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 A092055 #33 Dec 27 2024 03:51:04
%S A092055 1,4,20,120,816,5984,45760,357760,2829056,22500864,179481600,
%T A092055 1433753600,11461636096,91659526144,733141975040,5864598896640,
%U A092055 46914643623936,375308558925824,3002434111406080,24019335451770880,192154133857304576,1537230871833083904,12297838178567454720
%N A092055 a(n) = binomial(2 + 2^n,3).
%C A092055 a(n) = Sum_{i=1...(2^n)} i*(i+1)/2, this sequence is thus similar to A016131 as it is a sum of triangular numbers on the interval <1,2^n>, A016131 is a sum of triangular numbers on the interval <1,2^n - 1>. - _Ctibor O. Zizka_, Mar 03 2009
%C A092055 a(n) is the number of unordered (not necessarily distinct) triples of subsets taken from the power set of {1,2,...,n}. Cf. A007582 (pairs of such subsets). - _Geoffrey Critzer_, Jul 10 2013
%H A092055 G. C. Greubel, <a href="/A092055/b092055.txt">Table of n, a(n) for n = 0..1000</a>
%H A092055 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (14,-56,64).
%F A092055 a(n) = (2^(3*n-1) +3*2^(2*n-1) + 2^n)/3 = A092056(3, n).
%F A092055 a(n) = A000079(n)*A007581(n).
%F A092055 a(n) = 2*a(n-1) + 4^(n-1) + 8^(n-1).
%F A092055 a(n) = [x^3] 1/(1-x)^(2^n). - _Geoffrey Critzer_, Jul 11 2013
%F A092055 a(n) = 14*a(n-1) - 56*a(n-2) + 64*a(n-3). - _Colin Barker_, Sep 13 2014
%F A092055 G.f.: (1-10*x+20*x^2) / ((1-2*x)*(1-4*x)*(1-8*x)). - _Colin Barker_, Sep 13 2014
%F A092055 E.g.f.: (1/6)*(2*exp(2*x) + 3*exp(4*x) + exp(8*x)). - _G. C. Greubel_, Dec 27 2024
%e A092055 a(5) = C(2+2^5,3) = C(34,3) = 5984.
%p A092055 seq(binomial(2+2^n, 3), n=0..25); # _Zerinvary Lajos_, Feb 22 2008
%t A092055 nn=20;Table[Coefficient[Series[1/(1-x)^(2^n),{x,0,nn}],x^3],{n,0,nn}] (* _Geoffrey Critzer_, Jul 10 2013 *)
%t A092055 Binomial[2+2^Range[0,30], 3] (* _G. C. Greubel_, Dec 27 2024 *)
%o A092055 (PARI) Vec((1-10*x+20*x^2)/((1-2*x)*(1-4*x)*(1-8*x)) + O(x^100)) \\ _Colin Barker_, Sep 13 2014
%o A092055 (Magma) [Binomial(2^n+2, 3): n in [0..30]]; // _G. C. Greubel_, Dec 27 2024
%o A092055 (Python)
%o A092055 def A092055(n): return binomial(pow(2,n)+2,3)
%o A092055 print([A092055(n) for n in range(41)]) # _G. C. Greubel_, Dec 27 2024
%Y A092055 Cf. A000079, A007581, A092056, A016131.
%K A092055 nonn,easy
%O A092055 0,2
%A A092055 _Henry Bottomley_, Feb 19 2004