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 A301653 #11 Feb 16 2025 08:33:53
%S A301653 0,1,3,5,10,16,28,45,75,121,198,320,520,841,1363,2205,3570,5776,9348,
%T A301653 15125,24475,39601,64078,103680,167760,271441,439203,710645,1149850,
%U A301653 1860496,3010348,4870845,7881195,12752041,20633238,33385280,54018520,87403801,141422323,228826125,370248450
%N A301653 Expansion of x*(1 + 2*x)/((1 - x)*(1 + x)*(1 - x - x^2)).
%C A301653 Apparently (for n > 0), numbers that have a unique partition into a sum of distinct Lucas numbers (A000204).
%H A301653 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>
%H A301653 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,-1).
%F A301653 G.f.: x*(1 + 2*x)/((1 - x)*(1 + x)*(1 - x - x^2)).
%F A301653 a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4).
%F A301653 a(n) = Lucas(n+1) - (3 - (-1)^n)/2.
%F A301653 a(n) = floor(phi^(n+1)) - 1, where phi = (1 + sqrt(5))/2 is the golden ratio (A001622).
%F A301653 a(n) = Sum_{k>=0} A051601(n-k,k) (conjectured). - _Greg Dresden_, May 18 2023
%t A301653 CoefficientList[Series[x (1 + 2 x)/((1 - x) (1 + x) (1 - x - x^2)) , {x, 0, 40}], x]
%t A301653 LinearRecurrence[{1, 2, -1, -1}, {0, 1, 3, 5}, 41]
%t A301653 Table[LucasL[n + 1] - (3 - (-1)^n)/2, {n, 0, 40}]
%t A301653 Table[Floor[GoldenRatio^(n + 1)] - 1, {n, 0, 40}]
%o A301653 (PARI) a(n) = fibonacci(n) + fibonacci(n+2) + ((-1)^n - 3)/2; \\ _Altug Alkan_, Mar 25 2018
%Y A301653 Cf. A000071, A000204, A001622, A003263, A014217, A054770, A294203.
%K A301653 nonn,easy
%O A301653 0,3
%A A301653 _Ilya Gutkovskiy_, Mar 25 2018