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 1-6 of 6 results.

A161411 First differences of A160410.

Original entry on oeis.org

4, 12, 12, 36, 12, 36, 36, 108, 12, 36, 36, 108, 36, 108, 108, 324, 12, 36, 36, 108, 36, 108, 108, 324, 36, 108, 108, 324, 108, 324, 324, 972, 12, 36, 36, 108, 36, 108, 108, 324, 36, 108, 108, 324, 108, 324, 324, 972, 36, 108, 108, 324, 108, 324, 324, 972, 108, 324, 324
Offset: 1

Views

Author

Omar E. Pol, May 20 2009, Jun 13 2009, Jun 14 2009

Keywords

Comments

The rows of the triangle in A147582 converge to this sequence.
Contribution from Omar E. Pol, Mar 28 2011 (Start):
a(n) is the number of cells turned "ON" at n-th stage of the cellular automaton of A160410.
a(n) is also the number of toothpicks added at n-th stage to the toothpick structure of A160410.
(End)

Examples

			If written as a triangle:
.4;
.12;
.12,36;
.12,36,36,108;
.12,36,36,108,36,108,108,324;

Links

Crossrefs

Cf. A139250, A139251, A160413, A160415, A160417.
Cf. A000079, A048883, A147582, A160410, A160414.

Programs

  • Mathematica
    4*3^DigitCount[Range[0,100],2,1] (* Paolo Xausa, Sep 01 2023 *)

Formula

a(n) = A048883(n-1)*4.

Extensions

Edited by David Applegate and N. J. A. Sloane, Jul 13 2009

A161415 First differences of A160414.

Original entry on oeis.org

1, 8, 12, 28, 12, 36, 36, 92, 12, 36, 36, 108, 36, 108, 108, 292, 12, 36, 36, 108, 36, 108, 108, 324, 36, 108, 108, 324, 108, 324, 324, 908, 12, 36, 36, 108, 36, 108, 108, 324, 36, 108, 108, 324, 108, 324, 324, 972, 36, 108, 108, 324, 108, 324, 324, 972, 108, 324, 324
Offset: 1

Views

Author

Omar E. Pol, May 20 2009, Jun 13 2009

Keywords

Links

Crossrefs

Cf. A139250, A139251, A160411, A160413, A160414, A160417.
Cf. A160727.
Cf. A048883, A161411, A162349. [From Omar E. Pol, Nov 11 2009]

Programs

  • Maple
    Contribution from R. J. Mathar, Oct 16 2010: (Start)
isA000079 := proc(n) if type(n,'even') then nops(numtheory[factorset](n)) = 1 ; else false ; fi ; end proc:
A048883 := proc(n) 3^wt(n) ; end proc:
A161415 := proc(n) if n = 1 then 1; elif isA000079(n) then 4*A048883(n-1)-2*n ; else 4*A048883(n-1) ; end if; end proc: seq(A161415(n),n=1..90) ; (End)
  • Mathematica
    a[1] = 1; a[n_] := 4*3^DigitCount[n-1, 2, 1] - If[IntegerQ[Log[2, n]], 2n, 0];
Array[a, 60] (* Jean-François Alcover, Nov 17 2017, after N. J. A. Sloane *)

Formula

For n > 1, a(n) = 4*A048883(n-1), except a(n) = 4*A048883(n-1) - 2n if n is a power of 2. - N. J. A. Sloane, Jul 13 2009

Extensions

More terms from R. J. Mathar, Oct 16 2010

A160411 Number of cells turned "ON" at n-th stage of A160117.

Original entry on oeis.org

1, 8, 4, 28, 8, 52, 12, 76, 16, 100, 20, 124, 24, 148, 28, 172, 32, 196, 36, 220, 40, 244, 44, 268, 48, 292, 52, 316, 56, 340, 60, 364, 64, 388, 68, 412, 72, 436, 76, 460, 80, 484, 84, 508, 88, 532, 92, 556, 96
Offset: 1

Views

Author

Omar E. Pol, Jun 13 2009

Keywords

Comments

First differences of A160117.
It appears that one of the bisections is A008574. - Omar E. Pol, Sep 20 2011

Links

Crossrefs

Cf. A139251, A160117, A160411, A160413, A160415.

