A007728 -id:A007728 - cp's OEIS frontend

A072170 Square array T(n,k) (n >= 0, k >= 2) read by antidiagonals, where T(n,k) is the number of ways of writing n as Sum_{i>=0} c_i 2^i, c_i in {0,1,...,k-1}.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 2, 3, 2, 2, 1, 1, 1, 3, 3, 4, 2, 2, 1, 1, 1, 1, 4, 3, 4, 2, 2, 1, 1, 1, 4, 4, 5, 4, 4, 2, 2, 1, 1, 1, 3, 5, 4, 5, 4, 4, 2, 2, 1, 1, 1, 5, 5, 8, 5, 6, 4, 4, 2, 2, 1, 1, 1, 2, 6, 6, 8, 5, 6, 4, 4, 2, 2, 1, 1, 1, 5, 6, 9, 8, 9, 6, 6, 4, 4, 2, 2, 1, 1

Offset: 0

N. J. A. Sloane, Jun 29 2002

k-th column is k-th binary partition function.

The sequence data corresponds (via the table link) to the transpose of the array shown in example and given by the definition. - M. F. Hasler, Feb 14 2019

			Array begins: (rows n >= 0, columns k >= 2)
1 1 1 1 1 1 1 1 ...
1 1 1 1 1 1 1 1 ...
1 2 2 2 2 2 2 2 ...
1 1 2 2 2 2 2 2 ...
1 3 3 4 4 4 4 4 ...
1 2 3 3 4 4 4 4 ...
1 3 4 5 5 6 6 6 ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
B. Reznick, Some binary partition functions, in "Analytic number theory" (Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451-477, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990.

k=3 column is A002487, k=4 is A008619 (positive integers repeated), k = 5, 6, 7 are A007728, A007729, A007730, limiting (infinity) column is A000123 doubled up.

b:= proc(n, i, k) option remember;
      `if`(n=0, 1, `if`(i<0, 0, add(`if`(n-j*2^i<0, 0,
         b(n-j*2^i, i-1, k)), j=0..k-1)))
    end:
T:= (n, k)-> b(n, ilog2(n), k):
seq(seq(T(d+2-k, k), k=2..d+2), d=0..14); # Alois P. Heinz, Jun 21 2012

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 0, 0, Sum[If[n-j*2^i < 0, 0, b[n-j*2^i, i-1, k]], {j, 0, k-1}]]];
t[n_, k_] := b[n, Length[IntegerDigits[n, 2]] - 1, k];
Table[Table[t[d+2-k, k], {k, 2, d+2}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Jan 14 2014, translated from Alois P. Heinz's Maple code *)

M72170=[[]]; A072170(n,k,i=logint(n+!n,2),r=1)={if( !i, k>n, r&&(k<5||k>=n),if(k>4, A000123(n\2)-(k==n), k<3, 1, k<4, A002487(n), n\2+1), M72170[r=setsearch(M72170,[n,k,i,""],1)-1][^-1]==[n,k,i], M72170[r][4], M72170=setunion(M72170,[[n,k,i,r=sum(j=0,min(k-1,n>>i),A072170(n-j*2^i,k,i-1,0))]]);r)} \\ Code for k<5 (using A002487 for k=3) and k>=n (using A000123) is optional but makes it about 3x faster. - M. F. Hasler, Feb 14 2019

T(n,k) = T(n,n+1) = T(n,n)+1 = A000123(floor(n/2)) for all k >= n+1. - M. F. Hasler, Feb 14 2019

A309616 a(n) is the number of ways to represent 2*n in the decibinary system.

Original entry on oeis.org

1, 2, 4, 6, 10, 13, 18, 22, 30, 36, 45, 52, 64, 72, 84, 93, 110, 122, 140, 154, 177, 192, 214, 230, 258, 277, 304, 324, 356, 376, 405, 426, 464, 490, 528, 557, 604, 634, 678, 710, 765, 802, 854, 892, 952, 989, 1042, 1080, 1146, 1190, 1253, 1300, 1374, 1420, 1486, 1533, 1612, 1664

Offset: 0

Jonas Hollm, Aug 10 2019

It appears that a(n) is the number of decibinary numbers that can be constructed to represent the decimal numbers 2n-2 and 2n-1. To make this more clear let's consider n = 5: a(5) = 10 means that there are 10 decibinary numbers that represent the decimal numbers 2*5 - 2 = 8 and 2*5 - 1 = 9.

Furthermore, a(n) is the number of k such that A028897(k)=2*n.

			a(1) = 1.
a(2) = a(2-1) + a(ceiling(2/2)) = a(1) + a(1) = 1 + 1 = 2.
a(3) = a(3-1) + a(ceiling(3/2)) = a(2) + a(2) = 2 + 2 = 4.
a(4) = a(4-1) + a(ceiling(4/2)) = a(3) + a(2) = 4 + 2 = 6.
a(5) = a(5-1) + a(ceiling(5/2)) = a(4) + a(3) = 6 + 4 = 10.
a(6) = a(6-1) + a(ceiling(6/2)) - a(ceiling((6-5)/2)) = a(5) + a(3) - a(1) = 10 + 4 - 1 = 13.
a(7) = a(7-1) + a(ceiling(7/2)) - a(ceiling((7-5)/2)) = a(6) + a(4) - a(1) = 13 + 6 - 1 = 18.
a(8) = a(8-1) + a(ceiling(8/2)) - a(ceiling((8-5)/2)) = a(7) + a(4) - a(2) = 18 + 6 - 2 = 22.
a(9) = a(9-1) + a(ceiling(9/2)) - a(ceiling((9-5)/2)) = a(8) + a(5) - a(2) = 22 + 10 - 2 = 30.
a(10) = a(10-1) + a(ceiling(10/2)) - a(ceiling((10-5)/2)) = a(9) + a(5) - a(3) = 30 + 10 - 4 = 36.

HackerRank, Decibinary Numbers.

Cf. A007728: superseeker found that the deltas of the sequence a(n+1) - a(n) match transformations of the original query.

Cf. A028897.

Nest[Append[#1, #1[[-1]] + #1[[Ceiling[#2/2] ]] - If[#2 > 5, #1[[Ceiling[(#2 - 5)/2] ]], 0 ]] & @@ {#, Length@ # + 1} &, {1}, 57] (* Michael De Vlieger, Sep 29 2019 *)

a(1) = 1. a(n) = a(n-1) + a(ceiling(n/2)) if 1 < n <= 5.
Conjecture: a(n) = a(n-1) + a(ceiling(n/2)) - a(ceiling((n-5)/2)) if n > 5.
I think this sequence is closely related to the 10th binary partition function. The only difference is that every second number is omitted. At the moment, the 10th binary partition function is not in the OEIS. However, my experiments strongly suggest that the 10th binary partition function would indeed look like 1, 1, 2, 2, 4, 4, 6, 6, 10, 10, 13, 13, ...

Name corrected by Rémy Sigrist, Oct 15 2019

cp's OEIS Frontend

A072170 Square array T(n,k) (n >= 0, k >= 2) read by antidiagonals, where T(n,k) is the number of ways of writing n as Sum_{i>=0} c_i 2^i, c_i in {0,1,...,k-1}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula