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.

Showing 1-4 of 4 results.

A173787 Triangle read by rows: T(n,k) = 2^n - 2^k, 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 7, 6, 4, 0, 15, 14, 12, 8, 0, 31, 30, 28, 24, 16, 0, 63, 62, 60, 56, 48, 32, 0, 127, 126, 124, 120, 112, 96, 64, 0, 255, 254, 252, 248, 240, 224, 192, 128, 0, 511, 510, 508, 504, 496, 480, 448, 384, 256, 0, 1023, 1022, 1020, 1016, 1008, 992, 960, 896, 768, 512, 0
Offset: 0

Views

Author

Reinhard Zumkeller, Feb 28 2010

Keywords

Examples

			Triangle begins as:
   0;
   1,  0;
   3,  2,  0;
   7,  6,  4,  0;
  15, 14, 12,  8,  0;
  31, 30, 28, 24, 16, 0;

Links

Programs

  • Magma
    [2^n -2^k: k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 13 2021
  • Mathematica
    Table[2^n -2^k, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 13 2021 *)
  • Sage
    flatten([[2^n -2^k for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 13 2021

Formula

A000120(T(n,k)) = A025581(n,k).
Row sums give A000337.
Central terms give A020522.
T(2*n+1, n) = A006516(n+1).
T(2*n+3, n+2) = A059153(n).
T(n, k) = A140513(n,k) - A173786(n,k), 0 <= k <= n.
T(n, k) = A173786(n,k) - A059268(n+1,k+1), 0 < k <= n.
T(2*n, 2*k) = T(n,k) * A173786(n,k), 0 <= k <= n.
T(n, 0) = A000225(n).
T(n, 1) = A000918(n) for n>0.
T(n, 2) = A028399(n) for n>1.
T(n, 3) = A159741(n-3) for n>3.
T(n, 4) = A175164(n-4) for n>4.
T(n, 5) = A175165(n-5) for n>5.
T(n, 6) = A175166(n-6) for n>6.
T(n, n-4) = A110286(n-4) for n>3.
T(n, n-3) = A005009(n-3) for n>2.
T(n, n-2) = A007283(n-2) for n>1.
T(n, n-1) = A000079(n-1) for n>0.
T(n, n) = A000004(n).

A175164 a(n) = 16*(2^n - 1).

Original entry on oeis.org

0, 16, 48, 112, 240, 496, 1008, 2032, 4080, 8176, 16368, 32752, 65520, 131056, 262128, 524272, 1048560, 2097136, 4194288, 8388592, 16777200, 33554416, 67108848, 134217712, 268435440
Offset: 0

Views

Author

Reinhard Zumkeller, Feb 28 2010

Keywords

Links

Crossrefs

Sequences of the form m*(2^n - 1): A000225 (m=1), A000918 (m=2), A068156 (m=3), A028399 (m=4), A068293 (m=6), A159741 (m=8), this sequence (m=16), A175165 (m=32), A175166 (m=64).
Cf. A140504, A173787.

Programs

  • Magma
    I:=[0,16]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021
  • Mathematica
    16*(2^Range[0,40] - 1) (* G. C. Greubel, Jul 08 2021 *)
  • Python
    def A175164(n): return (1<Chai Wah Wu, Jun 27 2023
  • Sage
    [16*(2^n -1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

Formula

a(n) = 2^(n+4) - 16.
a(n) = A173787(n+4, 4).
a(2*n) = A140504(n+2)*A028399(n).
a(n) = 3*a(n-1) - 2*a(n-2), a(0)=0, a(1)=16. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 16*x/((1-x)*(1-2*x)).
E.g.f.: 16*(exp(2*x) - exp(x)). (End)

A175166 a(n) = 64*(2^n - 1).

Original entry on oeis.org

0, 64, 192, 448, 960, 1984, 4032, 8128, 16320, 32704, 65472, 131008, 262080, 524224, 1048512, 2097088, 4194240, 8388544, 16777152, 33554368, 67108800, 134217664, 268435392, 536870848, 1073741760
Offset: 0

Views

Author

Reinhard Zumkeller, Feb 28 2010

Keywords

Links

Crossrefs

Sequences of the form m*(2^n - 1): A000225 (m=1), A000918 (m=2), A068156 (m=3), A028399 (m=4), A068293 (m=6), A159741 (m=8), A175164 (m=16), A175165 (m=32), this sequence (m=64).
Cf. A173787, A175161.

Programs

  • Magma
    I:=[0,64]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021
  • Mathematica
    LinearRecurrence[{3,-2},{0,64},30] (* Harvey P. Dale, Apr 08 2015 *)
  • Python
    def A175166(n): return (1<Chai Wah Wu, Jun 27 2023
  • Sage
    [64*(2^n -1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

Formula

a(n) = 2^(n+6) - 64.
a(n) = A173787(n+6, 6).
a(2*n) = A175161(n)*A159741(n) for n > 0.
a(n) = 3*a(n-1) - 2*a(n-2), a(0)=0, a(1)=64. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 64*x/((1-x)*(1-2*x)).
E.g.f.: 64*(exp(2*x) - exp(x)). (End)

A341419 a(0) = 1, a(1) = 1, a(2^(n-1)..2^n-1) = fwht(0..2^(n-2)). Here "fwht" is the fast Walsh-Hadamard transform with natural ordering and without multiplication of any factors.

Original entry on oeis.org

1, 1, 2, 0, 4, 2, 0, -2, 8, 6, 8, -2, 0, -2, -8, -2, 16, 14, 24, -2, 32, 14, -8, -18, 0, -2, -8, -2, -32, -18, -8, 14, 32, 30, 56, -2, 96, 46, -8, -50, 128, 94, 120, -34, -32, -50, -136, -18, 0, -2, -8, -2, -32, -18, -8, 14, -128, -98, -136, 30, -32, 14, 120, 46, 64, 62
Offset: 0

Views

Author

Thomas Scheuerle, Mar 24 2021

Keywords

Comments

This sequence is a rough integer-valued approximation to one of the nontrivial solutions to f(n) = a*fwht(f(n)).

Links

Crossrefs

Cf. A159741, A175165.

Programs

  • MATLAB
    function a = A341419(max_n)
a(1) = 1;
a(2) = 1;
    while length(a) < max_n
        w = fwht(a,[],'hadamard')*length(a);
        %w = myfwht(a); % own implementation for documentation purpose
        a = [a w];
    end
end
function w = myfwht(in)
    h = 1;
    while h < length(in)
        for i = 1:h*2:length(in)
            for j = i:i+h-1
                x = in(j);
                y = in(j+h);
                in(j) = x+y;
                in(j+h) = x-y;
            end
        end
        h = h*2;
    end
    w = in;
end

Formula

a(2^n) = 2^n.
a(2^n + 1) = 2^n-2 for n > 0.
a(2^n + 2) = 8*(2^(n-2) - 1) = A159741(n-2) for n > 1.
a(2^n + 3) = -2 for n > 1.
a(2^n + 4) = 32*(2^(n-3) - 1) = A175165(n-3) for n > 2.
a(2^n + 5) = 2*(2^n - 9) for n > 2.
a(2^n + 6) = -8 for n > 2.
a(2^n + 7) = -2*(8 * 2^(n-3) - 7) for n > 2.
a(2^n + 8) = 64*(2^(n-3) - 2) for n > 3.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.