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A005997 Number of paraffins.
(Formerly M2832)
9
1, 3, 10, 20, 39, 63, 100, 144, 205, 275, 366, 468, 595, 735, 904, 1088, 1305, 1539, 1810, 2100, 2431, 2783, 3180, 3600, 4069, 4563, 5110, 5684, 6315, 6975, 7696, 8448, 9265, 10115, 11034, 11988, 13015, 14079, 15220, 16400, 17661, 18963, 20350, 21780, 23299 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

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

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

FORMULA

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

a(n) = A005999(n)+(n-1)^2. - Enrique Pérez Herrero, Mar 27 2012

a(n) = 1 + floor((n-1)/2) + 2*(C(n+1,3)-C(floor((n+1)/2),3)-C(ceiling((n+1)/2),3). - Enrique Pérez Herrero, Apr 22 2012

a(n) = (n+1)(2n^2-(-1)^n+1)/8. - Bruno Berselli, Apr 22 2012

a(n) = A004526(n) + 2*A111384(n). - Enrique Pérez Herrero, Apr 25 2012

MAPLE

a:= n-> (Matrix([[0, 0, -1, -5, -12, -26]]). Matrix(6, (i, j)-> if (i=j-1) then 1 elif j=1 then [2, 1, -4, 1, 2, -1][i] else 0 fi)^n)[1, 1]: seq (a(n), n=1..50); # Alois P. Heinz, Jul 31 2008

MATHEMATICA

A005997[n_]:=1+Floor[(n-1)/2]+2*(Binomial[n+1, 3]-Binomial[Floor[(n+1)/2], 3]-Binomial[Ceiling[(n+1)/2], 3]); Array[A005997, 37] (* Enrique Pérez Herrero, Apr 22 2012 *)

LinearRecurrence[{2, 1, -4, 1, 2, -1}, {1, 3, 10, 20, 39, 63}, 37] (* Bruno Berselli, Apr 22 2012 *)

CROSSREFS

Cf. A005999.

Sequence in context: A272764 A293765 A295953 * A213850 A081205 A273379

Adjacent sequences:  A005994 A005995 A005996 * A005998 A005999 A006000

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 19 12:29 EDT 2018. Contains 316360 sequences. (Running on oeis4.)