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.

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A344902 Number of open tours by a biased rook on a specific f(n) X 1 board, where f(n) = A070941(n) and cells are colored white or black according to the binary representation of 2n.

Original entry on oeis.org

1, 2, 4, 6, 8, 18, 18, 24, 16, 54, 54, 96, 54, 96, 96, 120, 32, 162, 162, 384, 162, 384, 384, 600, 162, 384, 384, 600, 384, 600, 600, 720, 64, 486, 486, 1536, 486, 1536, 1536, 3000, 486, 1536, 1536, 3000, 1536, 3000, 3000, 4320, 486, 1536, 1536, 3000, 1536
Offset: 0

Views

Author

Mikhail Kurkov, Jun 01 2021 [verification needed]

Keywords

Comments

A cell is colored white if the binary digit is 0 and a cell is colored black if the binary digit is 1. A biased rook on a white cell moves to the left to any cell or to the right only to a black cell. A biased rook on a black cell moves in any direction.

Links

Crossrefs

Cf. A000120, A023416, A070941, A073138, A284005, A329369, A329718.

Programs

  • Mathematica
    a[n_] := With[{s = DigitCount[n, 2]}, s[[1]]! * (1 + s[[1]])^(1 + s[[2]])]; a[0] = 1; Array[a, 50, 0] (* Amiram Eldar, Aug 03 2023 *)

Formula

a(n) = A000120(n)!*(1 + A000120(n))^(A023416(n) + 1) for n > 0 with a(0)=1.
a(2n) = (1 + A000120(n))*a(n) for n > 0 with a(0)=1.
From Mikhail Kurkov, Oct 16 2021: (Start)
Conjecture: a(n) = A284005(A073138(n)) for n >= 0 (noticed by Sequence Machine).
Proof: note that A073138(n) in binary is A000120(n) of ones followed by A023416(n) zeros. Then use the formula from "Comments on A284005". (End) [verification needed]

A361480 a(n) is the greatest integer whose binary expansion has the same multiset of run-lengths as that of n.

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 6, 7, 14, 13, 10, 13, 12, 13, 14, 15, 30, 29, 26, 27, 26, 21, 26, 29, 28, 27, 26, 27, 28, 29, 30, 31, 62, 61, 58, 59, 54, 53, 54, 59, 58, 53, 42, 53, 54, 53, 58, 61, 60, 59, 54, 51, 54, 53, 54, 59, 56, 59, 58, 59, 60, 61, 62, 63, 126, 125
Offset: 0

Views

Author

Rémy Sigrist, Mar 13 2023

Keywords

Links

Crossrefs

Cf. A073138, A335835, A361478, A361479.

Programs

  • PARI
    a(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); r=vecsort(r); my (h=0, t=#r+1, v=0); for (k=1, #r, v=(v+k%2)*2^if (k%2, r[t--], r[h++])-k%2;); return (v); }

Formula

a(a(n)) = a(n).
a(n) >= n.

A351988 In the factorial base expansion of n, arrange digits in decreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 8, 9, 14, 15, 12, 14, 14, 15, 16, 17, 18, 20, 20, 21, 22, 23, 24, 30, 30, 32, 54, 56, 30, 32, 32, 33, 56, 57, 54, 56, 56, 57, 62, 63, 78, 80, 80, 81, 86, 87, 48, 54, 54, 56, 60, 62, 54, 56, 56, 57, 62, 63, 60, 62, 62, 63, 64, 65, 84, 86
Offset: 0

Views

Author

Rémy Sigrist, Feb 27 2022

Keywords

Comments

This sequence is to factorial base what A004186 is to decimal base.

Examples

			For n = 1664:
- the factorial base expansion of 1664 is "214110",
- arranging these digits in decreasing order gives "421110",
- so a(1664) = 4*6! + 2*5! + 1*4! + 1*3! + 1*2! + 0*1! = 3152.

Links

Crossrefs

Cf. A004186 (decimal analog), A073138 (binary analog), A108731, A319651 (ternary analog), A351987.

Programs

  • Mathematica
    max = 5; b = MixedRadix[Range[max, 2, -1]]; a[n_] := FromDigits[Sort[IntegerDigits[n, b], Greater], b]; Array[a, max!, 0] (* Amiram Eldar, Feb 28 2022 *)
  • PARI
    a(n) = { my (dd=[]); for (r=2, oo, if (n==0, dd = vecsort(dd); return (sum(k=1, #dd, dd[k]*k!)), dd = concat(dd, n%r); n\=r)) }

Formula

a(a(n)) = a(n).
a(n) >= n with equality iff n belongs to A351987.
Previous Showing 21-23 of 23 results.

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