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.
Since 9 is 1001 in binary, the 9th row is 1,2,1. Since 11 is 1011 in binary, the 11th row is 1,1,2. Triangle begins: 1; 1,1; 2; 1,2; 1,1,1; 2,1; 3; 1,3;
import Data.List (group) a101211 n k = a101211_tabf !! (n-1) !! (k-1) a101211_row n = a101211_tabf !! (n-1) a101211_tabf = map (reverse . map length . group) $ tail a030308_tabf -- Reinhard Zumkeller, Dec 16 2013
# Maple program due to W. Edwin Clark: Runs := proc (L) local j, r, i, k; j := 1: r[j] := L[1]: for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1: r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc: RunLengths := proc (L) map(nops, Runs(L)) end proc: c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc: # Row n is obtained with the command c(n). - Emeric Deutsch, Jul 03 2017 # Maple program due to W. Edwin Clark, yielding the integer ind corresponding to a given composition (the index of the composition): ind := proc (x) local X, j, i: X := NULL: for j to nops(x) do if type(j, odd) then X := X, seq(1, i = 1 .. x[j]) end if: if type(j, even) then X := X, seq(0, i = 1 .. x[j]) end if end do: X := [X]: add(X[i]*2^(nops(X)-i), i = 1 .. nops(X)) end proc; # Clearly, ind(c(n))= n. - Emeric Deutsch, Jan 23 2018
Table[Length /@ Split@ IntegerDigits[n, 2], {n, 38}] // Flatten (* Michael De Vlieger, Jul 11 2017 *)
apply( {A101211_row(n)=Vecrev((n=vecextract([-1..exponent(n)], bitxor(2*n, bitor(n,1))))[^1]-n[^-1])}, [1..19]) \\ replacing older code by M. F. Hasler, Mar 24 2025
from itertools import groupby
def arow(n): return [len(list(g)) for k, g in groupby(bin(n)[2:])]
def auptorow(rows):
alst = []
for i in range(1, rows+1): alst.extend(arow(i))
return alst
print(auptorow(38)) # Michael S. Branicky, Oct 02 2021
Table begins as: Row n in Terms on n binary that row 1 1 1; 2 10 1,1; 3 11 2; 4 100 2,1; 5 101 1,1,1; 6 110 1,2; 7 111 3; 8 1000 3,1; 9 1001 1,2,1; 10 1010 1,1,1,1; 11 1011 2,1,1; 12 1100 2,2; 13 1101 1,1,2; 14 1110 1,3; 15 1111 4; 16 10000 4,1; etc. with the terms of row n appearing in reverse order compared how the runs of the same length appear in the binary expansion of n (Cf. A101211). From _Omar E. Pol_, Sep 08 2013: (Start) Illustration of initial terms: --------------------------------------- k m Diagram Composition --------------------------------------- . _ 1 1 |_|_ 1; 2 1 |_| | 1, 1, 2 2 |_ _|_ 2; 3 1 |_ | | 2, 1, 3 2 |_|_| | 1, 1, 1, 3 3 |_| | 1, 2, 3 4 |_ _ _|_ 3; 4 1 |_ | | 3, 1, 4 2 |_|_ | | 1, 2, 1, 4 3 |_| | | | 1, 1, 1, 1, 4 4 |_ _|_| | 2, 1, 1, 4 5 |_ | | 2, 2, 4 6 |_|_| | 1, 1, 2, 4 7 |_| | 1, 3, 4 8 |_ _ _ _|_ 4; 5 1 |_ | | 4, 1, 5 2 |_|_ | | 1, 3, 1, 5 3 |_| | | | 1, 1, 2, 1, 5 4 |_ _|_ | | 2, 2, 1, 5 5 |_ | | | | 2, 1, 1, 1, 5 6 |_|_| | | | 1, 1, 1, 1, 1, 5 7 |_| | | | 1, 2, 1, 1, 5 8 |_ _ _|_| | 