A007728 -id:A007728

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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}.

+10
13

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

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

LINKS

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.

FORMULA

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

EXAMPLE

Array begins: (rows n >= 0, columns k >= 2)

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 ...

MAPLE

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

MATHEMATICA

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 *)

PROG

(PARI) 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

CROSSREFS

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.

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Jun 29 2002

STATUS

approved

A309616

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

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..57.

HackerRank, Decibinary Numbers.

FORMULA

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, ...

EXAMPLE

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.

MATHEMATICA

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 *)

PROG

(C++) int a(int n) {

std::vector<int> seq;

int a = 1;

seq.push_back(a);

for (int i = 1; i < n; i++) {

a += seq.at(i / 2);

a -= (i >= 5) ? seq.at((i - 5) / 2) : 0;

seq.push_back(a);

}

return seq.back();

}

CROSSREFS

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

Cf. A028897.

KEYWORD

nonn,base

AUTHOR

Jonas Hollm, Aug 10 2019

EXTENSIONS

Name corrected by Rémy Sigrist, Oct 15 2019

STATUS

approved

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