A072170 -id:A072170 - cp's OEIS frontend

A007730 7th binary partition function.

1, 1, 2, 2, 4, 4, 6, 5, 9, 8, 12, 10, 16, 14, 19, 15, 24, 20, 28, 22, 34, 29, 39, 30, 46, 38, 52, 40, 59, 49, 64, 48, 72, 58, 78, 59, 87, 72, 94, 70, 104, 84, 113, 85, 124, 102, 132, 98, 144, 115, 153, 114, 166, 136, 176, 130, 189, 151, 200, 148, 212, 172, 220

Offset: 0

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000
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.

A column of A072170.

b:= proc(n, i) 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)), j=0..6)))
    end:
a:= n-> b(n, ilog2(n)):
seq(a(n), n=0..70);  # 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}]]];
a[n_] := b[n, Length[IntegerDigits[n, 2]] - 1, 7];
Table[a[n], {n, 0, 70} ] (* Jean-François Alcover, Jan 17 2014, after Alois P. Heinz *)

G.f.: Product_{k>=0} (1 - x^(7*2^k))/(1 - x^(2^k)). - Ilya Gutkovskiy, Jul 09 2019

More terms from Vladeta Jovovic, May 07 2004

A322010 Inverse permutation to A322000.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 14, 20, 26, 36, 3, 5, 7, 11, 15, 21, 27, 37, 46, 59, 8, 12, 16, 22, 28, 38, 47, 60, 72, 90, 17, 23, 29, 39, 48, 61, 73, 91, 108, 130, 30, 40, 49, 62, 74, 92, 109, 131, 152, 182, 50, 63, 75, 93, 110, 132, 153, 183, 212, 248, 76, 94, 111, 133, 154, 184, 213, 249

Offset: 0

M. F. Hasler, Feb 19 2019

nonn

a(n) is the position of n in the list A322000 of "decibinary numbers", i.e., integers sorted according to their decibinary value A028897(n) = Sum d[i]*2^i, where d[i] are the decimal digits of n.
For 0 <= m <= 9, we have a(n) = A322003(n) = A000123(n-1), because 1..9 are the first few terms of A322000 where the decibinary value increases.
We see that a(10..19) = a(2..9)+1 concatenated with (46, 49). Then, a(20..29) = a(12..19)+1 concatenated with (72, 90). Then, a(30..39) = a(22..29)+1 concatenated with (108, 130), and so on. This yields an alternate way to compute the sequence.

Cf. A322000, A028897, A322003, A072170(.,10), A000123.

vec_A322010=vecsort(A,,1)[1..vecmin(setminus([1..#A],Set(A)))-1] \\ Assumes the vector A = A322000(1..N) has been computed for some N. Exclude initial 0's to have correct (1-based) indices of the vectors.

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A007730 7th binary partition function.

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