A140064 a(n) = (9*n^2 - 5*n + 2)/2.

1, 3, 14, 34, 63, 101, 148, 204, 269, 343, 426, 518, 619, 729, 848, 976, 1113, 1259, 1414, 1578, 1751, 1933, 2124, 2324, 2533, 2751, 2978, 3214, 3459, 3713, 3976, 4248, 4529, 4819, 5118, 5426, 5743, 6069, 6404, 6748, 7101, 7463, 7834, 8214, 8603, 9001, 9408, 9824, 10249, 10683, 11126, 11578, 12039, 12509, 12988, 13476, 13973

Offset: 0

Author

Gary W. Adamson, May 03 2008

Keywords

• nonn
• easy

Comments

Originally this entry was defined by a(n) = (9*n^2 - 23*n + 16)/2 and had offset 1. The current, simpler definition seems preferable, since it matches the following two geometrical applications. This change will also require several changes to the rest of the entry. - N. J. A. Sloane, Jun 26 2025

The letter Wu, ᗐ, is like a V but with three arms instead of two. Wu is included in the Unified Canadian Aboriginal Syllabics section of Unicode. The Unicode symbol for Wu is 0x2a5b. Wu is also called a "Boolean OR with middle stem", and is also the alchemical symbol Dissolve-2.

The long-legged Wu is a pencil of three semi-infinite lines originating from a point (the "tip"). The angles between the three lines are arbitrary.

Theorem 1 (Edward Xiong, Jonathan Pei, and David Cutler, Jun 24 2025): a(n) is the maximum number of regions in the plane that can be formed from n copies of a long-legged Wu.

Theorem 2: a(n) is also the maximum number of regions in the plane that can be formed from n copies of a long-legged letter A.

For proofs of Theorems 1 and 2 see "The Pancake, Hatpin, and Wu Planar Graphs".

For analogous sequences for long-legged letters V and Z see A130883 and A117625.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..5000 [First 1000 terms from G. C. Greubel]
• N. J. A. Sloane, The long-legged letters A, I, V, X, and Z. The long-legged A is a long-legged V with a crossbar. The distances from the tip of the A to the end-points of the crossbar need not be equal.
• N. J. A. Sloane, 14 regions using 2 copies of the Wu graph
• N. J. A. Sloane, 34 regions using 3 copies of the Wu graph
• N. J. A. Sloane, Transforming a Hatpin graph to a Wu graph.
• N. J. A. Sloane, 14 regions using 2 long-legged A's [Crossbars are shown in red]
• N. J. A. Sloane, 34 regions using 3 long-legged A's [Crossbars are shown in red]
• N. J. A. Sloane, Transforming a long-legged A_n graph to a Wu_n graph, and vice-versa
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A007318, A064226, A130883, A117625.

A row of the array in A386478.

Programs

Magma

[ n eq 1 select 1 else Self(n-1)+9*n-16: n in [1..50] ];

Maple

seq((16-23*n+9*n^2)*1/2,n=1..40); # Emeric Deutsch, May 07 2008

Mathematica

Table[(9n^2-23n+16)/2,{n,40}] (* or *) LinearRecurrence[{3,-3,1},{1,3,14},40] (* Harvey P. Dale, Oct 01 2011 *)

PARI

x='x+O('x^50); Vec(x*(1+8*x^2)/(1-x)^3) \\ G. C. Greubel, Feb 18 2017

Formula

Binomial transform of [1, 2, 9, 0, 0, 0, ...].

a(n) = A000217(n) + 8*A000217(n-2). - R. J. Mathar, May 06 2008

O.g.f.: x*(1+8*x^2)/(1-x)^3. - R. J. Mathar, May 06 2008

a(n) = A064226(n-2), n>1. - R. J. Mathar, Jul 31 2008

a(n) = a(n-1) + 9*n - 16, a(1)=1. - Vincenzo Librandi, Nov 24 2010

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=3, a(3)=14. - Harvey P. Dale, Oct 01 2011

E.g.f.: exp(x)*(16 - 14*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022

Extensions

More terms from R. J. Mathar and Emeric Deutsch, May 06 2008

Edited by N. J. A. Sloane, Jun 21 2025 and Jun 26 2025