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 1-8 of 8 results.

A160727 a(n) = A161415(n+1)/4.

Original entry on oeis.org

2, 3, 7, 3, 9, 9, 23, 3, 9, 9, 27, 9, 27, 27, 73, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 227, 3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 243, 9, 27, 27, 81, 27, 81, 81, 243, 27, 81, 81, 243, 81, 243, 243, 697, 3, 9, 9, 27, 9
Offset: 1

Views

Author

Omar E. Pol, Jun 13 2009

Keywords

Examples

			From _Omar E. Pol_, Jan 01 2014: (Start)
Written as an irregular triangle in which row lengths is A000079 the sequence begins:
2;
3,7;
3,9,9,23;
3,9,9,27,9,27,27,73;
3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,227;
3,9,9,27,9,27,27,81,9,27,27,81,27,81,81,243,9,27,27,81,27, 81,81,243,27,81,81,243,81,243,243,697;
(End)

Links

Crossrefs

Cf. A152978, A153000, A160414, A161415.

Programs

  • Mathematica
    A160727[n_]:=3^DigitCount[n,2,1]-If[IntegerQ[Log2[n+1]],(n+1)/2,0];Array[A160727,100] (* Paolo Xausa, Sep 01 2023 *)

Formula

a(n) = A048883(n), except a(n) = A048883(n) - (n+1)/2 if n is a power of 2 minus 1. - Omar E. Pol, Jan 06 2014

Extensions

a(11)-a(58) from M. F. Hasler, Dec 03 2012
a(59)-a(68) from Omar E. Pol, Jan 06 2014

A160414 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (same as A160410, but a(1) = 1, not 4).

Original entry on oeis.org

0, 1, 9, 21, 49, 61, 97, 133, 225, 237, 273, 309, 417, 453, 561, 669, 961, 973, 1009, 1045, 1153, 1189, 1297, 1405, 1729, 1765, 1873, 1981, 2305, 2413, 2737, 3061, 3969, 3981, 4017, 4053, 4161, 4197, 4305, 4413, 4737, 4773, 4881, 4989, 5313, 5421, 5745
Offset: 0

Views

Author

Omar E. Pol, May 20 2009

Keywords

Comments

The structure has a fractal behavior similar to the toothpick sequence A139250.
First differences: A161415, where there is an explicit formula for the n-th term.
For the illustration of a(24) = 1729 (the Hardy-Ramanujan number) see the Links section.

Examples

			From _Omar E. Pol_, Sep 24 2015: (Start)
With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
9;
21,    49;
61,    97,  133,  225;
237,  273,  309,  417,  453, 561,  669,  961;
...
Right border gives A060867.
This triangle T(n,k) shares with the triangle A256530 the terms of the column k, if k is a power of 2, for example both triangles share the following terms: 1, 9, 21, 49, 61, 97, 225, 237, 273, 417, 961, etc.
.
Illustration of initial terms, for n = 1..10:
.       _ _ _ _                       _ _ _ _
.      |  _ _  |                     |  _ _  |
.      | |  _|_|_ _ _ _ _ _ _ _ _ _ _|_|_  | |
.      | |_|  _ _     _ _   _ _     _ _  |_| |
.      |_ _| |  _|_ _|_  | |  _|_ _|_  | |_ _|
.          | |_|  _ _  |_| |_|  _ _  |_| |
.          |   | |  _|_|_ _ _|_|_  | |   |
.          |  _| |_|  _ _   _ _  |_| |_  |
.          | | |_ _| |  _|_|_  | |_ _| | |
.          | |_ _| | |_|  _  |_| | |_ _| |
.          |  _ _  |  _| |_| |_  |  _ _  |
.          | |  _|_| | |_ _ _| | |_|_  | |
.          | |_|  _| |_ _| |_ _| |_  |_| |
.          |   | | |_ _ _ _ _ _ _| | |   |
.          |  _| |_ _| |_   _| |_ _| |_  |
.       _ _| | |_ _ _ _| | | |_ _ _ _| | |_ _
.      |  _| |_ _|   |_ _| |_ _|   |_ _| |_  |
.      | | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
.      | |_ _| |                     | |_ _| |
.      |_ _ _ _|                     |_ _ _ _|
.
After 10 generations there are 273 ON cells, so a(10) = 273.
(End)

Links

Crossrefs

Cf. A001235, A011541, A011782, A000225, A060867, A139250, A147562, A160117, A160118, A160410, A160412, A161415, A160720, A160727, A151725, A256530, A256534.

