A053539 a(n) = n * 8^(n-1).

0, 1, 16, 192, 2048, 20480, 196608, 1835008, 16777216, 150994944, 1342177280, 11811160064, 103079215104, 893353197568, 7696581394432, 65970697666560, 562949953421312, 4785074604081152, 40532396646334464, 342273571680157696, 2882303761517117440, 24211351596743786496

Offset: 0

Author

Barry E. Williams, Jan 15 2000

Keywords

• easy
• nonn

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).

Crossrefs

Binomial transform of A027473.

Cf. A001787, A053464, A053469, A053540.

Programs

GAP

List([0..20], n-> n*8^(n-1)); # G. C. Greubel, May 16 2019

Magma

[n*8^(n-1): n in [0..20]]; // Vincenzo Librandi, Feb 09 2011

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 *)

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

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)

Extensions

Offset corrected and name edited by Emeric Deutsch, Aug 06 2014