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 A088207 #41 Jan 12 2025 09:21:06
%S A088207 0,2,7,14,24,37,52,70,90,113,139,167,198,232,268,307,348,392,439,488,
%T A088207 540,594,651,711,773,838,906,976,1049,1124,1202,1283,1366,1452,1541,
%U A088207 1632,1726,1822,1921,2023,2127,2234,2343,2455,2570,2687,2807,2930,3055,3183
%N A088207 a(n) = Sum_{k=0..n} floor(k*phi^2) where phi=(1+sqrt(5))/2.
%C A088207 Partial sums of A001950.
%C A088207 A001950 is the upper Beatty sequence for the constant phi^2, where phi = (1 + sqrt(5))/2 and the sequence is generated by floor(n*phi). A054347 = partial sums of the lower Beatty sequence (A000201).
%C A088207 Conjecture: a(n)/A054347(n) tends to phi. Example: a(28)/A054347(28) = 1049/643 = 1.6314...
%C A088207 From _Michel Dekking_, Aug 19 2019: (Start)
%C A088207 Proof of Adamson's conjecture. We know that lim_{n->oo} A054347(n)/(n*(n+1)) = phi/2 (see A054347).
%C A088207 Using that floor(k*phi^2) = floor(k*phi)+k, for k=1,...,n, we obtain a(n)/A054347(n) = (A054347(n)+n*(n+1)/2)/A054347(n) = 1+(n*(n+1)/2)/A054347(n) -> 1+(2/phi)*(1/2), which equals phi. [parentheses corrected by _Jason Yuen_, Jan 12 2025]
%C A088207 (End)
%F A088207 a(n) = Sum_{k=1..n} floor(k*phi^2).
%F A088207 a(n) = floor((n*(n+1)/2)*phi^2 - n/2) + (0 or 1). - _Benoit Cloitre_, Sep 27 2003
%F A088207 a(n) = Sum_{k=1..n} floor(k*phi)+k = A054347(n)+n*(n+1)/2. - _Jason Yuen_, Jan 12 2025
%e A088207 A001950(1) = 2, then 5, 7, 10, 13, ...; partial sums are 2, 7, 14, 24, 37, ...
%t A088207 a[0] = 0; a[n_] := a[n] = (a[n - 1] + Floor[n*(1 + Sqrt[5])^2/4]); Table[ a[n], {n, 1, 50}] (* _Robert G. Wilson v_, Sep 27 2003 *)
%t A088207 Accumulate[Floor[GoldenRatio^2 Range[0,50]]] (* _Harvey P. Dale_, Aug 11 2021 *)
%o A088207 (Python)
%o A088207 from math import isqrt
%o A088207 from itertools import islice, count, accumulate
%o A088207 def A088207_gen(): # generator of terms
%o A088207 return accumulate((n+isqrt(5*n**2)>>1)+n for n in count(0))
%o A088207 A088207_list = list(islice(A088207_gen(),10)) # _Chai Wah Wu_, Aug 29 2022
%Y A088207 Cf. A001622 (phi), A001950, A054347, A000201, A000217 (triangular numbers).
%K A088207 nonn,easy
%O A088207 0,2
%A A088207 _Gary W. Adamson_, Sep 23 2003
%E A088207 More terms from _Robert G. Wilson v_ and _Benoit Cloitre_, Sep 27 2003