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 1-5 of 5 results.

A209721 1/4 the number of (n+1) X 3 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.

Original entry on oeis.org

3, 4, 5, 7, 9, 13, 17, 25, 33, 49, 65, 97, 129, 193, 257, 385, 513, 769, 1025, 1537, 2049, 3073, 4097, 6145, 8193, 12289, 16385, 24577, 32769, 49153, 65537, 98305, 131073, 196609, 262145, 393217, 524289, 786433, 1048577, 1572865, 2097153, 3145729
Offset: 1

Views

Author

R. H. Hardin, Mar 12 2012

Keywords

Comments

Column 2 of A209727.
From Richard Locke Peterson, Apr 26 2020: (Start)
The formula a(n) = 2*a(n-2)-1 also fits empirically. With the given initial numbers a(1)=3, a(2)=4, a(3)=5, this new formula implies the old empirical formula. (But it is not established that the old empirical formula is true, so it is not established that the new formula is true either.) Furthermore, if the initial numbers had somehow, for example, been 3,4,6 instead, the new formula no longer implies the old formula.
If the new formula actually is true, it follows that a(n) is the number of distinct integer triangles that can be formed with sides of length a(n-1) and a(n-2), since the greatest length the third side can have is a(n-1)+a(n-2)-1, and the least length would be a(n-1)-a(n-2)+1. (End)
Conjectures: a(n) = A029744(n+1)+1. Also, a(n) = positions of the zeros in A309019(n+2) - A002487(n+2). - George Beck, Mar 26 2022

Examples

			Some solutions for n=4
..2..1..2....1..2..1....0..2..1....2..0..1....1..2..0....2..1..2....0..1..0
..0..2..0....2..0..2....1..0..2....1..2..0....2..0..1....0..2..0....2..0..2
..1..0..1....0..1..0....0..2..1....2..0..1....1..2..0....1..0..1....1..2..1
..0..2..0....2..0..2....1..0..2....1..2..0....2..0..1....0..2..0....2..0..2
..1..0..1....0..1..0....0..2..1....2..0..1....1..2..0....2..1..2....1..2..1

Links

Crossrefs

Cf. A029744, A209727.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

Formula

Empirical: a(n) = a(n-1) +2*a(n-2) -2*a(n-3).
Empirical g.f.: x*(3+x-5*x^2)/((1-x)*(1-2*x^2)). [Colin Barker, Mar 23 2012]

A070875 Binary expansion is 1x100...0 where x = 0 or 1.

Original entry on oeis.org

5, 7, 10, 14, 20, 28, 40, 56, 80, 112, 160, 224, 320, 448, 640, 896, 1280, 1792, 2560, 3584, 5120, 7168, 10240, 14336, 20480, 28672, 40960, 57344, 81920, 114688, 163840, 229376, 327680, 458752, 655360, 917504, 1310720, 1835008, 2621440
Offset: 0

Views

Author

N. J. A. Sloane, May 19 2002

Keywords

Comments

A 2-automatic sequence. - Charles R Greathouse IV, Sep 24 2012
Third row in array A228405. - Richard R. Forberg, Sep 06 2013
Conjecture: a(n) = -1 + positions of the ones in A309019(n+2) - A002487(n+2). - George Beck, Mar 26 2022
Consecutive integers for which the number of its proper nondivisors of the form 2^k (k > 0) is 2; proper nondivisors are defined in A173540 (5 has two such nondivisors: 2 and 4, 7 has 2 and 4, 10 has 4 and 8, 14 has 4 and 8, 20 has 8 and 16,...). - Lechoslaw Ratajczak, Dec 17 2024

Links

Crossrefs

Cf. A070876, A123760, A063920.

Programs

  • Magma
    [n le 2 select 2*n+3 else 2*Self(n-2): n in [1..39]]; // Bruno Berselli, Mar 01 2011
  • Mathematica
    Flatten@ NestList[ 2# &, {5, 7}, 19] (* Or *)
CoefficientList[ Series[(5 + 7 x)/(1 - 2 x^2), {x, 0, 38}], x] (* Robert G. Wilson v, May 20 2002 *)
  • PARI
    a(n)=if(n%2,7,5)<<(n\2) \\ Charles R Greathouse IV, Sep 24 2012

