A151762 -id:A151762 - cp's OEIS frontend

A151759 G.f.: Theta^3, where Theta = Sum_{k>=0} x^(2^k).

0, 0, 0, 1, 3, 3, 4, 6, 3, 3, 6, 6, 4, 6, 6, 0, 3, 3, 6, 6, 6, 6, 6, 0, 4, 6, 6, 0, 6, 0, 0, 0, 3, 3, 6, 6, 6, 6, 6, 0, 6, 6, 6, 0, 6, 0, 0, 0, 4, 6, 6, 0, 6, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 3, 3, 6, 6, 6, 6, 6, 0, 6, 6, 6, 0, 6, 0, 0, 0, 6, 6, 6, 0, 6, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 4, 6, 6, 0, 6, 0, 0, 0, 6

Offset: 0

N. J. A. Sloane, Jun 22 2009

nonn

Number of ways to write n as an ordered sum of 3 powers of 2. - Ilya Gutkovskiy, Feb 02 2021

Seiichi Manyama, Table of n, a(n) for n = 0..10000

(Sum_{k>=0} x^(2^k))^m; A209229 (m=1), A073267 (m=2), this sequence (m=3), A151760 (m=4), A151761 (m=5), A151762 (m=6).

Cf. A000079, A151758.

b:= proc(n, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(t<1, 0, add(b(n-2^j, t-1), j=0..ilog2(n))))
    end:
a:= n-> b(n, 3):
seq(a(n), n=0..104);  # Alois P. Heinz, Feb 02 2021

b[n_, t_] := b[n, t] = If[n == 0, If[t == 0, 1, 0],
     If[t < 1, 0, Sum[b[n - 2^j, t - 1], {j, 0, Floor@Log2[n]}]]];
a[n_] := b[n, 3];
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Mar 08 2022, after Alois P. Heinz *)

A151761 G.f.: Theta^5, where Theta = Sum_{k>=0} x^(2^k).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 5, 10, 15, 25, 31, 30, 40, 50, 50, 60, 75, 65, 55, 70, 66, 70, 90, 100, 100, 90, 90, 100, 110, 100, 120, 120, 75, 65, 95, 70, 90, 110, 130, 100, 126, 110, 130, 140, 150, 140, 160, 120, 100, 90, 130, 100, 150, 140, 160, 120, 110, 100, 160, 120, 120, 120, 120

Offset: 0

N. J. A. Sloane, Jun 22 2009

nonn

Number of ways to write n as an ordered sum of 5 powers of 2. - Ilya Gutkovskiy, Feb 02 2021

Seiichi Manyama, Table of n, a(n) for n = 0..10000

(Sum_{k>=0} x^(2^k))^m; A209229 (m=1), A073267 (m=2), A151759 (m=3), A151760 (m=4), this sequence (m=5), A151762 (m=6).

Cf. A151758.

b:= proc(n, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
      `if`(t<1, 0, add(b(n-2^j, t-1), j=0..ilog2(n))))
    end:
a:= n-> b(n, 5):
seq(a(n), n=0..62);  # Alois P. Heinz, Feb 02 2021

b[n_, t_] := b[n, t] = If[n == 0, If[t == 0, 1, 0],
     If[t < 1, 0, Sum[b[n - 2^j, t - 1], {j, 0, Floor@Log2[n]}]]];
a[n_] := b[n, 5];
Table[a[n], {n, 0, 62}] (* Jean-François Alcover, Apr 25 2022, after Alois P. Heinz *)

A151758 G.f.: Theta^2-Theta, where Theta = Sum_{k>=0} x^(2^k).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jun 22 2009

sign

If we omit the "-x" term and divide by 2, we get the characteristic function of the numbers with binary weight 2 (A018900, A151774).

Cf. A000079, A073267, A151759, A151760, A151761, A151762, A018900, A151774.

w[n_] := IntegerDigits[n, 2] // Total;
a[n_] := If[n == 1, -1, 2 Boole[w[n] == 2]];
a /@ Range[0, 104] (* Jean-François Alcover, Mar 31 2021 *)

A342250 Number of ways to write n as an ordered sum of seven powers of 2.

Original entry on oeis.org

1, 7, 21, 42, 77, 126, 168, 218, 294, 357, 427, 546, 637, 672, 756, 840, 854, 966, 1134, 1218, 1302, 1408, 1484, 1554, 1680, 1827, 1995, 2002, 1925, 2016, 1988, 1904, 2142, 2352, 2282, 2352, 2534, 2520, 2604, 2954, 3080, 3276, 3262, 3234, 3150, 3248, 3164, 3402, 3640

Offset: 7

Ilya Gutkovskiy, Mar 07 2021

Robert Israel, Table of n, a(n) for n = 7..10000

Cf. A000079, A023359, A073267, A151759, A151760, A151761, A151762, A209229, A342247, A342251, A342252, A342254.

