This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A258109 #70 Oct 27 2023 21:47:14
%S A258109 1,5,18,57,169,482,1341,3669,9922,26609,70929,188226,497845,1313501,
%T A258109 3459042,9096393,23895673,62721698,164531565,431397285,1130708866,
%U A258109 2962826465,7761964833,20331456642,53249182309,139449644717,365166860706,956185155129,2503657040137
%N A258109 Number of balanced parenthesis expressions of length 2n and depth 3.
%C A258109 a(n) is the number of Dyck paths of length 2n and height 3. For example, a(3) = 1 because there is only one such Dyck path which is UUUDDD. - _Ran Pan_, Sep 26 2015
%C A258109 a(n) is the number of rooted plane trees with n+1 nodes and height 3 (see example for correspondence). - _Gheorghe Coserea_, Feb 04 2016
%D A258109 S. S. Skiena and M. A. Revilla, Programming Challenges: The Programming Contest Training Manual, Springer, 2006, page 140.
%H A258109 Alois P. Heinz, <a href="/A258109/b258109.txt">Table of n, a(n) for n = 3..1000</a> (first 40 terms from Gheorghe Coserea)
%H A258109 Kyu-Hwan Lee, Se-jin Oh, <a href="http://arxiv.org/abs/1601.06685">Catalan triangle numbers and binomial coefficients</a>, arXiv:1601.06685 [math.CO], 2016.
%H A258109 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-7,2).
%F A258109 a(n) = 2^(n-3) + 3 * a(n-1) - a(n-2).
%F A258109 a(n) = 5*a(n-1) - 7*a(n-2) + 2*a(n-3) for n>5. - _Colin Barker_, May 24 2015
%F A258109 G.f.: -x^3 / ((2*x-1)*(x^2-3*x+1)). - _Colin Barker_, May 24 2015
%F A258109 a(n) = A000045(2n-1) - A000079(n-1). - _Gheorghe Coserea_, Feb 04 2016
%F A258109 a(n) = 2^(-1-n)*(-5*4^n - (-5+sqrt(5))*(3+sqrt(5))^n + (3-sqrt(5))^n*(5+sqrt(5))) / 5. - _Colin Barker_, Jun 05 2017
%F A258109 a(n) = Sum_{i=1..n-1} A061667(i)*(n-1-i) - _Tim C. Flowers_, May 16 2018
%e A258109 For n=4, the a(4) = 5 solutions are
%e A258109 /\ /\
%e A258109 / \ \
%e A258109 LRLLLRRR /\/ \ \
%e A258109 ................................
%e A258109 /\ |
%e A258109 /\/ \ / \
%e A258109 LLRLLRRR / \ \
%e A258109 ................................
%e A258109 /\/\ |
%e A258109 / \ |
%e A258109 LLLRLRRR / \ / \
%e A258109 ................................
%e A258109 /\ |
%e A258109 / \/\ / \
%e A258109 LLLRRLRR / \ /
%e A258109 ................................
%e A258109 /\ /\
%e A258109 / \ /
%e A258109 LLLRRRLR / \/\ /
%p A258109 a:= proc(n) option remember; `if`(n<3, 0,
%p A258109 `if`(n=3, 1, 2^(n-3) +3*a(n-1) -a(n-2)))
%p A258109 end:
%p A258109 seq(a(n), n=3..30); # _Alois P. Heinz_, May 20 2015
%t A258109 Join[{1, 5}, LinearRecurrence[{5, -7, 2}, {18, 57, 169}, 30]] (* _Vincenzo Librandi_, Sep 26 2015 *)
%o A258109 (C)
%o A258109 typedef long long unsigned Integer;
%o A258109 Integer a(int n)
%o A258109 {
%o A258109 int i;
%o A258109 Integer pow2 = 1, a[3] = {0};
%o A258109 for (i = 3; i <= n; ++i) {
%o A258109 a[ i%3 ] = pow2 + 3 * a[ (i-1)%3 ] - a[ (i-2)%3 ];
%o A258109 pow2 = pow2 * 2;
%o A258109 }
%o A258109 return a[ (i-1)%3 ];
%o A258109 }
%o A258109 (PARI) Vec(-x^3/((2*x-1)*(x^2-3*x+1)) + O(x^100)) \\ _Colin Barker_, May 24 2015
%o A258109 (Magma) I:=[1,5,18,57,169]; [n le 5 select I[n] else 5*Self(n-1) - 7*Self(n-2) + 2*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Sep 26 2015
%o A258109 (PARI) a(n) = fibonacci(2*n-1) - 2^(n-1) \\ _Gheorghe Coserea_, Feb 04 2016
%Y A258109 Column k=3 of A080936.
%Y A258109 Column k=2 of A287213.
%Y A258109 Cf. A000045, A000079, A262600.
%K A258109 nonn,walk,easy
%O A258109 3,2
%A A258109 _Gheorghe Coserea_, May 20 2015