ) = A001353(n)
VIII: les(n) + tes(n) = ves(2+n) + jes(n)
IX: lim n |jes(n+1)/jes(n)| = lim n |les(n+1)/les(n)| = lim n |tes(n+1)/tes(n)| = lim n |ves(n+1)/ves(n)| = 2 + sqrt(3)
Comment from Roland Bacher, Sep 07 2004: These 4 sequences satisfy jes(n+1)=-4*jes(n)-jes(n-1), les(n+1)=les(n-1)+jes(n), ves(n+1)=les(n-1)-jes(n-1)+tes(n-1), tes(n+1)=les(n-1)+3*jes(n), plus initial conditions for n=0, 1.
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CROSSREFS
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Cf. A001353, A097948, A097949.
Sequence in context: A068797 A049951 A025263 this_sequence A101032 A025271 A100404
Adjacent sequences: A097944 A097945 A097946 this_sequence A097948 A097949 A097950
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KEYWORD
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sign
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AUTHOR
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njas, following a suggestion of Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Sep 06 2004
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