A151758 -id:A151758 - cp's OEIS frontend

A151760 G.f.: Theta^4, where Theta = Sum_{k>=0} x^(2^k).

0, 0, 0, 0, 1, 4, 6, 8, 13, 12, 10, 16, 18, 16, 20, 24, 13, 12, 18, 16, 22, 24, 28, 24, 18, 16, 28, 24, 20, 24, 24, 0, 13, 12, 18, 16, 30, 24, 28, 24, 22, 24, 36, 24, 28, 24, 24, 0, 18, 16, 28, 24, 28, 24, 24, 0, 20, 24, 24, 0, 24, 0, 0, 0, 13, 12, 18, 16, 30, 24, 28, 24, 30, 24, 36

Offset: 0

N. J. A. Sloane, Jun 22 2009

nonn

Number of ways to write n as an ordered sum of 4 powers of 2. - Ilya Gutkovskiy, Feb 02 2021

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A151758.

N:= 10: # for a(0) .. a(2^N)
g:= add(x^(2^i),i=0..N)^4:
S:= series(g,x,2^N+1):
seq(coeff(S,x,j),j=0..2^N); # Robert Israel, Mar 27 2020

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

Original entry on oeis.org

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 *)

A151762 G.f.: Theta^6, where Theta = Sum_{k>=0} x^(2^k).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 6, 15, 26, 45, 66, 76, 96, 126, 140, 165, 210, 221, 210, 240, 252, 246, 300, 346, 360, 366, 380, 396, 420, 440, 480, 525, 450, 405, 450, 416, 396, 510, 540, 510, 516, 582, 540, 636, 660, 720, 720, 706, 600, 630, 620, 636, 660, 800, 720, 756, 660, 720

Offset: 0

N. J. A. Sloane, Jun 22 2009

nonn

Number of ways to write n as an ordered sum of 6 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), A151761 (m=5), this sequence (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, 6):
seq(a(n), n=0..58);  # 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, 6];
Table[a[n], {n, 0, 58}] (* Jean-François Alcover, Apr 25 2022, after Alois P. Heinz *)

A151774 Characteristic function of numbers with binary weight 2 (A018900).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Jun 23 2009

a(A018900(n)) = 1; a(A161989(n)) = 0. - Reinhard Zumkeller, Jun 24 2009

R. Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for characteristic functions

Cf. A000120, A018900, A151758.

w[n_] := IntegerDigits[n, 2] // Total;
a[n_] := Boole[w[n] == 2];
a /@ Range[0, 105] (* Jean-François Alcover, Mar 31 2021 *)

a(n)=hammingweight(n)==2 \\ Charles R Greathouse IV, Sep 26 2015

Let Theta = Sum_{k >= 0} x^(2^k). G.f. is (x + Theta^2 - Theta)/2 (cf. A151758).

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A151760 G.f.: Theta^4, where Theta = Sum_{k>=0} x^(2^k).

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

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

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A151762 G.f.: Theta^6, where Theta = Sum_{k>=0} x^(2^k).

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Maple

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A151774 Characteristic function of numbers with binary weight 2 (A018900).

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