A075897 -id:A075897 - cp's OEIS frontend

A089052 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) = number of partitions of n into exactly k powers of 2.

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 1, 1, 1, 0, 0, 0, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 0, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 1, 2, 2, 3, 3, 2, 2, 2, 1, 1, 1, 0, 0, 0, 1, 2, 2, 3, 3, 2, 2, 2, 1, 1, 1

Offset: 0

N. J. A. Sloane, Dec 03 2003

			1
0 1
0 1 1
0 0 1 1
0 1 1 1 1
0 0 1 1 1 1
0 0 1 2 1 1 1
0 0 0 1 2 1 1 1
0 1 1 1 2 2 1 1 1
0 0 1 1 1 2 2 1 1 1
0 0 1 2 2 2 2 2 1 1 1
0 0 0 1 2 2 2 2 2 1 1 1

J. Jordan and R. Southwell, Further Properties of Reproducing Graphs, Applied Mathematics, Vol. 1 No. 5, 2010, pp. 344-350. doi: 10.4236/am.2010.15045. - From N. J. A. Sloane, Feb 03 2013

Alois P. Heinz, Rows n = 0..200, flattened
J. Jordan and R. Southwell, Further Properties of Reproducing Graphs, Applied Mathematics, Vol. 1 No. 5, 2010, pp. 344-350. doi: 10.4236/am.2010.15045. - From _N. J. A. Sloane_, Feb 03 2013

Columns give A036987, A075897 (essentially), A089049, A089050, A089051, A319922.
Row sums give A018819.
See A089053 for another version.

A089052 := proc(n, k)
    option remember;
    if k > n then
        return(0);
    end if;
    if k= 0 then
        if n=0 then
            return(1)
        else
            return(0);
        end if;
    end if;
    if n mod 2 = 1 then
            return procname(n-1, k-1);
    end if;
    procname(n-1, k-1)+procname(n/2, k);
end proc:

t[n_, k_] := t[n, k] = Which[k > n, 0, k == 0, If[n == 0, 1, 0], Mod[n, 2] == 1, t[n-1, k-1], True, t[n-1, k-1] + t[n/2, k]]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 14 2014, after Maple *)

T(2m, k) = T(m, k)+T(2m-1, k-1); T(2m+1, k) = T(2m, k-1).

G.f.: 1/Product_{k>=0} (1-y*x^(2^k)). - Vladeta Jovovic, Dec 03 2003

A089049 Number of ways of writing n as a sum of exactly 4 powers of 2.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 1, 3, 1, 1, 0, 2, 1, 2, 2, 3, 2, 3, 1, 3, 2, 3, 1, 3, 1, 1, 0, 2, 2, 3, 1, 3, 1, 1, 0, 3, 1, 1, 0, 1, 0, 0, 0, 2, 1, 2, 2, 3, 2, 3, 1, 3, 2, 3, 1, 3, 1, 1, 0, 3, 2, 3, 1, 3, 1, 1, 0, 3, 1, 1, 0, 1, 0, 0, 0, 2, 2, 3, 1, 3, 1, 1, 0, 3

Offset: 0

N. J. A. Sloane, Dec 03 2003

nonn

The powers do not need to be distinct.

Alois P. Heinz, Table of n, a(n) for n = 0..20000

A column of A089052. Cf. A036987, A075897, A089048, A089050, A089051, A089053.

A089050 Number of ways of writing n as a sum of exactly 5 powers of 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 2, 3, 2, 4, 3, 4, 3, 4, 2, 4, 3, 4, 3, 4, 1, 3, 2, 3, 2, 5, 3, 4, 3, 5, 3, 5, 3, 5, 3, 4, 1, 4, 2, 4, 3, 5, 3, 4, 1, 4, 3, 4, 1, 4, 1, 1, 0, 3, 2, 3, 2, 5, 3, 4, 3, 6, 3, 5, 3, 5, 3, 4, 1, 5, 3, 5, 3, 6, 3, 4, 1, 5, 3, 4, 1, 4, 1, 1, 0, 4, 2, 4, 3, 5, 3, 4, 1, 5

Offset: 0

N. J. A. Sloane, Dec 03 2003

nonn

The powers do not need to be distinct.

