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 A094374 #46 Sep 26 2024 02:55:55
%S A094374 1,3,8,21,56,153,428,1221,3536,10353,30548,90621,269816,805353,
%T A094374 2407868,7207221,21588896,64701153,193972388,581655021,1744440776,
%U A094374 5232273753,15694724108,47079978021,141231545456,423677859153,1271000023028,3812932960221,11438664662936
%N A094374 a(n) = (3^n-1)/2 + 2^n.
%C A094374 Binomial transform of A094373.
%C A094374 Row sums of A125103. - _Paul Barry_, Dec 04 2007
%C A094374 Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 2) x = y. - _Ross La Haye_, Jan 11 2008
%C A094374 a(n) is the number of words of length n over the alphabet {0,1,2} with an even number of occurrences of the substring 01. - _Daimon S. Mayorga_, Sep 10 2020
%H A094374 G. C. Greubel, <a href="/A094374/b094374.txt">Table of n, a(n) for n = 0..1000</a>
%H A094374 Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary Relations on the Power Set of an n-Element Set</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
%H A094374 A. Prasad, <a href="http://arxiv.org/abs/1407.5284">Equivalence classes of nodes in trees and rational generating functions</a>, arXiv preprint arXiv:1407.5284 [math.CO], 2014.
%H A094374 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).
%F A094374 G.f.: (1-3x+x^2)/((1-x)*(1-2x)*(1-3x)).
%F A094374 a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3).
%F A094374 a(n) = A003462(n) + A000079(n).
%F A094374 a(n) = Sum_{k=0..n} C(n,k)+2^k*C(n,k+1). - _Paul Barry_, Dec 04 2007
%F A094374 a(n) = StirlingS2(n+1,3) + 2*StirlingS2(n+1,2) + 1. - _Ross La Haye_, Jan 11 2008
%F A094374 E.g.f.: exp(2*x)*(1 + sinh(x)). - _G. C. Greubel_, Sep 26 2024
%t A094374 Table[(3^n-1)/2+2^n,{n,0,30}] (* or *) LinearRecurrence[{6,-11,6},{1,3,8},30] (* _Harvey P. Dale_, Jul 22 2013 *)
%o A094374 (PARI) a(n)=(3^n-1)/2+2^n \\ _Charles R Greathouse IV_, Oct 16 2015
%o A094374 (Magma) [(3^n-1)/2+2^n: n in [0..30]]; // _Vincenzo Librandi_, Nov 30 2015
%o A094374 (SageMath) [(3^n +2^(n+1) -1)//2 for n in range(31)] # _G. C. Greubel_, Sep 26 2024
%Y A094374 Cf. A000079, A000225, A000392, A003462, A094373, A125103.
%K A094374 easy,nonn
%O A094374 0,2
%A A094374 _Paul Barry_, Apr 28 2004