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

%I A140064 #134 Aug 06 2025 16:02:30
%S A140064 1,3,14,34,63,101,148,204,269,343,426,518,619,729,848,976,1113,1259,
%T A140064 1414,1578,1751,1933,2124,2324,2533,2751,2978,3214,3459,3713,3976,
%U A140064 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
%N A140064 a(n) = (9*n^2 - 5*n + 2)/2.
%C A140064 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
%C A140064 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.
%C A140064 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.
%C A140064 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.
%C A140064 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.
%C A140064 For proofs of Theorems 1 and 2 see "The Pancake, Hatpin, and Wu Planar Graphs".
%C A140064 For analogous sequences for long-legged letters V and Z see A130883 and A117625.
%H A140064 N. J. A. Sloane, <a href="/A140064/b140064.txt">Table of n, a(n) for n = 0..5000</a> [First 1000 terms from G. C. Greubel]
%H A140064 N. J. A. Sloane, <a href="/A140064/a140064.jpg">The long-legged letters A, I, V, X, and Z.</a> 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.
%H A140064 N. J. A. Sloane, <a href="/A140064/a140064_5.jpg">14 regions using 2 copies of the Wu graph</a>
%H A140064 N. J. A. Sloane, <a href="/A140064/a140064_6.jpg">34 regions using 3 copies of the Wu graph</a>
%H A140064 N. J. A. Sloane, <a href="/A140064/a140064_1.jpg">Transforming a Hatpin graph to a Wu graph.</a>
%H A140064 N. J. A. Sloane, <a href="/A140064/a140064_2.jpg">14 regions using 2 long-legged A's</a> [Crossbars are shown in red]
%H A140064 N. J. A. Sloane, <a href="/A140064/a140064_3.jpg">34 regions using 3 long-legged A's</a> [Crossbars are shown in red]
%H A140064 N. J. A. Sloane, <a href="/A140064/a140064_4.jpg">Transforming a long-legged A_n graph to a Wu_n graph, and vice-versa</a>
%H A140064 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A140064 Binomial transform of [1, 2, 9, 0, 0, 0, ...].
%F A140064 a(n) = A000217(n) + 8*A000217(n-2). - _R. J. Mathar_, May 06 2008
%F A140064 O.g.f.: x*(1+8*x^2)/(1-x)^3. - _R. J. Mathar_, May 06 2008
%F A140064 a(n) = A064226(n-2), n>1. - _R. J. Mathar_, Jul 31 2008
%F A140064 a(n) = a(n-1) + 9*n - 16, a(1)=1. - _Vincenzo Librandi_, Nov 24 2010
%F A140064 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
%F A140064 E.g.f.: exp(x)*(16 - 14*x + 9*x^2)/2. - _Stefano Spezia_, Dec 25 2022
%p A140064 seq((16-23*n+9*n^2)*1/2,n=1..40); # _Emeric Deutsch_, May 07 2008
%t A140064 Table[(9n^2-23n+16)/2,{n,40}] (* or *) LinearRecurrence[{3,-3,1},{1,3,14},40] (* _Harvey P. Dale_, Oct 01 2011 *)
%o A140064 (Magma) [ n eq 1 select 1 else Self(n-1)+9*n-16: n in [1..50] ];
%o A140064 (PARI) x='x+O('x^50); Vec(x*(1+8*x^2)/(1-x)^3) \\ _G. C. Greubel_, Feb 18 2017
%Y A140064 Cf. A000217, A007318, A064226, A130883, A117625.
%Y A140064 A row of the array in A386478.
%K A140064 nonn,easy
%O A140064 0,2
%A A140064 _Gary W. Adamson_, May 03 2008
%E A140064 More terms from _R. J. Mathar_ and _Emeric Deutsch_, May 06 2008
%E A140064 Edited by _N. J. A. Sloane_, Jun 21 2025 and Jun 26 2025