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 A036295 #77 Jun 16 2025 00:47:57
%S A036295 0,1,1,11,13,57,15,247,251,1013,509,4083,4089,16369,2047,65519,65527,
%T A036295 262125,131067,1048555,1048565,4194281,1048573,16777191,16777203,
%U A036295 67108837,33554425,268435427,268435441,1073741793,67108863,4294967263,4294967279,17179869149
%N A036295 Numerator of Sum_{i=1..n} i/2^i.
%C A036295 The fraction is twice the probability that the convex hull of n+2 points on a circle randomly chosen from a uniform distribution contains the center of the circle. This probability remains the same if the points are chosen from the circumference instead. - _Lewis Chen_, Jun 14 2025
%D A036295 C. C. Clawson, The Beauty and Magic of Numbers. New York: Plenum Press (1996): 95.
%H A036295 Colin Barker, <a href="/A036295/b036295.txt">Table of n, a(n) for n = 0..1000</a>
%H A036295 A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/12-3/horadam.pdf">Oresme numbers</a>, Fib. Quart., 12 (1974), 267-271.
%F A036295 a(n) = numerator(2-(n+2)/2^n).
%F A036295 If n+2=2^k*m with m odd, then a(n) = 2^(n+1-k) - m.
%F A036295 For n >= 1, a(n) = A000265(A000295(n+1)). - _Peter Munn_, May 30 2023
%F A036295 a(n) = A000295(n+1)/A006519(n+2). - _Ridouane Oudra_, Jul 16 2023
%F A036295 Numerators of coefficients in expansion of 2*x / ((1 - x) * (2 - x)^2). - _Ilya Gutkovskiy_, Aug 04 2023
%p A036295 seq(numer(2-(n+2)/2^n), n=0..50); # _Ridouane Oudra_, Jul 16 2023
%t A036295 a[n_] := Module[{k, m}, For[k = 0; m = n + 2, EvenQ[m], k++, m/=2]; 2^(n + 1 - k) - m]
%t A036295 Table[Numerator[Sum[i/2^i, {i, n}]], {n, 40}] (* _Alonso del Arte_, Aug 12 2012 *)
%o A036295 (PARI) concat(0, vector(100, n, numerator(sum(i=1, n, i/2^i)))) \\ _Colin Barker_, Nov 09 2014
%o A036295 (PARI) a(n) = numerator(2-(n+2)/2^n); \\ _Joerg Arndt_, Jul 17 2023
%o A036295 (Magma) [0] cat [Numerator(&+[i/2^i: i in [1..n]]): n in [1..40]]; // _Vincenzo Librandi_, Nov 09 2014
%Y A036295 Cf. A036296 (denominators).
%Y A036295 Cf. A000265, A000295, A006519.
%K A036295 nonn,easy,frac
%O A036295 0,4
%A A036295 _N. J. A. Sloane_