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A005994
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Alkane (or paraffin) numbers l(7,n).
(Formerly M2774)
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8
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1, 3, 9, 19, 38, 66, 110, 170, 255, 365, 511, 693, 924, 1204, 1548, 1956, 2445, 3015, 3685, 4455, 5346, 6358, 7514, 8814, 10283, 11921, 13755, 15785, 18040, 20520, 23256, 26248, 29529, 33099, 36993, 41211, 45790, 50730, 56070, 61810, 67991
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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COMMENTS
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Equals A000217 (1, 3, 6, 10, 15, ...) convolved with A193356 (1, 0, 3, 0, 5, ...). - Gary W. Adamson, Feb 16 2009
F(1,4,n) is the number of bracelets with 1 blue, 4 red and n black beads. If F(1,4,1)=3 and F(1,4,2)=9 taken as a base;
F(1,4,n) = n(n+1)(n+2)/6+F(1,2,n) + F(1,4,n-2). [F(1,2,n) is the number of bracelets with 1 blue, 2 red and n black beads. If F(1,2,1)=2 and F(1,2,2)=4 taken as a base F(1,2,n)=n+1+F(1,2,n-2)]. - Ata Aydin Uslu and Hamdi G. Ozmenekse, Jan 11 2012
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REFERENCES
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S. J. Cyvin et al., Polygonal systems including the corannulene and coronene homologs: novel applications of Pólya's theorem, Z. Naturforsch., 52a (1997), 867-873.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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"http://commons.wikimedia.org/wiki/File:Bracelet_Problem_(Bileklik_problemi).pdf" number of bracelets made with 1 blue, 4 red and n black beads [From Ata Aydin Uslu and Hamdi G. Ozmenekse, Jan 11 2012].
"http://commons.wikimedia.org/wiki/File:Bileklik_Problemi_(Bracelet_Problem).pdf" number of bracelets made with 1 blue, 2 red and n black beads [From Ata Aydin Uslu and Hamdi G. Ozmenekse, Jan 12 2012].
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FORMULA
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G.f.: (1+x^2)/((1-x)^3*(1-x^2)^2) = (1+x^2)/((1-x)^5*(1+x)^2).
l(c, r) = 1/2 C(c+r-3, r) + 1/2 d(c, r), where d(c, r) is C((c + r - 3)/2, r/2) if c is odd and r is even, 0 if c is even and r is odd, C((c + r - 4)/2, r/2) if c is even and r is even, C((c + r - 4)/2, (r - 1)/2) if c is odd and r is odd.
Euler transform of length 4 sequence [3, 3, 0, -1]. - Michael Somos, Mar 08 2007
a(n) = 3a(n-1) - a(n-2) - 5a(n-3) + 5a(n-4) + a(n-5) - 3a(n-6) + a(n-7), with a(0)=1, a(1)=3, a(2)=9, a(4)=19, a(5)=38, a(6)=66, a(7)=110. - Harvey P. Dale, May 02 2011
a(n) = (1/48)*(n+1)*(n+3)*((n+2)*(n+4)+3)+1/32*(2*n+5)*(1+(-1)^n). - Yosu Yurramendi, Jun 20 2013
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MAPLE
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a:= n -> (Matrix([[1, 0$4, 1, 3]]). Matrix(7, (i, j)-> if (i=j-1) then 1 elif j=1 then [3, -1, -5, 5, 1, -3, 1][i] else 0 fi)^n)[1, 1]: seq (a(n), n=0..40); # Alois P. Heinz, Jul 31 2008
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MATHEMATICA
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LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {1, 3, 9, 19, 38, 66, 110}, 50] (* or *) CoefficientList[Series[(1+x^2)/((1-x)^3(1-x^2)^2), {x, 0, 50}], x] (* Harvey P. Dale, May 02 2011 *)
nn=45; With[{a=Accumulate[Range[nn]], b=Riffle[Range[1, nn, 2], 0]}, Flatten[ Table[ListConvolve[Take[a, n], Take[b, n]], {n, nn}]]] (* Harvey P. Dale, Nov 11 2011 *)
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PROG
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(PARI) {a(n)=if(n<-4, n=-5-n); polcoeff( (1+x^2)/((1-x)^3*(1-x^2)^2)+x*O(x^n), n)} /* Michael Somos, Mar 08 2007 */
import Data.List (inits, intersperse)
a005994 n = a005994_list !! n
a005994_list = map (sum . zipWith (*) (intersperse 0 [1, 3 ..]) . reverse) $
tail $ inits $ tail a000217_list
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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