Programs

  • Maple
    read("transforms") ; 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:
A160414 := proc(n) add( A161415(k),k=1..n) ; end proc: seq(A160414(n),n=0..90) ; # R. J. Mathar, Oct 16 2010
  • Mathematica
    A160414list[nmax_]:=Accumulate[Table[If[n<2,n,4*3^DigitCount[n-1,2,1]-If[IntegerQ[Log2[n]],2n,0]],{n,0,nmax}]];A160414list[100] (* Paolo Xausa, Sep 01 2023, after R. J. Mathar *)
  • PARI
    my(s=-1, t(n)=3^norml2(binary(n-1))-if(n==(1<Altug Alkan, Sep 25 2015

Formula

a(n) = 1 + 4*A219954(n), n >= 1. - M. F. Hasler, Dec 02 2012
a(2^k) = (2^(k+1) - 1)^2. - Omar E. Pol, Jan 05 2013

Extensions

Edited by N. J. A. Sloane, Jun 15 2009 and Jul 13 2009
More terms from R. J. Mathar, Oct 16 2010

A256531 First differences of A256530.

Original entry on oeis.org

0, 1, 8, 12, 28, 12, 36, 60, 68, 12, 36, 60, 84, 108, 132, 156, 148, 12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 308, 12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372, 396, 420, 444, 468, 492, 516, 540, 564, 588, 612, 636, 660, 684, 708, 732, 628, 12, 36, 60, 84, 108
Offset: 0

Views

Author

Omar E. Pol, Apr 21 2015

Keywords

Comments

Number of cells turned ON at n-th stage of cellular automaton of A256530.
Similar to A261695 which shares infinitely many terms.

Examples

			With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
8;
12, 28;
12, 36, 60, 68;
12, 36, 60, 84, 108, 132, 156, 148;
12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 308;
...
The terms of the rows that start with 12 are also the initial terms of A073762, except the last term of every row, hence rows converge to A073762.

Links

Crossrefs

Cf. A011782, A073762, A139251, A147582, A161411, A161415, A256530, A261695.

Programs

  • Mathematica
    With[{z=7},Differences[Join[{0,0},Flatten[Array[(2^#-1)^2+12Range[0,2^(#-1)-1]^2&,z]]]]] (* Generates 2^z terms *) (* Paolo Xausa, Nov 15 2023, after Omar E. Pol *)

A162349 First differences of A160412.

Original entry on oeis.org

3, 9, 9, 27, 9, 27, 27, 81, 9, 27, 27, 81, 27, 81, 81, 243, 9, 27, 27, 81, 27, 81, 81, 243, 27, 81, 81, 243, 81, 243, 243, 729, 9, 27, 27, 81, 27, 81, 81, 243, 27, 81, 81, 243, 81, 243, 243, 729, 27, 81, 81, 243, 81, 243, 243, 729, 81, 243, 243, 729, 243, 729
Offset: 1

Views

Author

Omar E. Pol, Jul 14 2009

Keywords

Comments

Note that if A048883 is written as a triangle then rows converge to this sequence. - Omar E. Pol, Nov 15 2009

Links

Crossrefs

Cf. A063787, A152980, A160410, A160411, A160412, A160414, A160415, A160417, A160797.
Cf. A048883, A161415. - Omar E. Pol, Nov 15 2009

Programs

  • Mathematica
    a[n_] := 3^(1 + DigitCount[n - 1, 2, 1]); Array[a, 100] (* Amiram Eldar, Feb 02 2024 *)

Formula

a(n) = 3^A063787(n) = 3 * A048883(n-1). - Amiram Eldar, Feb 02 2024

Extensions

More terms from Omar E. Pol, Nov 15 2009
More terms from Colin Barker, Apr 19 2015
More terms from Amiram Eldar, Feb 02 2024

A261695 First differences of A256534.

Original entry on oeis.org

0, 4, 12, 12, 36, 12, 36, 60, 84, 12, 36, 60, 84, 108, 132, 156, 180, 12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372, 12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372, 396, 420, 444, 468, 492, 516, 540, 564, 588, 612, 636, 660, 684, 708, 732, 756, 12, 36, 60, 84, 108
Offset: 0

Views

Author

Omar E. Pol, Sep 24 2015

Keywords

Comments

Number of cells turned ON at n-th stage of cellular automaton of A256534.
Similar to A256531 which shares infinitely many terms.

