A131130 Binomial transform of [1,1,7,1,7,1,7,1,...].

1, 2, 10, 26, 58, 122, 250, 506, 1018, 2042, 4090, 8186, 16378, 32762, 65530, 131066, 262138, 524282, 1048570, 2097146, 4194298, 8388602, 16777210, 33554426, 67108858, 134217722, 268435450, 536870906, 1073741818, 2147483642

Offset

0,2

Comments

For n >= 3, number of vertices of 4,4'-bipyridinium dendrimers (see the Arjomanfar and Gholami reference, p. 71). - Emeric Deutsch, Apr 12 2015

Number of ways to color a (2n-1) X (2n-1) chess board in a "balanced" way. A coloring is called balanced if, within every square subgrid made up of k^2 cells for 1 <= k <= 2*n-1, the number of black cells differs from the number of white cells by at most one. It is problem 3 from the British Maths Olympiad 2020. - Ruediger Jehn, Jan 27 2021

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

A. Arjomanfar and N. Gholami, Computing the Szeged index of 4,4'-bipyridinium dendrimer, Iranian J. Math. Chem., 3, 2012, 67-72.

British Mathematical Olympiad, 2020 - Round 2, Problem 3.

Index to sequences related to Olympiads.

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Formula

Row sums of triangle A131131.

a(n) = 4*2^n - 6 for n >= 1; a(0)=1.

From Philippe Deléham, Jan 04 2009: (Start)

a(n) = 3*a(n-1) - 2*a(n-2), n > 2; a(0)=1, a(1)=2, a(2)=10.

G.f.: (1-x+6*x^2) / (1-3*x+2*x^2). (End)

a(n) = 2*a(n-1) + 6 for n > 1, a(0)=1, a(1)=2. - Philippe Deléham, Sep 25 2009

E.g.f.: 3 - 6*exp(x) + 4*exp(2*x). - Stefano Spezia, Feb 05 2021

Maple

1, seq(4*2^n -6, n = 1..30);

Mathematica

Join[{1}, LinearRecurrence[{3, -2}, {2, 10}, 30]] (* Harvey P. Dale, Mar 07 2014 *)
CoefficientList[Series[(1 -x +6x^2)/(1 -3x +2x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 08 2014 *)

Crossrefs

Cf. A095121 (bin transf 1,1,3,1,3,...), A131128 (bin transf 1,1,5,1,5,..), A131131.

Sequence in context: A212969 A277712 A009307 * A099969 A241688 A183331

Adjacent sequences: A131127 A131128 A131129 * A131131 A131132 A131133

Keyword

nonn,easy

Author

Gary W. Adamson, Jun 16 2007

Extensions

Edited by Emeric Deutsch, Jul 12 2007

Status

approved