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 A035038 #42 Mar 20 2023 09:11:26
%S A035038 0,0,0,0,0,0,1,8,37,130,386,1024,2510,5812,12911,27824,58651,121670,
%T A035038 249528,507624,1026876,2069256,4158861,8344056,16721761,33486026,
%U A035038 67025182,134116144,268313018,536724316,1073567387,2147277280,4294724471,8589650318,17179537972
%N A035038 a(n) = 2^n - C(n,0) - C(n,1) - ... - C(n,5).
%C A035038 Starting with "1", equals the eigensequence of a triangle with A000579 = binomial(n,6) = (1, 7, 28, 84, 210, ...) as the left column and the rest 1's. - _Gary W. Adamson_, Jul 24 2010
%H A035038 Alois P. Heinz, <a href="/A035038/b035038.txt">Table of n, a(n) for n = 0..1000</a>
%H A035038 J. Eckhoff, <a href="http://dx.doi.org/10.1007/BF01297698">Der Satz von Radon in konvexen Produktstrukturen II</a>, Monat. f. Math., 73 (1969), 7-30.
%F A035038 From _Paul Barry_, Aug 23 2004: (Start)
%F A035038 G.f.: x^6/((1-2*x)*(1-x)^6).
%F A035038 a(n) = Sum_{k=0..n} C(n, k+6) = Sum_{k=6..n} C(n, k).
%F A035038 a(n) = 2*a(n-1) + C(n-1, 5). (End)
%p A035038 a:= n-> (Matrix(7, (i,j)-> if (i=j-1) then 1 elif j=1 then [8,-27,50,-55, 36,-13,2][i] else 0 fi)^(n))[1,7]:
%p A035038 seq(a(n), n=0..30); # _Alois P. Heinz_, Aug 05 2008
%t A035038 Table[Sum[Binomial[n, k+6], {k,0,n}], {n,0,30}] (* _Zerinvary Lajos_, Jul 08 2009 *)
%t A035038 Table[2^n-Total[Binomial[n,Range[0,5]]],{n,0,40}] (* _Harvey P. Dale_, Oct 24 2017 *)
%o A035038 (Haskell)
%o A035038 a035038 n = a035038_list !! n
%o A035038 a035038_list = map (sum . drop 6) a007318_tabl
%o A035038 -- _Reinhard Zumkeller_, Jun 20 2015
%o A035038 (Magma) [n le 5 select 0 else (&+[Binomial(n,j): j in [6..n]]): n in [0..50]]; // _G. C. Greubel_, Mar 20 2023
%o A035038 (SageMath) [sum(binomial(n,j) for j in range(6,n+1)) for n in range(51)] # _G. C. Greubel_, Mar 20 2023
%Y A035038 Cf. A000079, A000225, A000295, A000579, A002663, A002664, A007318.
%Y A035038 Cf. A035039, A035040, A035041, A035042.
%K A035038 nonn,easy
%O A035038 0,8
%A A035038 _N. J. A. Sloane_