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 A235963 #62 Jun 02 2025 16:49:53
%S A235963 0,1,2,2,2,3,3,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,7,7,7,7,8,8,8,8,8,8,8,8,
%T A235963 8,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,
%U A235963 12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13
%N A235963 n appears (n+1)/(1 + (n mod 2)) times.
%C A235963 n appears A001318(n+1) - A001318(n) = A026741(n+1) times.
%C A235963 Sum_{k=0...a(n)} (-1)^ceiling(k/2)*p(n-G(k)) = 0 for n>0, where p(n)=A000041(n) is the partition function, and G(k)=A001318(k) denotes the generalized pentagonal numbers.
%C A235963 Row lengths of A238442, n >= 1. - _Omar E. Pol_, Dec 22 2016
%H A235963 G. C. Greubel, <a href="/A235963/b235963.txt">Table of n, a(n) for n = 0..5000</a>
%F A235963 Let t = (sqrt(n*8/3 + 1) - 1)/2 + 1/3 and let k = floor(t); then a(n) = 2k if t - k < 2/3, 2k+1 otherwise. - _Jon E. Schoenfield_, Jun 13 2017
%F A235963 a(n) = m if n+1>A001318(m) and a(n) = m-1 otherwise where m = floor(sqrt(8(n+1)/3)). - _Chai Wah Wu_, Nov 23 2024
%F A235963 From _Natalia L. Skirrow_, May 13 2025: (Start)
%F A235963 a(n) = A180447(n) + A085141(n).
%F A235963 a(n) = floor((s+1)/6) + floor((s-1)/6) where s=floor(sqrt(24*n+1)).
%F A235963 G.f.: (f(x,x^2)-1)/(1-x), where f is Ramanujan's bivariate theta function. (End)
%F A235963 Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/8 + (2-sqrt(2))*log(2)/8 + log(2+sqrt(2))/(2*sqrt(2)). - _Amiram Eldar_, May 28 2025
%e A235963 As a triangle:
%e A235963 0;
%e A235963 1;
%e A235963 2, 2, 2;
%e A235963 3, 3;
%e A235963 4, 4, 4, 4, 4;
%e A235963 5, 5, 5;
%e A235963 6, 6, 6, 6, 6, 6, 6;
%e A235963 7, 7, 7, 7;
%e A235963 8, 8, 8, 8, 8, 8, 8, 8, 8;
%e A235963 ...
%p A235963 T:= n-> n$(n+1)/(n mod 2+1):
%p A235963 seq(T(n), n=0..13); # _Alois P. Heinz_, Nov 23 2024
%t A235963 Table[Table[n, {(n + 1)/(1 + Mod[n, 2])}], {n, 0, 14}]//Flatten (* _T. D. Noe_, Jan 29 2014 *)
%o A235963 (Python)
%o A235963 from math import isqrt
%o A235963 def A235963(n): return (m:=isqrt((n+1<<3)//3))-(n+1<=(m*(3*m+4)+1 if m&1 else m*(3*m+2))>>3) # _Chai Wah Wu_, Nov 23 2024
%o A235963 (Python)
%o A235963 A235963=lambda n: ((s:=isqrt(24*n+1))+1)//6+(s-1)//6 # _Natalia L. Skirrow_, May 13 2025
%Y A235963 First differences are A080995.
%Y A235963 Cf. A000041, A001318, A026741, A085141, A180447, A238442.
%K A235963 nonn,tabf
%O A235963 0,3
%A A235963 _Mircea Merca_, Jan 17 2014