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 A026937 #52 Jun 30 2025 06:10:47
%S A026937 1,3,10,30,87,245,676,1836,4925,13079,34446,90090,234227,605865,
%T A026937 1560200,4002072,10230201,26069995,66251090,167941494,424753615,
%U A026937 1072057117,2700704172,6791746500,17052595573,42752015487,107035180630,267634562754,668407232235,1667467065425
%N A026937 a(n) = Sum_{k=0..n} (k+1)*T(n, n-k), where T is given by A008288.
%H A026937 Vincenzo Librandi, <a href="/A026937/b026937.txt">Table of n, a(n) for n = 0..1000</a>
%H A026937 Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See p. 23.
%H A026937 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-4,-1).
%F A026937 G.f.: (1-x)/(1 - 2*x - x^2)^2.
%F A026937 a(n) = Sum_{k=0..n+1} A000129(k)*A001333(n+1-k). - _Graeme McRae_, Aug 03 2006 and _Michel Marcus_, Aug 01 2023
%F A026937 a(n) = A006645(n+2) - A006645(n+1). - _R. J. Mathar_, Jan 27 2011
%F A026937 a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 20 2012
%F A026937 a(n) = ((n+2)/2)*A000129(n+1). - _G. C. Greubel_, May 25 2021
%F A026937 a(n) = ((n+2)/8)*((sqrt(2) + 2)*(1 + sqrt(2))^n - (sqrt(2) - 2)*(1 - sqrt(2))^n). - _Peter Luschny_, Jul 31 2023
%F A026937 a(n) = A361732(n+2)/2. - _R. J. Mathar_, Jun 30 2025
%p A026937 with (combinat):seq(add(fibonacci(n,2),k=0..n)/2,n=1..27); # _Zerinvary Lajos_, May 25 2008
%t A026937 CoefficientList[Series[(1-x)/(1-2x-x^2)^2,{x,0,40}],x] (* _Harvey P. Dale_, Mar 22 2011 *)
%t A026937 LinearRecurrence[{4,-2,-4,-1},{1,3,10,30},40] (* _Vincenzo Librandi_, Jun 20 2012 *)
%t A026937 Table[(1/2)*(n+2)*Fibonacci[n+1, 2], {n, 0, 40}] (* _G. C. Greubel_, May 25 2021 *)
%o A026937 (Magma) I:=[1, 3, 10, 30]; [n le 4 select I[n] else 4*Self(n-1)-2*Self(n-2)-4*Self(n-3)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Jun 20 2012
%o A026937 (PARI) my(x='x+O('x^40)); Vec((1-x)/(1-2*x-x^2)^2) \\ _Altug Alkan_, Sep 20 2018
%o A026937 (PARI) a(n) = my(w=quadgen(8)); (n/8)*((2+w)*(1+w)^n - (w-2)*(1-w)^n); \\ _Michel Marcus_, Jul 31 2023
%o A026937 (Sage) [(1/2)*(n+2)*lucas_number1(n+1,2,-1) for n in (0..40)] # _G. C. Greubel_, May 25 2021
%Y A026937 Cf. A000129, A001333, A006645, A008288, A364553.
%K A026937 nonn,easy
%O A026937 0,2
%A A026937 _Clark Kimberling_