Programs

  • Mathematica
    LinearRecurrence[{0,2,0,-1},{1,8,4,28,8,52},100] (* Paolo Xausa, Sep 01 2023 *)

Formula

G.f.: x*(x^2+1)*(4*x^3 + x^2 + 8*x + 1) / ((x-1)^2*(x+1)^2). - Colin Barker, Mar 04 2013
From Colin Barker, Apr 06 2013: (Start)
a(n) = -11 - 9*(-1)^n + (7 + 5*(-1)^n)*n for n > 2.
a(n) = 2*a(n-2) - a(n-4) for n > 6. (End)

Extensions

a(10)-a(27) from Omar E. Pol, Mar 26 2011
More terms from Colin Barker, Apr 06 2013

A160415 First differences of A160118.

Original entry on oeis.org

1, 8, 4, 28, 4, 28, 12, 84, 4, 28, 12, 84, 12, 84, 36, 252, 4, 28, 12, 84, 12, 84, 36, 252, 12, 84, 36, 252, 36, 252, 108, 756, 4, 28, 12, 84, 12, 84, 36, 252, 12, 84, 36, 252, 36, 252, 108, 756, 12, 84, 36, 252, 36, 252, 108, 756, 36, 252, 108, 756, 108, 756, 324
Offset: 1

Views

Author

Omar E. Pol, Jun 13 2009

Keywords

Comments

Number of cells turned "ON" at n-th stage of the cellular automaton of A160118.

Examples

			From _Omar E. Pol_, Mar 21 2011: (Start)
If written as a triangle begins:
1,
8,
4,28,
4,28,12,84,
4,28,12,84,12,84,36,252,
4,28,12,84,12,84,36,252,12,84,36,252,36,252,108,756,
(End)

Links

Crossrefs

Cf. A139251, A160118, A160411, A160413, A160417.

Programs

  • Mathematica
    With[{d = 2}, wt[n_] := DigitCount[n, 2, 1]; f[n_] := If[OddQ[n], 3^d + (2^d) * Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 1)/2}] + (2^d)*(3^d - 2) * Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 3)/2}], 3^d + (2^d) * Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}] + (2^d)*(3^d - 2) * Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}]]; f[0] = 0; f[1] = 1; Differences[Array[f, 100, 0]]] (* Amiram Eldar, Feb 02 2024 *)

Extensions

More terms (a(8)-a(38)) from Nathaniel Johnston, Nov 14 2010
21 terms corrected between a(13) and a(38), and more terms (a(39)-a(48)) from Omar E. Pol, Mar 21 2011
More terms from Amiram Eldar, Feb 02 2024

A160417 a(n) = A160415(n+1)/4.

Original entry on oeis.org

2, 1, 7, 1, 7, 3, 21, 1, 7, 3, 21, 3, 21, 9, 63, 1, 7, 3, 21, 3, 21, 9, 63, 3, 21, 9, 63, 9, 63, 27, 189, 1, 7, 3, 21, 3, 21, 9, 63, 3, 21, 9, 63, 9, 63, 27, 189, 3, 21, 9, 63, 9, 63, 27, 189, 9, 63, 27, 189, 27, 189, 81, 567, 1, 7, 3, 21, 3, 21, 9, 63, 3, 21
Offset: 1

Views

Author

Omar E. Pol, Jun 13 2009

Keywords

Crossrefs

Cf. A152978, A160118, A160413, A160415.

Programs

  • Mathematica
    With[{d = 2}, wt[n_] := DigitCount[n, 2, 1]; f[n_] := If[OddQ[n], 3^d + (2^d) * Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 1)/2}] + (2^d)*(3^d - 2) * Sum[(2^d - 1)^(wt[k] - 1), {k, 1, (n - 3)/2}], 3^d + (2^d) * Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}] + (2^d)*(3^d - 2) * Sum[(2^d - 1)^(wt[k] - 1), {k, 1, n/2 - 1}]]; f[0] = 0; f[1] = 1; Differences[Array[f, 100]]/4] (* Amiram Eldar, Feb 02 2024 *)

Extensions

More terms from Max Alekseyev, Dec 12 2011
More terms from Amiram Eldar, Feb 02 2024

A161417 First differences of A160416.

Original entry on oeis.org

1, 7, 3, 21, 7, 41, 9, 57, 13
Offset: 1

Views

Author

Omar E. Pol, May 20 2009, Jun 14 2009

Keywords

Links

Crossrefs

Cf. A139250, A139251, A153006, A152980, A160411, A160413, A160415, A160416.
Cf. A160797.
Showing 1-6 of 6 results.

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