cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A110205 Triangle, read by rows, where T(n,k) equals the sum of cubes of numbers < 2^n having exactly k ones in their binary expansion.

Original entry on oeis.org

1, 9, 27, 73, 368, 343, 585, 3825, 6615, 3375, 4681, 36394, 88536, 86614, 29791, 37449, 332883, 1024002, 1449198, 970677, 250047, 299593, 2979420, 10970133, 20078192, 19714083, 9974580, 2048383, 2396745, 26298405, 112122225, 250021125, 320944275, 239783895, 97221555, 16581375
Offset: 1

Views

Author

Paul D. Hanna, Jul 16 2005

Keywords

Comments

Compare to triangle A110200 (sum of squares).

Examples

			Row 4 is formed by sums of cubes of numbers < 2^4:
  T(4,1) = 1^3 + 2^3 + 4^3 + 8^3 = 585;
  T(4,2) = 3^3 + 5^3 + 6^3 + 9^3 + 10^3 + 12^3 = 3825;
  T(4,3) = 7^3 + 11^3 + 13^3 + 14^3 = 6615;
  T(4,4) = 15^3 = 3375.
Triangle begins:
        1;
        9,       27;
       73,      368,       343;
      585,     3825,      6615,      3375;
     4681,    36394,     88536,     86614,     29791;
    37449,   332883,   1024002,   1449198,    970677,    250047;
   299593,  2979420,  10970133,  20078192,  19714083,   9974580,  2048383;
  2396745, 26298405, 112122225, 250021125, 320944275, 239783895, 97221555, 16581375; ...
Row g.f.s are:
  row 1: (1 + 2*x + 1*x^2)/(1+x)^2;
  row 2: (9 + 36*x + 27*x^2)/(1+x);
  row 3: (73 + 368*x + 343*x^2);
  row 4: (585 + 3240*x + 3375*x^2)*(1+x).
G.f. for row n is:
  ((8^n-1)/7 + ((2^n-1)*(4^n-1)-(8^n-1)/7)*x + (2^n-1)^3*x^2)*(1+x)^(n-3).

Links

Crossrefs

Cf. A110206 (row sums), A110207 (central terms), A023001 (column 1).

Programs

  • Magma
    b:= func< n,k | Binomial(n-3, k) >;
A110205:= func< n,k | (8^n-1)/7*(b(n,k-1) -b(n,k-2)) +(2^n-1)^2*((2^n+1)*b(n,k-2) +(2^n-1)*b(n,k-3)) >;
[A110205(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 03 2024
  • Mathematica
    b[n_, k_]= Binomial[n-3, k];
T[n_, k_]:= (8^n-1)/7*(b[n,k-1] -b[n,k-2]) + (2^n-1)^2*((2^n+1)*b[n,k-2] + (2^n-1)*b[n,k-3]);
A110205[n_, k_]:= If[n<3, T[n,k]/2, T[n,k]];
Table[A110205[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Oct 03 2024 *)
  • PARI
    T(n,k)=(8^n-1)/7*binomial(n-3,k-1)+((2^n-1)*(4^n-1)-(8^n-1)/7)*binomial(n-3,k-2) +(2^n-1)^3*binomial(n-3,k-3)
  • PARI
    /* Sum of cubes of numbers<2^n with k 1-bits: */
T(n,k)=local(B=vector(n+1));if(n
  • SageMath
    def b(n,k): return binomial(n-3, k)
def A110205(n,k): return (8^n-1)/7*(b(n,k-1) - b(n,k-2)) + (2^n-1)^2*((2^n+1)*b(n,k-2) + (2^n-1)*b(n,k-3))
flatten([[A110205(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 03 2024

Formula

T(n, k) = (8^n-1)/7*C(n-3, k-1) + ((2^n-1)*(4^n-1)-(8^n-1)/7)*C(n-3, k-2) + (2^n-1)^3*C(n-3, k-3).
G.f. for row n: ((8^n-1)/7 + ((2^n-1)*(4^n-1)-(8^n-1)/7)*x + (2^n-1)^3*x^2)*(1+x)^(n-3).

A110207 a(n) = sum of cubes of numbers < 2^n having exactly floor(n/2) + 1 ones in their binary expansion.

Original entry on oeis.org

1, 27, 368, 6615, 88536, 1449198, 20078192, 320944275, 4584724120, 72867715074, 1064153845776, 16896536592390, 250629464211504, 3980364331323996, 59709362473177824, 948742244639103915, 14352114907032903000
Offset: 1

Views

Author

Paul D. Hanna, Jul 16 2005

Keywords

Comments

a(n) equals the largest term in row n of triangle A110205.

