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 A181510 #36 Jan 21 2025 18:20:01
%S A181510 6,18,34,54,78,106,138,174,214,258,306,358,414,474,538,606,678,754,
%T A181510 834,918,1006,1098,1194,1294,1398,1506,1618,1734,1854,1978,2106,2238,
%U A181510 2374,2514,2658,2806,2958,3114,3274,3438,3606,3778,3954,4134,4318,4506,4698,4894
%N A181510 Number of permutations of the multiset {1,1,2,2,3,3,...,n+1,n+1} avoiding the permutation patterns {132, 231, 2134}.
%C A181510 a(n) is also the surface ares of the n-th solid in the following recursive construction:
%C A181510 The first solid is a unit cube (hence a(1)=6).
%C A181510 To form the n-th solid from the (n-1)st solid, construct a row of 2n-1 cubes, then center the (n-1)st solid on top of this row. (For example, the second solid is a row of 3 unit cubes, with a single unit cube centered on top of the middle cube. This construction has surface area a(2)=18.)
%C A181510 The sequence provides all nonnegative integers m such that 2*m + 13 is a square. - _Bruno Berselli_, Mar 01 2013
%H A181510 G. C. Greubel, <a href="/A181510/b181510.txt">Table of n, a(n) for n = 1..5000</a>
%H A181510 Lara K. Pudwell, <a href="http://www.jstor.org/stable/10.4169/002557010x521868">Stacking Blocks and Counting Permutations</a>, Mathematics Magazine, Vol. 83 (2010), pp. 297-302.
%H A181510 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A181510 a(n) = 2*n^2 + 6*n - 2.
%F A181510 From _Bruno Berselli_, Oct 29 2010: (Start)
%F A181510 G.f.: 2*x*(3-x^2)/(1-x)^3.
%F A181510 a(n) - 3*a(n-1) + 3*a(n-2) - a(n-3) = 0 for n > 3.
%F A181510 a(n) = 2*A014209(n) = 2*A082111(n-1) + 4 = A051936(2*n+2) + n + 4. (End)
%F A181510 Sum_{n>=1} 1/a(n) = 2/3 + Pi*tan(sqrt(13)*Pi/2)/(2*sqrt(13)). - _Amiram Eldar_, Dec 23 2022
%F A181510 E.g.f.: 2*(exp(x)*(x^2 + 4*x - 1) + 1). - _Elmo R. Oliveira_, Nov 17 2024
%e A181510 For n=1, the permutations of {1,1,2,2} avoiding the patterns {132, 231, 2134} are {1122, 1212, 1221, 2112, 2121, 2211}.
%e A181510 For n=2, the permutations of {1,1,2,2,3,3} avoiding the patterns {132, 231, 2134} are {112233, 121233, 122133, 211233, 212133, 221133, 311223, 312123, 312213, 321123, 321213, 322113, 331122, 331212, 331221, 332112, 332121, 332211}.
%t A181510 a[n_] := 2*n^2 + 6*n - 2; Array[a, 60] (* _Amiram Eldar_, Dec 23 2022 *)
%t A181510 LinearRecurrence[{3,-3,1},{6,18,34},50] (* _Harvey P. Dale_, May 10 2023 *)
%t A181510 ((2*Range[1,60]+3)^2 -13)/2 (* _G. C. Greubel_, Jan 21 2025 *)
%o A181510 (PARI) a(n)=2*n^2+6*n-2 \\ _Charles R Greathouse IV_, Jun 17 2017
%o A181510 (Magma)
%o A181510 A181510:= func< n | ((2*n+3)^2 -13)/2 >;
%o A181510 [A181510(n): n in [1..60]]; // _G. C. Greubel_, Jan 21 2025
%o A181510 (Python)
%o A181510 def A181510(n): return (pow(2*n+3,2) -13)//2
%o A181510 print([A181510(n) for n in range(1,61)]) # _G. C. Greubel_, Jan 21 2025
%Y A181510 Cf. A014209, A051936, A082111.
%K A181510 nonn,easy
%O A181510 1,1
%A A181510 _Lara Pudwell_, Oct 25 2010