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 A027974 #29 Jun 11 2025 00:55:36
%S A027974 1,5,14,35,81,180,389,825,1726,3575,7349,15020,30561,61965,125294,
%T A027974 252795,509161,1024100,2057549,4130225,8284926,16609455,33282989,
%U A027974 66669660,133507081,267285605,535010414,1070731475,2142612801,4287086100,8577182549,17159235945
%N A027974 a(n) = Sum_{k=1..n+1} A027960(n+1, n+1+k).
%C A027974 The former name, a(n) = Sum_{i=0..n} Sum_{j=0..i} T(i,j), T given by A027960, was in error with the data given. [This double summation gives A023537(n+1), or A027960(n+2, n+4) for n >= 0]. - _G. C. Greubel_, Jun 08 2025
%H A027974 G. C. Greubel, <a href="/A027974/b027974.txt">Table of n, a(n) for n = 0..1000</a>
%H A027974 Philipp Emanuel Weidmann, <a href="http://worldwidemann.com/the-sequencer-oeis-survey/">The Sequencer OEIS Survey</a>
%H A027974 Susanne Wienand, <a href="https://oeis.org/w/images/b/bc/Suggestion_for_a_proof_%28conjecture_of_sequencer%29.pdf">Suggestion for a proof of Philipp Emanuel Weidman's conjecture concerning A027983</a>
%H A027974 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-2).
%F A027974 a(n) = 2^(n+3) - Fibonacci(n+5) - Fibonacci(n+3).
%F A027974 a(n) = A101220(4, 2, n+1).
%F A027974 G.f.: (1+2*x)/((1-2*x)*(1-x-x^2)). - _R. J. Mathar_, Sep 22 2008
%F A027974 a(n) = 2*a(n-1) + A000032(n+1). - _David A. Corneth_, Apr 16 2015
%F A027974 a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3). - _Colin Barker_, Feb 17 2016
%F A027974 From _G. C. Greubel_, Jun 08 2025: (Start)
%F A027974 a(n) = 2^(n+3) - A000032(n+4).
%F A027974 E.g.f.: 8*exp(2*x) - exp(x/2)*( 7*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2) ). (End)
%p A027974 with(combinat); f:=fibonacci; seq(2^(n+3) - f(n+5) - f(n+3), n=0..30); # _G. C. Greubel_, Sep 26 2019
%t A027974 Table[2^(n+3) - LucasL[n+4], {n,0,30}] (* _G. C. Greubel_, Sep 26 2019 *)
%o A027974 (PARI) vector(31, n, f=fibonacci; 2^(n+2) - f(n+4) - f(n+2)) \\ _G. C. Greubel_, Sep 26 2019
%o A027974 (Magma) [2^(n+3) - Lucas(n+4): n in [0..30]]; // _G. C. Greubel_, Sep 26 2019
%o A027974 (SageMath)
%o A027974 def A027974(n): return 2**(n+3) - lucas_number2(n+4,1,-1)
%o A027974 [A027974(n) for n in range(31)] # _G. C. Greubel_, Sep 26 2019; Jun 08 2025
%o A027974 (GAP) List([0..30], n-> 2^(n+3) - Lucas(1,-1,n+4)[2]); # _G. C. Greubel_, Sep 26 2019
%Y A027974 Cf. A000032, A000045, A027960, A101220.
%K A027974 nonn,easy
%O A027974 0,2
%A A027974 _Clark Kimberling_