Links

Crossrefs

Cf. A110205 (triangle), A110206 (row sums).

Programs

  • Magma
    b:= func< n,k | Binomial(n-3, Floor(n/2) - k) >;
A110207:= func< n | (8^n-1)/7*(b(n,0) - b(n,1)) + (2^n-1)^2*((2^n+1)*b(n,1) + (2^n-1)*b(n,2)) >;
[A110207(n): n in [1..30]]; // G. C. Greubel, Oct 03 2024
  • Mathematica
    b[n_, k_]:= Binomial[n-3, Floor[n/2]-k];
f[n_]:= (8^n-1)/7*(b[n,0] - b[n,1]) + (2^n-1)^2*((2^n+1)*b[n,1] + (2^n - 1)*b[n,2]);
A110207[n_]:= If[n<3, f[n]/2, f[n]];
Table[A110207[n], {n,30}] (* G. C. Greubel, Oct 03 2024 *)
  • PARI
    {a(n)=(8^n-1)/7*binomial(n-3,n\2)+((2^n-1)*(4^n-1)-(8^n-1)/7)*binomial(n-3,n\2-1) +(2^n-1)^3*binomial(n-3,n\2-2)}
  • SageMath
    def b(n,k): return binomial(n-3, (n//2) - k)
def A110207(n): return (8^n-1)/7*(b(n,0) - b(n,1)) + (2^n-1)^2*((2^n+1)*b(n,1) + (2^n-1)*b(n,2))
[A110207(n) for n in range(1,31)] # G. C. Greubel, Oct 03 2024

Formula

a(n) = (8^n-1)/7*C(n-3, floor(n/2)) + ((2^n-1)*(4^n-1)-(8^n-1)/7)*C(n-3, floor(n/2)-1) + (2^n-1)^3*C(n-3, floor(n/2)-2).

A176793 A symmetrical triangle read by rows: T(n, k) = 2^n*(q^k - 1)*(q^(n - k) - 1) + 1, where q = 2.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 25, 25, 1, 1, 113, 145, 113, 1, 1, 481, 673, 673, 481, 1, 1, 1985, 2881, 3137, 2881, 1985, 1, 1, 8065, 11905, 13441, 13441, 11905, 8065, 1, 1, 32513, 48385, 55553, 57601, 55553, 48385, 32513, 1, 1, 130561, 195073, 225793, 238081, 238081, 225793, 195073, 130561, 1
Offset: 0

Views

Author

Roger L. Bagula, Apr 26 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,      1;
  1,      5,      1;
  1,     25,     25,      1;
  1,    113,    145,    113,      1;
  1,    481,    673,    673,    481,      1;
  1,   1985,   2881,   3137,   2881,   1985,      1;
  1,   8065,  11905,  13441,  13441,  11905,   8065,      1;
  1,  32513,  48385,  55553,  57601,  55553,  48385,  32513,      1;
  1, 130561, 195073, 225793, 238081, 238081, 225793, 195073, 130561,    1;

Links

Crossrefs

Cf. A000012 (q=1), this sequence (q=2), A176794 (q=3), A176795 (q=4).
Cf. A110206.

Programs

  • Magma
    f:= func< n,k,q | 1 + (q^k-1)*(q^(n-k)-1)*2^n >;
A176793:= func< n,k | f(n,k,2) >;
[A176793(n,k): k in [0..n], n in [0..13]]; // G. C. Greubel, Oct 02 2024
  • Mathematica
    T[n_, k_, q_] := 2^n*(q^k - 1)*(q^(n - k) - 1) + 1;
Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten
  • SageMath
    def f(n, k, q): return 1 + (q^k -1)*(q^(n-k) -1)*2^n
def A176793(n,k): return f(n,k,2)
flatten([[A176793(n, k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Oct 02 2024

Formula

T(n, k) = 1 - (f(n+1, 2*k+1, q) - f(n+1, 1, q)) - (f(n+1, 2*n-2*k+1, q) - f(n+1, 2*n+1, q)), where f(n, k, q) = 2^(n-1) * q^((k-1)/2), and q = 2.
From G. C. Greubel, Oct 02 2024: (Start)
T(n, k) = 2^n*(2^k - 1)*(2^(n-k) - 1) + 1.
T(2*n, n) = 1 + 4^n - 2*8^n + 16^n = 1 + 4*A110206(n).
Sum_{k=0..n} T(n, k) = 4^n*(n-3) + 2^n*(n+3) + (n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*(1+(-1)^n)*(1 - (2/3)*binomial(2^n, 2)). (End)

Extensions

Edited by G. C. Greubel, Oct 02 2024
Showing 1-3 of 3 results.

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