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A122046
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Partial sums of floor(n^2/8).
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5
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0, 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; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Degree of the polynomial P(n+1,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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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
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]
Brian OSullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=4]
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FORMULA
| a(n) = sum(k=0..n,floor(k^2/8)).
a(n) = round((2*n^3+3*n^2-8*n)/48) = round((4*n^3+6*n^2-16*n-9)/96) = floor((2*n^3+3*n^2-8*n+3)/48) = ceil((2*n^3+3*n^2-8*n-12)/48) . - M. Merca
a(n) = a(n-8) + (n-4)^2 + n , n>8 . - M. Merca
a(n+1) = \frac{1}{4\sqrt{2}}\cos((2n+1)\pi / 4)+\frac{1}{96}(2n+3)(2n^2+6n-5)+\frac{1}{32}(-1)^n. a(n+1)=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.: x(-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
From Johannes W. Meijer, May 20 2011: (Start)
a(n+3) = A144678(n)+A144678(n-1)+A144678(n-2)+A144678(n-3)
a(n+3) = sum(min(6-k+1,k+1)* A190718(n+k-6),k=0..6) (End)
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EXAMPLE
| a(6) = 10 = 0 + 0 + 0 + 1 + 2 + 3 + 4.
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MAPLE
| a(n):=round((2*n^(3)+3*n^(2)-8*n)/(48)) (from M. Merca)
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
| 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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PROG
| (MAGMA) [Round((2*n^3+3*n^2-8*n)/48): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011
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CROSSREFS
| Cf. A014125, A122047.
Sequence in context: A088637 A066377 A173653 * A078663 A173691 A025222
Adjacent sequences: A122043 A122044 A122045 * A122047 A122048 A122049
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KEYWORD
| nonn
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AUTHOR
| Roger Bagula (rlbagulatftn(AT)yahoo.com), Sep 13 2006
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Sep 17 2006, Jul 11 2008, Jul 12 2008
More terms from R. J. Mathar, Jul 11 2008, Jul 15 2008
More formulae and better name from Mircea Merca (mircea(AT)teacher.com), Nov 19 2010
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