Alois P. Heinz, Table of n, a(n) for n = 0..65536

A column of A089052. Cf. A036987, A075897, A089048, A089049, A089051, A089053.

A089051 Number of ways of writing n as a sum of exactly 6 powers of 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 3, 4, 3, 4, 4, 5, 4, 6, 4, 5, 4, 6, 4, 6, 4, 5, 3, 5, 3, 6, 5, 6, 4, 7, 5, 7, 5, 8, 5, 7, 4, 7, 4, 6, 4, 8, 5, 7, 4, 7, 4, 7, 4, 7, 4, 5, 1, 5, 3, 5, 3, 7, 5, 6, 4, 9, 6, 8, 5, 9, 5, 7, 4, 8, 5, 8, 5, 10, 6, 8, 4, 9, 5, 8, 4, 8, 4, 5, 1, 7, 4, 6, 4, 9, 5, 7, 4, 9

Offset: 0

N. J. A. Sloane, Dec 03 2003

nonn

The powers do not need to be distinct.

Alois P. Heinz, Table of n, a(n) for n = 0..65536

A column of A089052. Cf. A036987, A075897, A089048, A089049, A089050, A089053.

A089048 Number of ways of writing n as a sum of exactly 3 powers of 2.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 2, 1, 1, 0, 1, 0, 0, 0, 1, 1, 2, 1, 2, 1, 1, 0, 2, 1, 1, 0, 1, 0, 0, 0, 2, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 1, 2, 1, 1, 0, 2, 1, 1, 0, 1, 0, 0, 0, 2, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1, 0, 1, 0, 0, 0, 1

Offset: 0

N. J. A. Sloane, Dec 03 2003

nonn

The powers do not need to be distinct.

Amiram Eldar, Table of n, a(n) for n = 0..10000

A column of A089052.

Cf. A000120, A036987, A075897, A089049, A089050, A089051, A089053.

f := proc(n,k) option remember; if k > n then RETURN(0); fi; if k= 0 then if n=0 then RETURN(1) else RETURN(0); fi; fi; if n mod 2 = 1 then RETURN(f(n-1,k-1)); fi; f(n-1,k-1)+f(n/2,k); end; # present sequence is f(n,3)

a[n_] := If[n < 3, 0, ((1 - Mod[n, 2])*(1 - Mod[DigitCount[n, 2, 1], 2]) + 1)*If[Floor[(1/4)*DigitCount[n, 2, 1]] == 0, 1, 0]];
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Feb 13 2018, after Reinhard Zumkeller *)

For n > 2: a(n) = (1 + (1 - A000120(n) mod 2)*(1 - n mod 2)) * 0^floor(A000120(n)/4). - Reinhard Zumkeller, Dec 14 2003

A255264 Total number of ON cells in the "Ulam-Warburton" two-dimensional cellular automaton of A147562 after A048645(n) generations.

Original entry on oeis.org

1, 5, 9, 21, 25, 37, 85, 89, 101, 149, 341, 345, 357, 405, 597, 1365, 1369, 1381, 1429, 1621, 2389, 5461, 5465, 5477, 5525, 5717, 6485, 9557, 21845, 21849, 21861, 21909, 22101, 22869, 25941, 38229, 87381, 87385, 87397, 87445, 87637

Offset: 1

Omar E. Pol, Feb 19 2015

It appears that these are the terms of A147562, A162795, A169707, A255366, A256250, A256260, whose indices have binary weight 1 or 2.

			Also, written as an irregular triangle in which row lengths are the terms of A028310 the sequence begins:
      1;
      5;
      9,    21;
     25,    37,    85;
     89,   101,   149,   341;
    345,   357,   405,   597,  1365;
   1369,  1381,  1429,  1621,  2389,  5461;
   5465,  5477,  5525,  5717,  6485,  9557, 21845;
  21849, 21861, 21909, 22101, 22869, 25941, 38229, 87381;
  ...
Right border gives the positive terms of A002450.
It appears that the second leading diagonal gives the odd terms of A206374.

