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Search: id:A122046
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| A122046 |
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Degree of the polynomial P(n,x), defined by P(n,x) = [x^(n-1)*P(n-1,x)*P(n-4,x)+P(n-2,x)*P(n-3,x)]/P(n-5,x) with P(1,x)=P(0,x)=P(-1,x)=P(-2,x)=P(-3,x)=1. |
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+0
2
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| 0, 0, 1, 3, 6, 10, 16, 24, 34, 46, 61, 79, 100, 124, 152, 184, 220, 260, 305, 355, 410, 470, 536, 608, 686, 770, 861, 959, 1064, 1176, 1296, 1424, 1560, 1704, 1857, 2019, 2190, 2370, 2560, 2760, 2970, 3190, 3421, 3663, 3916, 4180, 4456, 4744, 5044, 5356, 5681, 6019, 6370
(list; graph; listen)
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OFFSET
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0,4
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LINKS
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A. N. W. Hone, Comments on A122046
A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painleve transcendent, Proceedings of SIDE 6, Helsinki, Finland, 2004. [Set a(n)=d(n+3) on p. 8]
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FORMULA
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Comment from A. N. W. Hone (A.N.W.Hone(AT)kent.ac.uk): a(n) = \frac{1}{4\sqrt{2}}\cos((2n+1)\pi / 4)+\frac{1}{96}(2n+3)(2n^2+6n-5)+\frac{1}{32}(-1)^n. For proof see the link "Comments on A122046".
a(n)=A057077(n+1)/8+A090294(n-1)/32+(-1)^n/32. - A. N. W. Hone (A.N.W.Hone(AT)kent.ac.uk), Jul 15 2008
a(n)=3a(n-1)-3a(n-2)+a(n-3)+a(n-4)-3a(n-5)+3a(n-6)-a(n-7). - A. N. W. Hone (A.N.W.Hone(AT)kent.ac.uk), Jul 15 2008
O.g.f.: (-15x+47x^2+5-5x^3+15x^5-15x^6+5x^7-5x^4)/(32(1+x^2)(x-1)^4/(1+x)). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 15 2008
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MAPLE
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P := proc(n) option remember ;
if n <= 1 then
RETURN(1) ;
else
(P(n-1)*P(n-4)*q^(n-1)+P(n-2)*P(n-3))/P(n-5) ;
expand(%) ;
factor(%) ;
fi ;
end:
for n from 0 to 80 do
bag := P(n) ;
printf("%d %d\n", n, degree(bag, q)) ;
od: (Maple program from R. J. Mathar)
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MATHEMATICA
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p[n_] := p[n] = Cancel[Simplify[ (x^(n - 1)p[n - 1]p[n - 4] + p[n - 2]*p[n - 3])/p[n - 5]]]; p[ -5] = 1; p[ -4] = 1; p[ -3] = 1; p[ -2] = 1; p[ -1] = 1; Table[Exponent[p[n], x], {n, 0, 20}]
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CROSSREFS
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Cf. A014125, A122047.
Sequence in context: A121776 A088637 A066377 this_sequence A078663 A025222 A011902
Adjacent sequences: A122043 A122044 A122045 this_sequence A122047 A122048 A122049
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KEYWORD
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nonn
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AUTHOR
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Roger Bagula (rlbagulatftn(AT)yahoo.com), Sep 13 2006
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EXTENSIONS
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Edited by njas, Sep 17 2006, Jul 11 2008, Jul 12 2008
More terms from R. J. Mathar, Jul 11 2008, Jul 15 2008
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