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 A224227 #45 Jan 05 2025 19:51:40
%S A224227 0,0,0,1,2,7,16,38,82,173,352,701,1368,2628,4980,9329,17302,31811,
%T A224227 58040,105178,189446,339373,604964,1073593,1897488,3341160,5863080,
%U A224227 10256065,17888138,31115071,53985856,93447278,161397754,278184461,478550344,821734901,1408610088,2410719084,4119433884,7029086705,11977419742,20382654971,34643298728,58811818210
%N A224227 a(n) = (1/50)*((15*n^2-20*n+4)*Fibonacci(n) - (5*n^2-6*n)*A000032(n)).
%C A224227 The right-hand side of a binomial-coefficient identity.
%C A224227 From _Steven Finch_, Mar 22 2020: (Start)
%C A224227 a(n+2) is the total binary weight squared of all A000045(n+2) binary sequences of length n not containing any adjacent 1's.
%C A224227 The only three 2-bitstrings without adjacent 1's are 00, 01 and 10. The bitsums squared of these are 0, 1 and 1. Adding these give a(4)=2.
%C A224227 The only five 3-bitstrings without adjacent 1's are 000, 001, 010, 100 and 101. The bitsums squared of these are 0, 1, 1, 1 and 4. Adding these give a(5)=7.
%C A224227 The only eight 4-bitstrings without adjacent 1's are 0000, 0001, 0010, 0100, 1000, 0101, 1010 and 1001. The bitsums squared of these are 0, 1, 1, 1, 1, 4, 4, and 4. Adding these give a(6)=16. (End)
%H A224227 Steven Finch, <a href="https://arxiv.org/abs/2003.09458">Cantor-solus and Cantor-multus Distributions</a>, arXiv:2003.09458 [math.CO], 2020.
%H A224227 N. Gauthier (Proposer), <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Problems/November2012advanced.pdf">Problem H-703</a>, Fib. Quart., 50 (2012), 379-381.
%H A224227 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-5,0,3,1).
%F A224227 a(n) = Sum_{k=0..n-1} k^2*binomial(n-k-1,k).
%F A224227 G.f.: -x^3*(x^2-x+1)/(x^2+x-1)^3. - _Mark van Hoeij_, Apr 10 2013
%F A224227 a(n+3) = A001628(n) - A001628(n-1) + A001628(n-2). - _R. J. Mathar_, May 23 2014
%F A224227 E.g.f.: 2*exp(x/2)*(sqrt(5)*(2 + 5*x^2)*sinh(sqrt(5)*x/2) - 5*x*cosh(sqrt(5)*x/2))/125. - _Stefano Spezia_, Mar 20 2023
%t A224227 LinearRecurrence[{3,0,-5,0,3,1},{0,0,0,1,2,7},50] (* _Harvey P. Dale_, Jan 22 2016 *)
%t A224227 Table[((15 n^2 - 20 n + 4) Fibonacci[n] - (5 n - 6) n LucasL[n])/50, {n, 0, 30}] (* _Vladimir Reshetnikov_, Oct 10 2016 *)
%o A224227 (PARI) concat([0,0,0],Vec((x^2-x+1)/(x^2+x-1)^3+O(x^96))) \\ _Charles R Greathouse IV_, Mar 19 2014
%Y A224227 Cf. A000032, A000045, A001628.
%K A224227 nonn,easy
%O A224227 0,5
%A A224227 _N. J. A. Sloane_, Apr 09 2013