A101946 a(n) = 6*2^n - 3*n - 5.

1, 4, 13, 34, 79, 172, 361, 742, 1507, 3040, 6109, 12250, 24535, 49108, 98257, 196558, 393163, 786376, 1572805, 3145666, 6291391, 12582844, 25165753, 50331574, 100663219, 201326512, 402653101, 805306282, 1610612647, 3221225380, 6442450849, 12884901790

Offset: 0

Author

Gary W. Adamson, Dec 22 2004

Keywords

• nonn
• easy

Comments

Sequence generated from a 3 X 3 matrix, companion to A101945.

Characteristic polynomial of M = x^3 - 4x^2 + 5x - 2.

Sequence can also be generated by the same method as A061777 with slightly different rules. Refer to A061777, which is the "vertex to vertex" expansion version. For this case, the expandable vertices of the existing generation will contact the sides of the new ones, i.e., "vertex to side" expansion version. Let us assign the label "1" to the triangle at the origin; at n-th generation add a triangle at each expandable vertex, i.e., each vertex where the added generations will not overlap the existing ones, although overlaps among new generations are allowed. The non-overlapping triangles will have the same label value as a predecessor; for the overlapping ones, the label value will be sum of label values of predecessors. a(n) is the sum of all label values at n-th generation. The triangles count is A005448. See illustration. - Kival Ngaokrajang, Sep 26 2014

The number of ways to select 0 or more nodes of a 2 X n rectangular grid such that the selected nodes are connected, but do not fill any 2 X 2 square. This question arises in logic puzzles such as Nurikabe. - Hugo van der Sanden, Feb 22 2024

Examples

			a(4) = 79 = 4*34 - 5*13 + 2*4 = 4*a(3) - 5*a(2) + 2*a(1).
a(4) = right term in M^4 * [1 1 1], since M^4 * [1 1 1] = [1 46 a(4)], where 46 = A033484(4).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Joseph Breen and Emma Copeland, Non-orientable Nurikabe, arXiv:2506.12612 [math.CO], 2025. See pp. 1-2, 4.
• Madeleine Goertz and Aaron Williams, The Quaternary Gray Code and How It Can Be Used to Solve Ziggurat and Other Ziggu Puzzles, arXiv:2411.19291 [math.CO], 2024. See pp. 1, 3, 5, 7, 17, 38, 40.
• Kival Ngaokrajang, Illustration of initial terms
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Cf. A033484, A101945.

Cf. A000225, A005448, A061777.

Programs

Magma

[6*2^n -3*n-5: n in [0..40]]; // G. C. Greubel, Feb 06 2022

Mathematica

a[0]=1; a[1]=4; a[2]=13; a[n_]:= a[n]= 4a[n-1] -5a[n-2] +2a[n-3]; Table[ a[n], {n, 0, 30}] (* Or *)
a[n_] := (MatrixPower[{{1, 0, 0}, {2, 2, 0}, {1, 2, 1}}, n].{{1}, {1}, {1}})[[3, 1]]; Table[ a[n], {n, 0, 30}] (* Robert G. Wilson v, Jan 12 2005 *)
Table[6*2^n-3n-5,{n,0,40}] (* or *) LinearRecurrence[{4,-5,2},{1,4,13},40] (* Harvey P. Dale, Jun 03 2017 *)

PARI

a(n) = if (n<1, 1, 5*(2^n-1)+a(n-1))\\ Kival Ngaokrajang, Sep 26 2014

PARI

Vec(-(2*x^2+1)/((x-1)^2*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 26 2014

Sage

[3*(2^(n+1) -n-2) +1 for n in (0..40)] # G. C. Greubel, Feb 06 2022

Formula

a(0)=1, a(1)=4, a(2)=13 and for n>2, a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).

a(n) = right term in M^n * [1 1 1], where M = the 3X3 matrix [1 0 0 / 2 2 0 / 1 2 1]. M^n * [1 1 1] = [1 A033484(n) a(n)].

a(0) = 1, for n >= 1, a(n) = 3*A000225(n) + a(n-1). - Kival Ngaokrajang, Sep 26 2014

G.f.: (1+2*x^2)/((1-x)^2*(1-2*x)). - Colin Barker, Sep 26 2014

E.g.f.: 6*exp(2*x) - (5+3*x)*exp(x). - G. C. Greubel, Feb 06 2022

Extensions

New definition from Ralf Stephan, May 17 2007