Cf. A002450, A028310, A032925, A048645, A075897, A139250, A147562, A162795, A169707, A206374, A209492, A255263, A255366, A256250, A256260.

a(n) = A147562(A048645(n)).
Conjecture 1: a(n) = A162795(A048645(n)).
Conjecture 2: a(n) = A169707(A048645(n)).
Conjecture 3: a(n) = A255366(A048645(n)).
Conjecture 4: a(n) = A256250(A048645(n)).
Conjecture 5: a(n) = A256260(A048645(n)).
a(n) = A032925(A209492(n-1)) (conjectured). - Jon Maiga, Dec 17 2021

A342247 Number of partitions of n into seven powers of 2.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 3, 4, 4, 4, 4, 5, 4, 6, 5, 6, 6, 7, 5, 8, 6, 7, 6, 8, 5, 7, 5, 7, 6, 9, 6, 9, 7, 9, 7, 11, 8, 10, 7, 10, 7, 10, 6, 11, 8, 10, 7, 12, 7, 10, 7, 11, 7, 10, 5, 9, 5, 8, 5, 10, 7, 10, 6, 11, 9, 12, 8, 14, 9, 11, 7, 13, 8, 12, 8, 14, 10, 13, 8, 15, 9, 13

Offset: 7

Ilya Gutkovskiy, Mar 07 2021

nonn

Column k=7 of A089052.

Cf. A000079, A000123, A018819, A075897, A089048, A089049, A089050, A089051, A209229, A319922, A342248, A342249, A342250.

A342248 Number of partitions of n into eight powers of 2.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 4, 5, 5, 6, 6, 7, 6, 8, 7, 8, 8, 10, 7, 10, 8, 9, 7, 10, 7, 10, 9, 11, 9, 12, 9, 13, 11, 14, 10, 14, 10, 13, 10, 14, 11, 15, 10, 15, 12, 15, 10, 17, 11, 14, 10, 15, 9, 13, 8, 14, 10, 14, 10, 16, 11, 16, 12, 18, 14, 18, 11, 18, 13, 17, 12, 20

Offset: 8

Ilya Gutkovskiy, Mar 07 2021

nonn

Column k=8 of A089052.

Cf. A000079, A000123, A018819, A075897, A089048, A089049, A089050, A089051, A209229, A319922, A342247, A342249, A342251.

A342249 Number of partitions of n into nine powers of 2.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 5, 5, 5, 6, 6, 8, 7, 8, 8, 9, 8, 11, 10, 11, 10, 12, 9, 12, 10, 12, 10, 14, 11, 14, 12, 15, 13, 17, 14, 18, 14, 17, 13, 18, 14, 19, 15, 19, 15, 20, 15, 21, 17, 21, 14, 21, 15, 19, 13, 20, 14, 19, 14, 22, 16, 21, 16, 24, 18, 24, 18, 25

Offset: 9

Ilya Gutkovskiy, Mar 07 2021

nonn

Column k=9 of A089052.

Cf. A000079, A000123, A018819, A075897, A089048, A089049, A089050, A089051, A209229, A319922, A342247, A342248, A342252.

cp's OEIS Frontend

A089052 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) = number of partitions of n into exactly k powers of 2.

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A089049 Number of ways of writing n as a sum of exactly 4 powers of 2.

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A089050 Number of ways of writing n as a sum of exactly 5 powers of 2.

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A089051 Number of ways of writing n as a sum of exactly 6 powers of 2.

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A089048 Number of ways of writing n as a sum of exactly 3 powers of 2.

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A255264 Total number of ON cells in the "Ulam-Warburton" two-dimensional cellular automaton of A147562 after A048645(n) generations.

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A342247 Number of partitions of n into seven powers of 2.

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A342248 Number of partitions of n into eight powers of 2.

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A342249 Number of partitions of n into nine powers of 2.

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