Formula

A093873(a(n)) = 2. - Reinhard Zumkeller, Oct 13 2006
For n>1, a(n+1) = a(n) + A000010(a(n)). - Stefan Steinerberger, Dec 20 2007
From Bruno Berselli, Mar 01 2011: (Start)
G.f.: (5+7*x)/(1-2*x^2).
a(n) = (6-(-1)^n)*2^((2*n+(-1)^n-1)/4). Therefore: a(n) = 5*2^(n/2) for n even, otherwise a(n) = 7*2^((n-1)/2).
a(n) = 2*a(n-2) for n>1. (End)
a(n+1) = A063757(n) + 6. - Philippe Deléham, Apr 13 2013
a(n) = sqrt(2*a(n-1) - (-2)^(n-1)). - Richard R. Forberg, Sep 06 2013
a(n+3) = a(n+2)*a(n+1)/a(n). - Richard R. Forberg, Sep 06 2013
For n>1, a(n) = 2*phi(a(n)) + phi(phi(a(n))). - Ivan Neretin, Feb 28 2016
a(2n) = A020714(n), a(2n+1) = A005009(n); for n>0. - Yosu Yurramendi, Jun 01 2016
From Ilya Gutkovskiy, Jun 02 2016: (Start)
E.g.f.: 7*sinh(sqrt(2)*x)/sqrt(2) + 5*cosh(sqrt(2)*x).
a(n) = 2^((n-3)/2)*(5*sqrt(2)*(1 + (-1)^n) + 7*(1 - (-1)^n)). (End)
Sum_{n>=0} 1/a(n) = 24/35. - Amiram Eldar, Mar 28 2022

Extensions

Extended by Robert G. Wilson v, May 20 2002

A309020 Expansion of x * Product_{k>=0} (1 + x^(2^k) + x^(2^(k+1)) - x^(2^(k+2))).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 3, 2, 2, 2, 1, 1, 0, 1, 2, 2, 4, 3, 3, 2, 1, 2, 2, 2, 1, 1, 0, 1, 0, 0, 0, 1, 3, 2, 3, 2, 4, 4, 5, 3, 2, 3, 2, 2, 0, 1, 1, 2, 3, 2, 2, 2, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, -1, 0, 0, 0, 1, 1, 4, 3, 4, 2, 2, 3, 3, 2, 3, 4, 6, 4, 5, 5, 4, 3, 0, 2
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 06 2019

Keywords

Crossrefs

Cf. A002487, A005590, A309019, A309021, A309022.

Programs

  • Mathematica
    nmax = 90; CoefficientList[Series[x Product[(1 + x^(2^k) + x^(2^(k + 1)) - x^(2^(k + 2))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], a[(n - 1)/2] + a[(n + 1)/2] - a[(n - 3)/2]]; Table[a[n], {n, 0, 90}]

Formula

a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = a(n) + a(n+1) - a(n-1).

A309021 Expansion of x * Product_{k>=0} (1 + x^(2^k) - x^(2^(k+1)) - x^(2^(k+2))).

Original entry on oeis.org

0, 1, 1, 0, 1, -2, 0, 0, 1, -3, -2, 1, 0, 2, 0, 1, 1, -4, -3, 0, -2, 6, 1, 1, 0, 1, 2, -2, 0, -1, 1, 0, 1, -6, -4, 0, -3, 7, 0, 1, -2, 8, 6, -3, 1, -6, 1, -2, 0, 0, 1, 1, 2, -5, -2, 0, 0, 1, -1, 2, 1, 0, 0, 0, 1, -7, -6, 1, -4, 10, 0, 1, -3, 10, 7, -4, 0, -6, 1, -3, -2, 9, 8, 0, 6, -17, -3, -2, 1, -4, -6
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 06 2019

Keywords

Crossrefs

Cf. A002487, A005590, A309019, A309020, A309022.

Programs

  • Mathematica
    nmax = 90; CoefficientList[Series[x Product[(1 + x^(2^k) - x^(2^(k + 1)) - x^(2^(k + 2))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], a[(n + 1)/2] - a[(n - 1)/2] - a[(n - 3)/2]]; Table[a[n], {n, 0, 90}]

Formula

a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = a(n+1) - a(n) - a(n-1).

A309022 Expansion of x * Product_{k>=0} (1 - x^(2^k) - x^(2^(k+1)) - x^(2^(k+2))).

Original entry on oeis.org

0, 1, -1, -2, 1, -2, 2, 4, -1, -1, 2, 3, -2, 4, -4, -7, 1, -4, 1, 4, -2, 2, -3, -7, 2, 3, -4, -6, 4, -7, 7, 12, -1, 2, 4, 4, -1, 7, -4, -7, 2, 0, -2, -3, 3, -6, 7, 12, -2, 8, -3, -9, 4, -5, 6, 14, -4, -5, 7, 10, -7, 12, -12, -20, 1, -9, -2, 3, -4, -2, -4, -9, 1, 4, -7, -10, 4, -10, 7, 13, -2, 5, 0, -4, 2, -1, 3, 8, -3, -6, 6
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 06 2019

Keywords

Crossrefs

Cf. A002487, A177219, A309019, A309020, A309021.

Programs

  • Mathematica
    nmax = 90; CoefficientList[Series[x Product[(1 - x^(2^k) - x^(2^(k + 1)) - x^(2^(k + 2))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], -a[n/2], a[(n + 1)/2] - a[(n - 1)/2] - a[(n - 3)/2]]; Table[a[n], {n, 0, 90}]

Formula

a(0) = 0, a(1) = 1; a(2*n) = -a(n), a(2*n+1) = a(n+1) - a(n) - a(n-1).
Showing 1-5 of 5 results.

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