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.

A154473 a(n) = A014486(A154472(n)).

Original entry on oeis.org

842, 11090, 13202, 46882, 60994, 231272004, 198873570440, 266349291297936, 64442911458703648, 3667589230123774528, 3336154829743802737792, 17601566387699271821281536, 1023499990310357893964861952
Offset: 0

Views

Author

Antti Karttunen, with terms a(0)-a(100) also independently computed by Wouter Meeussen, with the given Mathematica program, Jan 11 2009

Keywords

Comments

This sequence gives the parenthesis expressions shown at the upper right corner image of the page 103 of NKS, with the left brackets (black squares) converted to 1's and the right brackets (white squares) converted to 0's and then interpreting each such number as a binary number and converted to decimal. A154474 shows the corresponding binary representations. Compare to A080070, A122242, A122245.

Links

Crossrefs

Cf. A154475, A154476.

Programs

  • Mathematica
    init=e[e[e][e]][e][e]
toDeca[ w_ ]:=FromDigits[ ToExpression[ Characters[ ToString[ w ] ]/.{"e"->Sequence[], "["->"1","]"->"0"} ],2 ]
toDeca /@ NestList[ #/.e[x_ ][y_ ]->x[x[y]]&, init, 100]

A154471 Function A154470 iterated, starting from the initial value 31706.

Original entry on oeis.org

31706, 4517553, 4875253, 59657666, 65204804, 467824043836025, 289931140991491544956, 232993060651625904999520564, 12090001045837621170309278896817, 41831072194327417802054794318226030
Offset: 0

Views

Author

Antti Karttunen, Jan 11 2009

Keywords

Comments

Note how A014486(31706) = 2988236 and (A014486->parenthesization 2988236) = (() (() (()) (())) (()) (())), from which, when after converting ()'s to e's we get: (e (e (e) (e)) (e) (e)), corresponding to the initial state e[e[e][e]][e][e] of Wolfram's system. A154472 gives the corresponding sequence with ()'s removed.

Links

Crossrefs

Cf. A014486, A154470, A154472.

Programs

Formula

a(0) = 31706, a(n) = A154470(a(n-1)).

A154475 Number of opening (equally: closing) brackets in each term of Wolfram's Symbolic Rewriting system A154473-A154474.

Original entry on oeis.org

5, 7, 7, 8, 8, 14, 19, 24, 28, 31, 36, 42, 45, 47, 49, 50, 50, 50, 51, 51, 51, 54, 55, 55, 55, 56, 56, 56, 58, 60, 61, 61, 61, 62, 62, 62, 65, 66, 66, 66, 67, 67, 67, 70, 72, 74, 75, 75, 75, 76, 76, 76, 79, 80, 80, 80, 81, 81, 81, 83, 85, 86, 86, 86, 87, 87, 87, 92, 93, 93
Offset: 0

Views

Author

Antti Karttunen, Jan 11 2009

Keywords

Comments

2*a(n) gives the number of bits in A154474(n).

Examples

			The iteration starts from the initial term e[e[e][e]][e][e], which contains 5 ['s (and also 5 ]'s), thus a(0)=5.

Links

Crossrefs

a(n) = A029837(1+A154473(n))/2. a(n) = A154476(n)-1.

Formula

a(n) = A072643(A154472(n)).

A154476 Number of e's in each iteration of Wolfram's e[x_][y_] -> x[x[y]] symbolic rewriting system, starting from the initial state e[e[e][e]][e][e].

Original entry on oeis.org

6, 8, 8, 9, 9, 15, 20, 25, 29, 32, 37, 43, 46, 48, 50, 51, 51, 51, 52, 52, 52, 55, 56, 56, 56, 57, 57, 57, 59, 61, 62, 62, 62, 63, 63, 63, 66, 67, 67, 67, 68, 68, 68, 71, 73, 75, 76, 76, 76, 77, 77, 77, 80, 81, 81, 81, 82, 82, 82, 84, 86, 87, 87, 87, 88, 88, 88, 93, 94, 94
Offset: 0

Views

Author

Antti Karttunen, Jan 11 2009

Keywords

Examples

			The iteration starts from the initial term e[e[e][e]][e][e], which contains 6 e's, thus a(0)=6.

Links

Crossrefs

a(n) = A154475(n)+1.

Formula

a(n) = A072643(A154471(n)) - A072643(A154472(n)).
Showing 1-4 of 4 results.

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