OFFSET
0,3
COMMENTS
The Szeged index of the hypercube Q_n (see the Ashrafi et al. reference, p. 45, last line). - Emeric Deutsch, Aug 06 2014
For n > 3, 2*a(n) is the number of spanning trees in a superprism on 2*n vertices (see Bogdanowicz). - Stefano Spezia, May 05 2024
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
A. R. Ashrafi, B. Manoochehrian, and H. Yousefi-Azari, On Szeged polynomial of a graph, Bull. Iranian Math. Soc., 33, 2007, 37-46. - Emeric Deutsch, Aug 06 2014
Zbigniew R. Bogdanowicz, The number of spanning trees in a superprism, Discrete Math. Lett. 13 (2024) 66-73. See Theorem 3.1.
Frank Ellermann, Illustration of binomial transforms.
Index entries for linear recurrences with constant coefficients, signature (16,-64).
FORMULA
a(n) = 16*a(n-1) - 64*a(n-2), with a(0)=0, a(1)=1. - Emeric Deutsch, Aug 06 2014
From G. C. Greubel, May 16 2019: (Start)
G.f.: x/(1-8*x)^2.
E.g.f.: x*exp(8*x). (End)
From Amiram Eldar, Oct 28 2020: (Start)
Sum_{n>=1} 1/a(n) = 8*log(8/7).
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(9/8). (End)
MAPLE
a := proc(n) option remember; if n<2 then n else 16*a(n-1)-64*a(n-2) end if end proc: seq(a(n), n = 0 .. 20); # Emeric Deutsch, Aug 06 2014
MATHEMATICA
Table[n 8^(n-1), {n, 0, 20}] (* or *) LinearRecurrence[{16, -64}, {0, 1}, 20] (* Harvey P. Dale, Feb 01 2017 *)
PROG
(Magma) [n*8^(n-1): n in [0..20]]; // Vincenzo Librandi, Feb 09 2011
(PARI) a(n) = n*8^(n-1); \\ Joerg Arndt, Aug 07 2014
(Sage) [n*8^(n-1) for n in (0..20)] # G. C. Greubel, May 16 2019
(GAP) List([0..20], n-> n*8^(n-1)) # G. C. Greubel, May 16 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, Jan 15 2000
EXTENSIONS
Offset corrected and name edited by Emeric Deutsch, Aug 06 2014
STATUS
approved