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 1-10 of 10 results.

A030497 Position of n-th 1 in A030496.

Original entry on oeis.org

1, 3, 4, 8, 10, 14, 16, 18, 21, 27, 30, 35, 38, 41, 45, 51, 54, 59, 62, 65, 69, 75, 78, 83, 86, 89, 93, 96, 100, 104, 108, 112, 116, 124, 128, 134, 138, 142, 147, 154, 158, 164, 168, 172, 177, 184, 188, 194, 198, 202, 207, 211, 216, 221, 226, 231, 236, 244
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A030496.

Extensions

a(34) onward corrected by Sean A. Irvine, Apr 02 2020

A030498 Position of n-th 2 in A030496.

Original entry on oeis.org

2, 5, 7, 11, 12, 15, 20, 22, 26, 31, 33, 37, 43, 46, 50, 55, 57, 61, 67, 70, 72, 76, 80, 84, 88, 92, 99, 102, 107, 110, 113, 117, 123, 129, 132, 137, 144, 148, 153, 159, 162, 167, 174, 178, 181, 186, 191, 196, 201, 206, 214, 218, 224, 228, 232, 237, 243, 249
Offset: 1

Views

Author

Clark Kimberling

Keywords

Extensions

a(33) onward corrected by Sean A. Irvine, Apr 02 2020

A030499 Position of n-th 3 in A030496.

Original entry on oeis.org

6, 9, 13, 17, 19, 23, 25, 29, 34, 39, 42, 47, 48, 52, 56, 60, 64, 68, 74, 79, 81, 85, 91, 94, 98, 103, 105, 109, 115, 118, 122, 127, 133, 139, 143, 149, 151, 156, 161, 166, 171, 176, 183, 189, 192, 197, 204, 208, 213, 219, 222, 227, 234, 238, 242, 247, 253
Offset: 1

Views

Author

Clark Kimberling

Keywords

Extensions

a(31) onward corrected by Sean A. Irvine, Apr 02 2020

A030500 Position of n-th 4 in A030496.

Original entry on oeis.org

24, 28, 32, 36, 40, 44, 49, 53, 58, 63, 66, 71, 73, 77, 82, 87, 90, 95, 97, 101, 106, 111, 114, 119, 121, 126, 131, 136, 141, 146, 152, 157, 163, 169, 173, 179, 182, 187, 193, 199, 203, 209, 212, 217, 223, 229, 233, 239, 240, 245, 250, 255, 260, 265, 270, 275
Offset: 1

Views

Author

Clark Kimberling

Keywords

Extensions

a(25) onward corrected by Sean A. Irvine, Apr 02 2020

A030501 Position of n-th 5 in A030496.

Original entry on oeis.org

120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 241, 246, 251, 256, 261, 266, 272, 277, 283, 289, 293, 299, 302, 307, 313, 319, 323, 329, 332, 337, 343, 349, 353, 359, 361, 366, 371, 376
Offset: 1

Views

Author

Clark Kimberling

Keywords

Extensions

a(25) onward corrected by Sean A. Irvine, Apr 02 2020

A030502 n-th partial sum of A030496.

Original entry on oeis.org

1, 3, 4, 5, 7, 10, 12, 13, 16, 17, 19, 21, 24, 25, 27, 28, 31, 32, 35, 37, 38, 40, 43, 47, 50, 52, 53, 57, 60, 61, 63, 67, 69, 72, 73, 77, 79, 80, 83, 87, 88, 91, 93, 97, 98, 100, 103, 106, 110, 112, 113, 116, 120, 121, 123, 126, 128, 132
Offset: 1

Views

Author

Clark Kimberling

Keywords

A030298 List of permutations of 1,2,3,...,n for n=1,2,3,..., in lexicographic order.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 3, 1, 3, 2, 2, 1, 3, 2, 3, 1, 3, 1, 2, 3, 2, 1, 1, 2, 3, 4, 1, 2, 4, 3, 1, 3, 2, 4, 1, 3, 4, 2, 1, 4, 2, 3, 1, 4, 3, 2, 2, 1, 3, 4, 2, 1, 4, 3, 2, 3, 1, 4, 2, 3, 4, 1, 2, 4, 1, 3, 2, 4, 3, 1, 3, 1, 2, 4, 3, 1, 4, 2, 3, 2, 1, 4, 3, 2, 4, 1, 3, 4, 1, 2, 3, 4, 2, 1, 4, 1, 2, 3, 4, 1, 3, 2, 4, 2
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

Contains every finite sequence of distinct numbers, infinitely many times.

Examples

			The permutations can be written as
  1,
  12, 21,
  123, 132, 213, 231, 312, 321, etc.
Write them in order and insert commas.

Links

Crossrefs

A030299 gives the initial portion of these same permutations as decimally encoded numbers.
Cf. A098280, A098281, A030299, A170942.
Cf. A001563 (row lengths), A001286 (row sums).
Cf. A030496 for another ordering.
The same information is essentially given in A055089, but in more compact way, by skipping those permutations which start with a fixed element (cf. A220696).
A220660(n) tells the zero-based rank r of the n-th permutation in this sequence, among all finite permutations of the same size.
A220663(n) tells the zero-based position (from the left) of that a(n) in that permutation of rank r.
A084557(n) tells that the n-th term a(n) belongs to the a(n):th lexicographically ordered permutation from the start (its "global rank").
A220660(A084557(n)) tells the "local rank" of the permutation (amongst the permutations of the same size) to which the n-th term a(n) belongs.
(A130664(n),A084555(n)) = (1,1),(2,3),(4,5),(6,8),(9,11),(12,14),... gives the starting and ending offsets of the n-th permutation in this list.

