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:=proc(n) local nn,nd: nn:=convert(n,base,2): nd:={seq(nn[j]-nn[j-1],j=2..nops(nn))}: if n=2 then 2 elif nd={0,1} then n else fi end: seq(a(n),n=1..2100); # Emeric Deutsch, Apr 21 2006
Take[Sort[Flatten[Table[(2^x - 1)*(2^y), {x, 32}, {y, 32}]]], 54] (* Alonso del Arte, Apr 21 2006 *)
Select[Range[2500],Length[Split[IntegerDigits[#,2]]]==2&] (* or *) Select[Range[2500],SequenceCount[IntegerDigits[#,2],{1,0}]>0 && SequenceCount[ IntegerDigits[#,2],{0,1}]==0&] (* Harvey P. Dale, Oct 04 2024 *)
def ok(n): b = bin(n)[2:]; return "10" in b and "01" not in b print([m for m in range(2045) if ok(m)]) # Michael S. Branicky, Feb 04 2021
def a_next(a_n): t = a_n >> 1; return (a_n | t) + (t & 1) a_n = 2; a = [] for i in range(54): a.append(a_n); a_n = a_next(a_n) # Falk Hüffner, Feb 19 2022
83 = 1010011 has 1's followed by 0's in positions 2 and 5 (reading from the right), so a(83)=7.
a049502 = f 0 1 where
f m i x = if x <= 4
then m else f (if mod x 4 == 1
then m + i else m) (i + 1) $ div x 2
-- Reinhard Zumkeller, Jun 17 2015
A049502 := proc(n) local a,ndgs,p ; a := 0 ; ndgs := convert(n,base,2) ; for p from 1 to nops(ndgs)-1 do if op(p,ndgs)- op(p+1,ndgs) = 1 then a := a+p ; end if; end do: a ; end proc: # R. J. Mathar, Oct 17 2012
Table[Total[Flatten[Position[Partition[Reverse[IntegerDigits[n,2]],2,1],?(#=={1,0}&)]]],{n,0,110}] (* _Harvey P. Dale, Oct 05 2013 *) Table[Total[SequencePosition[Reverse[IntegerDigits[n,2]],{1,0}][[All,1]]],{n,0,120}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 26 2020 *)
a(n)=if(n<5, return(0)); sum(i=0,exponent(n)-1, (bittest(n,i) && !bittest(n,i+1))*(i+1)) \\ Charles R Greathouse IV, Jan 30 2023
def m(n):
x=bin(int(n))[2:][::-1]
s=0
for i in range(1,len(x)):
if x[i-1]=="1" and x[i]=="0":
s+=i
return s
for i in range(101):
print(str(i)+" "+str(m(i))) # Indranil Ghosh, Dec 22 2016
The binary indices of 27 are {1,2,4,5}, with maximal anti-runs ((1),(2,4),(5)), with lengths (1,2,1). After removing duplicates, this is our row 10.
The binary indices of 53 are {1,3,5,6}, with maximal anti-runs ((1,3,5),(6)), with lengths (3,1). After removing duplicates, this is our row 16.
Triangle begins:
0: .
1: 1
2: 1 1
3: 2
4: 1 1 1
5: 1 2
6: 2 1
7: 1 1 1 1
8: 3
9: 1 1 2
10: 1 2 1
11: 2 1 1
12: 1 1 1 1 1
13: 1 3
14: 2 2
15: 1 1 1 2
16: 3 1
17: 1 1 2 1
18: 1 2 1 1
19: 2 1 1 1
20: 1 1 1 1 1 1
DeleteDuplicates[Table[Length/@Split[Join@@Position[Reverse[IntegerDigits[n,2]],1],#2!=#1+1&],{n,0,100}]]
See link.
import List (elemIndices) a187769 n k = a187769_tabf !! n !! k a187769_row n = a187769_tabf !! n a187769_tabf = [0] : [elemIndices (b, len - b) $ takeWhile ((<= len) . uncurry (+)) $ zip a000120_list a023416_list | len <- [1 ..], b <- [1 .. len]] a187769_list = concat a187769_tabf
{{0}}~Join~Table[SortBy[Range[2^n, 2^(n + 1) - 1], DigitCount[#, 2, 1] &], {n, 0, 8}] // Flatten (* Michael De Vlieger, Jan 03 2025 *)
Triangle starts: 1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 4, 2, 1, 1, 16, 8, 4, 2, 1, 1, 32, 16, 8, 4, 2, 1, 1, 64, 32, 16, 8, 4, 2, 1, 1, 128, 64, 32, 16, 8, 4, 2, 1, 1, 256, 128, 64, 32, 16, 8, 4, 2, 1, 1, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 1, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 1, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, ... - _Joerg Arndt_, May 04 2014 When viewed as a triangular array, row 8 of A023758 is 128 192 224 240 248 252 254 255 so row 8 here is 1 64 32 16 8 4 2 1 From _Mats Granvik_, Jan 19 2009: (Start) Except for the first term the table can also be formatted as: 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 4, 2, 1, 1, 16, 8, 4, 2, 1, 1, ... (End)
a057728 n k = a057728_tabl !! (n-1) !! (k-1) a057728_row n = a057728_tabl !! (n-1) a057728_tabl = iterate (\row -> zipWith (+) (row ++ [0]) ([0] ++ tail row ++ [1])) [1] -- Reinhard Zumkeller, Aug 08 2013
nn=10;Map[Select[#,#>0&]&,CoefficientList[Series[(x-x^2)/(1-2x)/(1-y x),{x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Jan 28 2014 *)
Module[{nn=12,ts},ts=2^Range[0,nn];Table[Join[{1},Reverse[Take[ts,n]]],{n,0,nn}]]//Flatten (* Harvey P. Dale, Jan 15 2022 *)
T(n, k) := if k = 0 then 1 else 2^(n - k - 1)$ create_list(T(n, k), n, 0, 12, k, 0, n - 1); /* Franck Maminirina Ramaharo, Jan 09 2019 */
102 in binary is 1100110, we rewrite it from the left so that first two 1's stay same ("11"), then "001" is rewritten to "0", the last 1 to "1", and we ignore the last 0, thus getting 1101, which is binary expansion of 13, thus a(102) = 13.
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Reverse[Length/@Split[bpe[n],#2!=#1+1&]]],{n,0,100}] (* Gus Wiseman, Jul 25 2025 *)
import re
def a(n): return int(re.sub("0+1", "0", bin(n)[2:].rstrip("0")), 2) if n else 0
print([a(n) for n in range(109)]) # Michael S. Branicky, Jul 25 2025
(define (A209859 n) (let loop ((n n) (s 0) (i (A053644 n))) (cond ((zero? n) s) ((> i n) (if (> (/ i 2) n) (loop n s (/ i 2)) (loop (- n (/ i 2)) (* 2 s) (/ i 4)))) (else (loop (- n i) (+ (* 2 s) 1) (/ i 2))))))
The binary indices of 27 are {1,2,4,5}, with maximal runs ((1,2),(4,5)), with lengths (2,2), which is the 10th composition in standard order, so a(27) = 10.
The binary indices of 100 are {3,6,7}, with maximal runs ((3),(6,7)), with lengths (1,2), which is the 6th composition in standard order, so a(100) = 6.
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Length/@Split[bpe[n],#2==#1+1&]],{n,0,100}]
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