A097813 a(n) = 3*2^n - 2*n - 2.

1, 2, 6, 16, 38, 84, 178, 368, 750, 1516, 3050, 6120, 12262, 24548, 49122, 98272, 196574, 393180, 786394, 1572824, 3145686, 6291412, 12582866, 25165776, 50331598, 100663244, 201326538, 402653128, 805306310, 1610612676, 3221225410, 6442450880, 12884901822, 25769803708

Offset: 0

Author

Paul Barry, Aug 25 2004

Keywords

• easy
• nonn

Comments

An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 58, 154, 178 and 184, lead to this sequence. For the central square these vectors lead to the companion sequence A033484. - Johannes W. Meijer, Aug 15 2010

a(n) is also the number of order-preserving partial isometries of an n-chain, i.e., the row sums of A183153 and A183154. - Abdullahi Umar, Dec 28 2010

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• F. Al-Kharousi, R. Kehinde and A. Umar, Combinatorial results for certain semigroups of partial isometries of a finite chain, The Australasian Journal of Combinatorics, Volume 58 (3) (2014), 363-375.
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Cf. A000295, A033484, A079583, A175654, A183153, A183154.

Programs

Magma

[3*2^n -2*(n+1): n in [0..40]]; // G. C. Greubel, Dec 30 2021

Mathematica

Table[3 2^n-2n-2,{n,0,40}] (* or *) LinearRecurrence[{4,-5,2},{1,2,6},40] (* Harvey P. Dale, Oct 25 2011 *)

PARI

a(n)=3*2^n-2*n-2 \\ Charles R Greathouse IV, Oct 07 2015

Sage

[3*2^n -2*(n+1) for n in (0..40)] # G. C. Greubel, Dec 30 2021

Formula

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

a(n) = 2*a(n-1) + 2*n - 2, for n>0, with a(0)=1.

a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).

From G. C. Greubel, Dec 30 2021: (Start)

a(n) = 2^n + 2*A000295(n).

E.g.f.: 3*exp(2*x) - 2*(1 + x)*exp(x). (End)