A018214 - OEIS

%I #31 Sep 08 2022 08:44:44

%S 1,6,36,146,511,1512,4032,9752,21942,46252,92504,176484,323554,572264,

%T 981024,1634776,2656511,4218786,6562556,10016006,15024009,22177584,

%U 32258304,46282704,65567164,91792792,127097712,174169352

%N Alkane (or paraffin) numbers l(13,n).

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

%D Winston C. Yang (paper in preparation).

%H Vincenzo Librandi, <a href="/A018214/b018214.txt">Table of n, a(n) for n = 0..1000</a>

%H N. J. A. Sloane, <a href="/classic.html#LOSS">Classic Sequences</a>

%H S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (6, -10, -10, 50, -34, -66, 110, 0, -110, 66, 34, -50, 10, 10, -6, 1).

%F 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.

%F G.f.: -(5*x^4+10*x^2+1)/((x-1)^11*(x+1)^5). [_Colin Barker_, Aug 06 2012]

%F a(n) = (1/(2*10!))*(n+2)*(n+4)*(n+6)*(n+8)*(n+10)*((n+1)*(n+3)*(n+5)*(n+7)*(n+9) + 1*3*5*7*9)- (1/6)*(1/2^8)*(n^4+22*n^3+170*n^2+539*n+579)*(1/2)*(1-(-1)^n). [_Yosu Yurramendi_, Jun 23 2013]

%t CoefficientList[Series[-(5 x^4 + 10 x^2 + 1)/((x - 1)^11 (x + 1)^5), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 16 2013 *)

%t LinearRecurrence[{6, -10, -10, 50, -34, -66, 110, 0, -110, 66, 34, -50, 10, 10, -6, 1},{1, 6, 36, 146, 511, 1512, 4032, 9752, 21942, 46252, 92504, 176484, 323554, 572264, 981024, 1634776},28] (* _Ray Chandler_, Sep 23 2015 *)

%o (Magma) [(1/(2*Factorial(10)))*(n+2)*(n+4)*(n+6)*(n+8)*(n+10)*((n+1)*(n+3)*(n+5)*(n+7)*(n+9)+1*3*5*7*9)-(1/6)*(1/2^8)*(n^4+22*n^3+170*n^2+539*n+579)*(1/2)*(1-(-1)^n): n in [0..40]]; // _Vincenzo Librandi_, Oct 16 2013

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Winston C. Yang (yang(AT)math.wisc.edu)