A096299 -id:A096299 - cp's OEIS frontend

A095684 Triangle read by rows. There are 2^(m-1) rows of length m, for m = 1, 2, 3, ... The rows are in lexicographic order. The rows have the property that the first entry is 1, the second distinct entry (reading from left to right) is 2, the third distinct entry is 3, etc.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 1, 2, 2, 2, 1, 2, 2, 3, 1, 2, 3, 3, 1, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 1, 1, 2, 2, 2, 1, 1, 2, 2, 3, 1, 1, 2, 3, 3, 1, 1, 2, 3, 4, 1, 2, 2, 2, 2, 1, 2, 2, 2, 3, 1, 2, 2, 3, 3

Offset: 1

N. J. A. Sloane, Jun 25 2004

Row k is the unique multiset that covers an initial interval of positive integers and has multiplicities equal to the parts of the k-th composition in standard order (graded reverse-lexicographic, A066099). This composition is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. For example, the 13th composition is (1,2,1), so row 13 is {1,2,2,3}. - Gus Wiseman, Apr 26 2020

			1, 11, 12, 111, 112, 122, 123, 1111, 1112, 1122, 1123, 1222, 1223, 1233, ...
The 8 strings of length 4 are 1111, 1112, 1122, 1123, 1222, 1223, 1233, 1234.
From _Gus Wiseman_, Apr 26 2020: (Start)
The triangle read by columns begins:
  1:{1}  2:{1,1}  4:{1,1,1}   8:{1,1,1,1}  16:{1,1,1,1,1}
         3:{1,2}  5:{1,1,2}   9:{1,1,1,2}  17:{1,1,1,1,2}
                  6:{1,2,2}  10:{1,1,2,2}  18:{1,1,1,2,2}
                  7:{1,2,3}  11:{1,1,2,3}  19:{1,1,1,2,3}
                             12:{1,2,2,2}  20:{1,1,2,2,2}
                             13:{1,2,2,3}  21:{1,1,2,2,3}
                             14:{1,2,3,3}  22:{1,1,2,3,3}
                             15:{1,2,3,4}  23:{1,1,2,3,4}
                                           24:{1,2,2,2,2}
                                           25:{1,2,2,2,3}
                                           26:{1,2,2,3,3}
                                           27:{1,2,2,3,4}
                                           28:{1,2,3,3,3}
                                           29:{1,2,3,3,4}
                                           30:{1,2,3,4,4}
                                           31:{1,2,3,4,5}
(End)

J. C. Kieffer, W. Szpankowski and E.-H. Yang, Problems on sequences: information theory and computer science interface, IEEE Trans. Inform. Theory, 50 (No. 7, 2004), 1385-1392.

See A096299 for another version.
The number of distinct parts in row n is A000120(n), also the maximum part.
Row sums are A029931.
Heinz numbers of rows are A057335.
Row lengths are A070939.
Row products are A284001.
The version for prime indices is A305936.
There are A333942(n) multiset partitions of row n.
Multisets of compositions are counted by A034691.
Combinatory separations of normal multisets are A269134.
All of the following pertain to compositions in standard order (A066099):
- Necklaces are A065609.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Dealings are counted by A333939.
- Distinct parts are counted by A334028.
Cf. A318284, A333764, A333940, A334030.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ptnToNorm[y_]:=Join@@Table[ConstantArray[i,y[[i]]],{i,Length[y]}];
Table[ptnToNorm[stc[n]],{n,15}] (* Gus Wiseman, Apr 26 2020 *)

A110382 Numbers which are sum of distinct unary numbers (containing only ones), i.e., numbers which are sum of distinct numbers of the form (10^k - 1)/9.

Original entry on oeis.org

1, 11, 12, 111, 112, 122, 123, 1111, 1112, 1122, 1123, 1222, 1223, 1233, 1234, 11111, 11112, 11122, 11123, 11222, 11223, 11233, 11234, 12222, 12223, 12233, 12234, 12333, 12334, 12344, 12345, 111111, 111112, 111122, 111123, 111222, 111223

Offset: 1

Amarnath Murthy, Jul 25 2005

Not the same as A096299, since a(1023) = 1234567900 which is not in lexicographic order. - Ralf Stephan, May 17 2007

Georg Fischer, Table of n, a(n) for n = 1..16384 [First 1023 terms from David A. Corneth]

Cf. A096299.

f:= proc(n) local L,i:
  L:= convert(n,base,2);
  add(L[i]*(10^i-1)/9, i=1..nops(L))
end proc:
map(f, [$1..100]); # Robert Israel, Feb 03 2025

Nest[Append[#1, 10 #1[[Floor[#2/2] ]] + DigitCount[#2, 2, 1]] & @@ {#, Length[#] + 1} &, {1}, 36] (* Michael De Vlieger, Mar 12 2021 *)

a(n) = sum(k=0, log(n)\log(2), hammingweight(n\(2^k))*10^k); \\ Michel Marcus, May 09 2019

a(n) = my(b = Vecrev(binary(n))); sum(i = 1, #b, b[i] * (10^i-1)) / 9 \\ David A. Corneth, May 19 2019

G.f.: 1/(1-x) * Sum_{k>=0} (10^(k+1) - 1)/9 * x^2^k/(1 + x^2^k). - Ralf Stephan, May 17 2007
a(n) = 10*a(floor(n/2)) + A000120(n) = Sum_{k=0..floor(log_2(n))} A000120(floor(n/(2^k)))*10^k. - Mikhail Kurkov, May 08 2019
a(n) = a(floor(n/2)) + A007088(n) = (10*A007088(n) - A000120(n))/9. - Mikhail Kurkov, Mar 03 2021

a(1024) ff. corrected by Georg Fischer, Feb 03 2025

cp's OEIS Frontend

A096104 Digit reversal of A096299(n).

Views

Author

Keywords

Comments

Crossrefs

A095684 Triangle read by rows. There are 2^(m-1) rows of length m, for m = 1, 2, 3, ... The rows are in lexicographic order. The rows have the property that the first entry is 1, the second distinct entry (reading from left to right) is 2, the third distinct entry is 3, etc.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A110382 Numbers which are sum of distinct unary numbers (containing only ones), i.e., numbers which are sum of distinct numbers of the form (10^k - 1)/9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions