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 A279100 #5 Feb 16 2025 08:33:37
%S A279100 1,3,6,11,18,30,48,78,125,202,325,525,847,1369,2212,3577,5784,9356,
%T A279100 15134,24484,39611,64088,103691,167771,271453,439215,710658,1149863,
%U A279100 1860510,3010362,4870860,7881210,12752057,20633254,33385297,54018537,87403819,141422341,228826144,370248469,599074596
%N A279100 a(n) = Sum_{k=0..n} ceiling(phi^k), where phi is the golden ratio (A001622).
%C A279100 Partial sums of A169986.
%H A279100 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldenRatio.html">Golden Ratio</a>
%H A279100 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-3,0,1).
%F A279100 G.f.: (1 + x - x^2 - x^3 - x^4)/((1 - x)^2*(1 - 2*x^2 - x^3)).
%F A279100 a(n) = 2*a(n-1) + a(n-2) - 3*a(n-3) + a(n-5).
%F A279100 a(n) = (10*n - 5*(-1)^n + 2^(1-n)*sqrt(5)*(5 + 3*sqrt(5))*(1 + sqrt(5))^n + sqrt(5)*2^(1-n)*(3*sqrt(5) - 5) *(1 - sqrt(5))^n - 35)/20.
%F A279100 a(n) ~ phi^(n+2).
%t A279100 Accumulate[Table[Ceiling[GoldenRatio^n], {n, 0, 40}]]
%t A279100 LinearRecurrence[{2, 1, -3, 0, 1}, {1, 3, 6, 11, 18}, 41]
%Y A279100 Cf. A001622, A020956, A169986.
%K A279100 nonn,easy
%O A279100 0,2
%A A279100 _Ilya Gutkovskiy_, Dec 06 2016