Examples

			With the positive terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
4;
12;
12, 36;
12, 36, 60, 84;
12, 36, 60, 84, 108, 132, 156, 180;
12, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372;
...

Links

Crossrefs

Cf. A011782, A139251, A147582, A161411, A161415, A241717, A256531, A256534.

Formula

It appears that a(n) = 4 * A241717(n-1), n >= 1.

A350633 First differences of A350632.

Original entry on oeis.org

0, 1, 8, 12, 24, 12, 28, 36, 56, 12, 28, 36, 76, 44, 92, 92, 164, 16, 28, 36, 76, 44, 92, 92, 196, 76, 108, 116, 204, 180, 236, 228, 424, 36, 28, 36, 76, 44, 92, 92, 196, 76, 108, 116, 204, 180, 236, 228, 492, 156, 156, 156, 252, 220, 284, 268, 540, 316, 348
Offset: 0

Views

Author

Rémy Sigrist, Jan 08 2022

Keywords

Comments

Equivalently, a(n) is the number of cells turned ON at stage n of the cellular automaton described in A350632.

Examples

			The first 5 generations can be depicted as follows:
         . . . . . . . . . . .
         . 5 5 . . . . . 5 5 .
         . 5 4 4 4 4 4 4 4 5 .
         . . 4 3 3 . 3 3 4 . .
         . . 4 3 2 2 2 3 4 . .
         . . 4 . 2 1 2 . 4 . .
         . . 4 3 2 2 2 3 4 . .
         . . 4 3 3 . 3 3 4 . .
         . 5 4 4 4 4 4 4 4 5 .
         . 5 5 . . . . . 5 5 .
         . . . . . . . . . . .
- so a(0) = 0,
     a(1) = 1,
     a(2) = 8,
     a(3) = 12,
     a(4) = 24,
     a(5) = 12.

Links

Crossrefs

Cf. A161415, A256531, A350632.

A161341 First differences of A161340.

Original entry on oeis.org

1, 26, 56, 260, 56, 392, 392, 2192, 56, 392, 392, 2744, 392, 2744, 2744, 16952, 56, 392, 392, 2744, 392, 2744, 2744, 19208, 392, 2744, 2744, 19208, 2744, 19208, 19208, 125336, 56, 392, 392, 2744, 392, 2744, 2744, 19208, 392, 2744, 2744, 19208, 2744, 19208, 19208
Offset: 1

Views

Author

Omar E. Pol, Jun 14 2009

Keywords

Examples

			From _Omar E. Pol_, Mar 15 2020: (Start)
Written as an irregular triangle in which row lengths give A011782 the sequence begins:
1;
26;
56, 260;
56, 392, 392, 2192;
56, 392, 392, 2744, 392, 2744, 2744, 16952;
56, 392, 392, 2744, 392, 2744, 2744, 19208, 392, 2744, 2744, 19208, 2744, ...
(End)

Crossrefs

Cf. A011782, A160414, A161340, A161415.

Programs

  • PARI
    f(n) = 8*7^hammingweight(n-1); \\ A160429
ispow2(n) = my(k); (n==2) || (ispower(n,,&k) && (k==2));
a(n) = if (n==1, 1, if (ispow2(n), f(n) - 3*n*(3*n - 1), f(n))); \\ Michel Marcus, Mar 15 2020

Formula

a(n) = A160429(n) for n>1 and n not a power of 2.
a(n) = A160429(n) - 3n*(3n - 1) for n>1 and n a power of 2.

Extensions

Formula and more terms from Nathaniel Johnston, Nov 15 2010
More terms from Jinyuan Wang, Mar 14 2020

A161325 Partial sums of A160414.

Original entry on oeis.org

0, 1, 10, 31, 80, 141, 238, 371, 596, 833, 1106, 1415, 1832, 2285, 2846, 3515, 4476, 5449, 6458, 7503, 8656, 9845, 11142, 12547, 14276, 16041, 17914, 19895, 22200, 24613, 27350, 30411, 34380, 38361, 42378, 46431, 50592, 54789, 59094, 63507, 68244, 73017, 77898, 82887, 88200, 93621, 99366
Offset: 1

Views

Author

Omar E. Pol, Jun 14 2009

Keywords

Crossrefs

Cf. A160414, A161415.

Extensions

More terms from Nathaniel Johnston, Nov 14 2010
Showing 1-8 of 8 results.

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