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A005994 Alkane (or paraffin) numbers l(7,n).
(Formerly M2774)
5
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)
OFFSET

0,2

COMMENTS

Equals (1, 3, 6, 10, 15,...) convolved with (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

a(A254338(n)) = 6 for n > 0. - Reinhard Zumkeller, Feb 27 2015

REFERENCES

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

N. J. A. Sloane, Classic Sequences

Index entries for linear recurrences with constant coefficients, signature (3, -1, -5, 5, 1, -3, 1).

"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].

FORMULA

G.f.: (1+x^2)/((1-x)^3*(1-x^2)^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.

a(-5-n)=a(n). - Michael Somos, Mar 08 2007

Euler transform of length 4 sequence [ 3, 3, 0, -1]. - Michael Somos, Mar 08 2007

a(0)=1, a(1)=3, a(2)=9, a(4)=19, a(5)=38, a(6)=66, a(7)=110,

a(n)=3a(n-1)-a(n-2)-5a(n-3)+5a(n-4)+a(n-5)-3a(n-6)+a(n-7). - Harvey P. Dale, May 02 2011

a(n) = A006009(n)/2 - A000332(n+4) = (1/2)*Sum(i=1..n+1, (i+1)*floor((i+1)^2/2)) - C(n+4,4). - Enrique Pérez Herrero, May 11 2012

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

MAPLE

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

MATHEMATICA

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] (* From 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}]]] (* From Harvey P. Dale, Nov 11 2011 *)

PROG

(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 */

(Haskell)   Following Gary W. Adamson.

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

-- Reinhard Zumkeller, Feb 27 2015

CROSSREFS

Cf. A006009, A005997.

Cf. A000217, A005408.

Sequence in context: A146050 A147500 A115238 * A080010 A135117 A038163

Adjacent sequences:  A005991 A005992 A005993 * A005995 A005996 A005997

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)

STATUS

approved

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Last modified December 10 11:50 EST 2016. Contains 279001 sequences.