N:= 100:
S:= add(x^(2^j),j=0..ilog2(N-6))^7:
[seq](coeff(S,x,j),j=7..N); # Robert Israel, Feb 26 2023

nmax = 55; CoefficientList[Series[Sum[x^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}]^7, {x, 0, nmax}], x] // Drop[#, 7] &

G.f.: ( Sum_{k>=0} x^(2^k) )^7.

A342251 Number of ways to write n as an ordered sum of eight powers of 2.

Original entry on oeis.org

1, 8, 28, 64, 126, 224, 336, 464, 645, 840, 1044, 1344, 1666, 1904, 2192, 2528, 2730, 3024, 3528, 3920, 4284, 4768, 5168, 5488, 5965, 6552, 7140, 7616, 7834, 8176, 8400, 8352, 8862, 9632, 9800, 10080, 10788, 10976, 11152, 12208, 13090, 13664, 14392, 14672, 14868, 15008, 15344

Offset: 8

Ilya Gutkovskiy, Mar 07 2021

nonn

Robert Israel, Table of n, a(n) for n = 8..10000

Cf. A000079, A023359, A073267, A151759, A151760, A151761, A151762, A209229, A342248, A342250, A342252, A342254.

N:= 100:
S:= add(x^(2^j),j=0..ilog2(N-7))^8:
seq(coeff(S,x,j),j=8..N); # Robert Israel, Feb 26 2023

nmax = 54; CoefficientList[Series[Sum[x^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}]^8, {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: ( Sum_{k>=0} x^(2^k) )^8.

A342252 Number of ways to write n as an ordered sum of nine powers of 2.

Original entry on oeis.org

1, 9, 36, 93, 198, 378, 624, 927, 1341, 1849, 2412, 3159, 4074, 4950, 5904, 7032, 8010, 9018, 10488, 11970, 13356, 15108, 16848, 18315, 20085, 22257, 24444, 26671, 28674, 30510, 32208, 33282, 34974, 37590, 39384, 40986, 43668, 45468, 46512, 49620, 53298, 55890, 59304, 62442

Offset: 9

Ilya Gutkovskiy, Mar 07 2021

nonn

Cf. A000079, A023359, A073267, A151759, A151760, A151761, A151762, A209229, A342249, A342250, A342251, A342254.

nmax = 52; CoefficientList[Series[Sum[x^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}]^9, {x, 0, nmax}], x] // Drop[#, 9] &

G.f.: ( Sum_{k>=0} x^(2^k) )^9.

A342254 Number of ways to write n as an ordered sum of ten powers of 2.

Original entry on oeis.org

1, 10, 45, 130, 300, 612, 1095, 1750, 2655, 3850, 5281, 7110, 9460, 12060, 14940, 18352, 21850, 25380, 29790, 34740, 39672, 45480, 51885, 57870, 64375, 72090, 80145, 88630, 97660, 106380, 114736, 122260, 130050, 139740, 148990, 157572, 168240, 178200, 185490, 196200, 210082

Offset: 10

Ilya Gutkovskiy, Mar 07 2021

nonn

Cf. A000079, A023359, A073267, A151759, A151760, A151761, A151762, A209229, A319922, A342250, A342251, A342252.

nmax = 50; CoefficientList[Series[Sum[x^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}]^10, {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: ( Sum_{k>=0} x^(2^k) )^10.

A347787 Number of compositions (ordered partitions) of n into at most 6 powers of 2.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 18, 30, 48, 68, 94, 118, 140, 168, 202, 224, 258, 292, 302, 302, 336, 352, 370, 424, 470, 472, 490, 504, 532, 544, 584, 600, 618, 532, 526, 542, 544, 536, 674, 664, 666, 656, 754, 704, 820, 824, 904, 840, 830, 712, 794, 744, 820, 824, 984

Offset: 0

Ilya Gutkovskiy, Sep 13 2021

nonn

Cf. A000079, A023359, A151762, A347784, A347785, A347786.

Table[Length@Flatten[Permutations/@IntegerPartitions[n,6,Select[Range@n,IntegerQ@Log2@#&]],1],{n,0,54}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

cp's OEIS Frontend

A151759 G.f.: Theta^3, where Theta = Sum_{k>=0} x^(2^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A151761 G.f.: Theta^5, where Theta = Sum_{k>=0} x^(2^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A151758 G.f.: Theta^2-Theta, where Theta = Sum_{k>=0} x^(2^k).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A342250 Number of ways to write n as an ordered sum of seven powers of 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A342251 Number of ways to write n as an ordered sum of eight powers of 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A342252 Number of ways to write n as an ordered sum of nine powers of 2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A342254 Number of ways to write n as an ordered sum of ten powers of 2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A347787 Number of compositions (ordered partitions) of n into at most 6 powers of 2.

Views

Author

Keywords

Crossrefs

Programs

Mathematica