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 A270737 #21 Jul 13 2025 21:01:13
%S A270737 1,3,5,10,20,42,91,195,415,880,1864,3952,8385,17795,37765,80138,
%T A270737 170044,360810,765595,1624515,3447071,7314368,15520400,32932800,
%U A270737 69880225,148279107,314634021,667623210,1416632420,3005958090,6378354619
%N A270737 a(n) = ((n+2)/2)*Sum_{k=0..n/2} (Sum_{i=0..n-2*k} (binomial(k+1,n-2*k-i)*binomial(k+i,k))*F(k+1)/(k+1)), where F(k) is Fibonacci numbers.
%H A270737 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,0,2,1).
%F A270737 G.f.: (-x^2+x+1)/(-x^6-2*x^5-2*x+1).
%F A270737 a(n) = 2*a(n-1) + 2*a(n-5) + a(n-6). - _G. C. Greubel_, Mar 25 2016
%t A270737 Table[((n + 2)/2) Sum[Sum[(Binomial[k + 1, n - 2 k - i] Binomial[k + i, k])/(k + 1) Fibonacci[k + 1], {i, 0, n - 2 k}], {k, 0, n/2}], {n, 0, 30}] (* or *)
%t A270737 CoefficientList[Series[(-x^2 + x + 1)/(-x^6 - 2 x^5 - 2 x + 1), {x, 0, 30}], x] (* _Michael De Vlieger_, Mar 25 2016 *)
%t A270737 LinearRecurrence[{2, 0, 0, 0, 2, 1}, {1, 3, 5, 10, 20, 42}, 100] (* _G. C. Greubel_, Mar 25 2016 *)
%o A270737 (Maxima)
%o A270737 a(n):=(n+2)/2*(sum(sum(binomial(k+1,n-2*k-i)*binomial(k+i,k),i,0,n-2*k)*fib(k+1)/(k+1),k,0,n/2));
%o A270737 (PARI) my(x='x+O('x^40)); Vec((-x^2+x+1)/(-x^6-2*x^5-2*x+1)) \\ _Altug Alkan_, Mar 22 2016
%Y A270737 Cf. A000045, A113413.
%K A270737 nonn,easy
%O A270737 0,2
%A A270737 _Vladimir Kruchinin_, Mar 22 2016