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.
a(2) = 4+5 = 9= 3^2. a(8)= 1+2+3+4+5+6+7+8 = 36 = 6^2.
11 in binary is 1011, which ends with two 1's.
N:= 1000: # to get all terms up to N Odds:= [seq(2*i+1,i=0..floor((N-1)/2)]: f:= proc(n) local L,x; L:= convert(n,base,2); x:= ListTools:-Search(0,L); if x = 0 then type(nops(L),even) else type(x,odd) fi end proc: A131323:= select(f,Odds); # Robert Israel, Apr 02 2014
Select[Range[500], OddQ[ # ] && EvenQ[FactorInteger[ # + 1][[1, 2]]] &] (* Stefan Steinerberger, Dec 18 2007 *) en1Q[n_]:=Module[{ll=Last[Split[IntegerDigits[n,2]]]},Union[ll] =={1} &&EvenQ[Length[ll]]]; Select[Range[1,501,2],en1Q] (* Harvey P. Dale, May 18 2011 *)
is(n)=n%2 && valuation(n+1,2)%2==0 \\ Charles R Greathouse IV, Aug 20 2013
from itertools import count, islice def A131323_gen(startvalue=3): # generator of terms >= startvalue return map(lambda n:(n<<1)+1,filter(lambda n:(~(n+1)&n).bit_length()&1,count(max(startvalue>>1,1)))) A131323_list = list(islice(A131323_gen(),30)) # Chai Wah Wu, Sep 11 2024
def A131323(n): def bisection(f,kmin=0,kmax=1): while f(kmax) > kmax: kmax <<= 1 kmin = kmax >> 1 while kmax-kmin > 1: kmid = kmax+kmin>>1 if f(kmid) <= kmid: kmax = kmid else: kmin = kmid return kmax def f(x): c, s = n+x, bin(x+1)[2:] l = len(s) for i in range(l&1,l,2): c -= int(s[i])+int('0'+s[:i],2) return c return bisection(f,n,n)<<1|1 # Chai Wah Wu, Jan 29 2025
f:= proc(n) local y,F,t,k,v;
if n::odd then
F:= ifactors(n)[2];
y:= mul(t[1]^ceil(t[2]/3),t=F);
k:= 1+floor((n*(n-1)/2)^(1/3)/y);
(k*y)^3/n - (n-1)/2;
else
v:= padic:-ordp(n,2);
if v mod 3 <> 1 then return 0 fi;
F:= ifactors(n/2^v)[2];
y:= mul(t[1]^ceil(t[2]/3),t=F)*2^((v-1)/3);
k:= 1 + floor((n*(n-1)/2)^(1/3)/y);
if k::even then k:= k+1 fi;
(k*y)^3/n - (n-1)/2;
fi
end proc:
map(f, [$1..100]); # Robert Israel, Nov 15 2023
f[n_] := Module[{y, F, t, k, v},
If[OddQ[n],
F = FactorInteger[n];
y = Product[t[[1]]^Ceiling[t[[2]]/3], {t, F}];
k = 1 + Floor[(n*(n-1)/2)^(1/3)/y];
(k*y)^3/n - (n-1)/2
,
v = IntegerExponent[n, 2];
If[Mod[v, 3] != 1, Return[0]];
F = FactorInteger[n/2^v];
y = Product[t[[1]]^Ceiling[t[[2]]/3], {t, F}]*2^((v-1)/3);
k = 1 + Floor[(n*(n-1)/2)^(1/3)/y];
If[EvenQ[k], k = k+1];
(k*y)^3/n - (n-1)/2]];
Map[f, Range[100]] (* Jean-François Alcover, Jul 09 2024, after Robert Israel *)
n = 4: the least k >= 2 such that (4 + 1)*(2*k + 4)/2 = 5*k + 10 is a triangular number is k = 7, thus a(4) = 7. n = 5: the least k >= 2 such that (5 + 1)*(2*k + 5)/2 = 6*k + 15 is a triangular number is k = 5, thus a(5) = 5.
a[n_] := Module[{k = 2}, While[! IntegerQ[Sqrt[4*(n + 1)*(2*k + n) + 1]], k++]; k]; Array[a, 100, 0] (* Amiram Eldar, Dec 30 2024 *)
a(n) = my(k=2); while (!ispolygonal((n + 1)*(2*k + n)/2, 3), k++); k; \\ Michel Marcus, Dec 30 2024
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