3, 1, 1, 5 9 |_ | | 3, 2, 5 10 |_|_ | | 1, 2, 2, 5 11 |_| | | | 1, 1, 1, 2, 5 12 |_ _|_| | 2, 1, 2, 5 13 |_ | | 2, 3, 5 14 |_|_| | 1, 1, 3, 5 15 |_| | 1, 4, 5 16 |_ _ _ _ _| 5; . Also irregular triangle read by rows in which row k lists the compositions of k, k >= 1. Triangle begins: [1]; [1,1], [2]; [2,1], [1,1,1], [1,2],[3]; [3,1], [1,2,1], [1,1,1,1], [2,1,1], [2,2], [1,1,2], [1,3], [4]; [4,1], [1,3,1], [1,1,2,1], [2,2,1], [2,1,1,1], [1,1,1,1,1], [1,2,1,1], [3,1,1], [3,2], [1,2,2], [1,1,1,2], [2,1,2], [2,3], [1,1,3], [1,4], [5]; Row k has length A001792(k-1). Row sums give A001787(k), k >= 1. (End)
import Data.List (group) a227736 n k = a227736_tabf !! (n-1) !! (k-1) a227736_row n = a227736_tabf !! (n-1) a227736_tabf = map (map length . group) $ tail a030308_tabf -- Reinhard Zumkeller, Aug 11 2014
Array[Length /@ Reverse@ Split@ IntegerDigits[#, 2] &, 34] // Flatten (* Michael De Vlieger, Dec 11 2020 *)
apply( {A227736_row(n, r=[1], b=n%2)=while(n\=2, n%2==b && r[#r]++ || [b=1-b, r=concat(r,1)]); r}, [1..22]) \\ M. F. Hasler, Mar 11 2025
def A227736_row(n): return[len(list(g))for _,g in groupby(bin(n)[:1:-1])] from itertools import groupby # M. F. Hasler, Mar 11 2025
(define (A227736 n) (A227186bi (A227737 n) (A227740 n))) ;; The Scheme-function for A227186bi has been given in A227186.
1 has 1 run in its binary representation "1". 2 has 2 runs in its binary representation "10". 3 has 1 run in its binary representation "11". 4 has 2 runs in its binary representation "100". 5 has 3 runs in its binary representation "101". Thus a(1) = 1, a(2) = 1+2 = 3, a(3) = 1+2+1 = 4, a(4) = 1+2+1+2 = 6, a(5) = 1+2+1+2+3 = 9.
Accumulate[Join[{0},Table[Length[Split[IntegerDigits[n,2]]],{n,110}]]] (* Harvey P. Dale, Jul 29 2013 *)
a(n) = my(v=binary(n+1),d=0,e=4); for(i=1,#v, if(v[i], v[i]=#v-i+d;d+=e;e=0, e=4)); fromdigits(v,2)>>1; \\ Kevin Ryde, Aug 27 2021
1 has one run in its binary representation "1", thus 1 occurs once. 2 has two runs in its binary representation "10", thus 2 occurs twice. 3 has one run in its binary representation "11", thus 3 occurs only once. 4 has two runs in its binary representation "100", thus 4 occurs twice. 5 has three runs in its binary representation "101", thus 5 occurs three times. The sequence thus begins as 1, 2,2, 3, 4,4, 5,5,5, ...
Table[ConstantArray[n, Length@ Split@ IntegerDigits[n, 2]], {n, 26}] // Flatten (* Michael De Vlieger, May 09 2017 *)
Table[PadRight[{},Length[Split[IntegerDigits[n,2]]],n],{n,40}]//Flatten (* Harvey P. Dale, Jul 23 2021 *)
Table[Range[0, #] &@ Total@ Flatten@ Map[Abs@ Differences@ # &,
Partition[IntegerDigits[n, 2], 2, 1]], {n, 34}] // Flatten (* Michael De Vlieger, May 09 2017 *)
(define (A227740 n) (- n (+ 1 (A173318 (- (A227737 n) 1)))))
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