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

Offset

0,2

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).

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

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 *)

Prog

(PARI) a(n)=if(n<0, 0, 2^(n+1)+n-1)
(Magma) [2^(n+1)+n-1: n in [0..35]]; // Vincenzo Librandi, Jul 20 2011

Crossrefs

Cf. A130301, A160692, A247983.

Sequence in context: A320851 A027053 A122039 * A352667 A229072 A261983

Adjacent sequences: A083703 A083704 A083705 * A083707 A083708 A083709

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 15 2003

Status

approved