A083706 a(n) = 2^(n+1) + n - 1.

1, 4, 9, 18, 35, 68, 133, 262, 519, 1032, 2057, 4106, 8203, 16396, 32781, 65550, 131087, 262160, 524305, 1048594, 2097171, 4194324, 8388629, 16777238, 33554455, 67108888, 134217753, 268435482, 536870939, 1073741852, 2147483677, 4294967326, 8589934623, 17179869216

Offset: 0

Author

N. J. A. Sloane, Jun 15 2003

Keywords

• nonn
• easy

Comments

Is A247983(n+1) = A247983(n) if and only if n is in A083706? - Clark Kimberling, Sep 28 2014

a(n) is the least number of nodes in a height-n 2-3-4 tree, if using the top-down insertion algorithm and there have been no deletions. - Daniel S. Roche, Oct 05 2014

Also the number of independent vertex sets and vertex covers in the n-crown graph. - Andrew Howroyd, May 14 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Crown Graph.
• Eric Weisstein's World of Mathematics, Independent Vertex Set.
• Eric Weisstein's World of Mathematics, Vertex Cover.
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Cf. A083706, A130301, A160692, A247983.

Programs

Magma

[2^(n+1)+n-1: n in [0..35]]; // Vincenzo Librandi, Jul 20 2011

Mathematica

Table[2^(n + 1) + n - 1, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 19 2011 *)
LinearRecurrence[{4, -5, 2}, {4, 9, 18}, {0, 20}] (* Eric W. Weisstein, Sep 21 2017 *)
CoefficientList[Series[(-1 + 2 x^2)/((-1 + x)^2 (-1 + 2 x)), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 21 2017 *)

PARI

a(n)=if(n<0,0,2^(n+1)+n-1)

Formula

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

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

Row sums of A130301. - Gary W. Adamson, May 20 2007

From Elmo R. Oliveira, Mar 06 2025: (Start)

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

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