Programs

  • Haskell
    import Data.List (permutations, sort)
a030298 n k = a030298_tabf !! (n-1) (k-1)
a030298_row = concat . sort . permutations . enumFromTo 1
a030298_tabf = map a030298_row [1..]
-- Reinhard Zumkeller, Mar 29 2012
(MIT/GNU Scheme, with Antti Karttunen's intseq-library):
;; Note that in Scheme, vector indexing is zero-based.
;; Requires also A055089permvec from A055089.
(define (A030298 n) (vector-ref (A030298permvec (A084556 (A084557 n)) (A220660 (A084557 n))) (A220663 n)))
(define (A030298permvec size rank) (vector-reverse (vector1invert (A055089permvec size rank))))
(define (vector1invert vec) (make-initialized-vector (vector-length vec) (lambda (i) (1+ (- (vector-length vec) (vector-ref vec i))))))
(define (vector-reverse vec) (make-initialized-vector (vector-length vec) (lambda (i) (vector-ref vec (- (vector-length vec) i 1)))))
  • Mathematica
    f[n_] := Permutations[Range@ n, {n}]; Array[f, 4] // Flatten (* Robert G. Wilson v, Dec 18 2012 *)
  • Python
    from itertools import permutations, count, chain, islice
def A030298_gen(): # generator of terms
    return chain.from_iterable(p for l in count(2) for p in permutations(range(1,l)))
A030298_list = list(islice(A030298_gen(),30)) # Chai Wah Wu, Mar 21 2022

Formula

Start with 1, then 12 and 21, then the 6 permutations of 123 in lexical order: 123, 132, 213, 231, 312, 321 and so on.

Extensions

Entry revised by N. J. A. Sloane, Feb 02 2006
Keyword tabf added by Reinhard Zumkeller, Mar 29 2012

A084556 n occurs n! times.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 02 2003

Keywords

Comments

Also minimum i such that A007489(i) >= n.
For n>=1, a(n) gives the length of the n-th permutation in the sequences like A030298 & A030496.

Links

Crossrefs

First differences of A084555. Used to compute A084557. Differs from A084506 first time at the 130th term, where A084506(130) = 6, while A084556(130) = 5. Cf. also A002024, A072643, A072649, A090529.

Programs

  • Mathematica
    Flatten[ Table[#, {#!}] & /@ Range[0, 5]]

A084555 Partial sums of A084556.

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 14, 17, 20, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99, 103, 107, 111, 115, 119, 124, 129, 134, 139, 144, 149, 154, 159, 164, 169, 174, 179, 184, 189, 194, 199, 204, 209, 214, 219, 224, 229, 234, 239, 244, 249
Offset: 0

Views

Author

Antti Karttunen, Jun 02 2003

Keywords

Comments

For n>=1, (A130664(n),a(n)) = (1,1),(2,3),(4,5),(6,8),(9,11),(12,14),... gives the starting and ending offsets of the n-th permutation in the sequences like A030298 & A030496.
Note: this sequence is related to (ordinary) permutations. For a similar sequence related to juggling with three objects, see A084505.

Links

Crossrefs

Differs from A084505 first time at the 130th term, where A084505(130) = 605, while A084555(130) = 604.

Programs

  • Mathematica
    Accumulate@ Flatten[ Table[#, {#!}] & /@ Range[0, 5]]

Formula

a(0)=0; for n >= 1, a(n) = a(n-1) + A084556(n). Also a(n) = A130664(n+1) - 1. - Antti Karttunen, Dec 18 2012

Extensions

Moved the misplaced Mathematica code from A084505. - Antti Karttunen, Oct 24 2012

A130664 a(1)=1. a(n) = a(n-1) + (number of terms from among a(1) through a(n-1) which are factorials).

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 15, 18, 21, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250
Offset: 1

Views

Author

Leroy Quet, Jun 21 2007

Keywords

Comments

Also this is an irregular array where row n contains the n! consecutive multiples of n starting with n!.
For n >= 1, (a(n), A084555(n)) = (1,1), (2,3), (4,5), (6,8), (9,11), (12,14), ... gives the starting and ending offsets of the n-th permutation in the sequences like A030298 and A030496. Gives also the fixed points of A220662; we have A220662(a(n)) = a(n). - Antti Karttunen, Dec 18 2012

Examples

			When interpreted as an irregular table, the rows begin as:
  1;
  2, 4;
  6, 9, 12, 15, 18, 21;

Links

Crossrefs

Cf. A000142, A084555, A220662.

Programs

  • Maple
    A[1]:= 1:
nextf:= 2!:
m:= 1:
for n from 2 to 100 do
  A[n]:= A[n-1]+m;
  if A[n] = nextf then
     m:= m+1;
      nextf:= (m+1)!;
  fi;
od:
seq(A[i],i=1..100); # Robert Israel, Apr 28 2016
  • Mathematica
    Table[Range[n!, (n + 1)! - 1, n], {n, 5}] // Flatten (* Michael De Vlieger, Aug 29 2017 *)
  • Scheme
    (define (A130664 n) (+ 1 (A084555(- n 1))))

Formula

a(n) = A084555(n-1) + 1.
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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