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 1-5 of 5 results.

A257503 Square array A(row,col) read by antidiagonals: A(1,col) = A256450(col-1), and for row > 1, A(row,col) = A255411(A(row-1,col)); Dispersion of factorial base shift A255411 (array transposed).

Original entry on oeis.org

1, 2, 4, 3, 12, 18, 5, 16, 72, 96, 6, 22, 90, 480, 600, 7, 48, 114, 576, 3600, 4320, 8, 52, 360, 696, 4200, 30240, 35280, 9, 60, 378, 2880, 4920, 34560, 282240, 322560, 10, 64, 432, 2976, 25200, 39600, 317520, 2903040, 3265920, 11, 66, 450, 3360, 25800, 241920, 357840, 3225600, 32659200, 36288000, 13, 70, 456, 3456, 28800, 246240, 2540160, 3588480, 35925120, 399168000, 439084800
Offset: 1

Views

Author

Antti Karttunen, Apr 27 2015

Keywords

Comments

The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
The first row (A256450) contains all the numbers which have at least one 1-digit in their factorial base representation (see A007623), after which the successive rows are obtained from the terms on the row immediately above by shifting their factorial representation one left and then incrementing the nonzero digits in that representation with a factorial base shift-operation A255411.

Examples

			The top left corner of the array:
     1,     2,     3,     5,      6,      7,      8,      9,     10,     11,     13
     4,    12,    16,    22,     48,     52,     60,     64,     66,     70,     76
    18,    72,    90,   114,    360,    378,    432,    450,    456,    474,    498
    96,   480,   576,   696,   2880,   2976,   3360,   3456,   3480,   3576,   3696
   600,  3600,  4200,  4920,  25200,  25800,  28800,  29400,  29520,  30120,  30840
  4320, 30240, 34560, 39600, 241920, 246240, 272160, 276480, 277200, 281520, 286560
  ...

Links

Crossrefs

Transpose: A257505.
Inverse permutation: A257504.
Row index: A257679, Column index: A257681.
Row 1: A256450, Row 2: A257692, Row 3: A257693.
Columns 1-3: A001563, A062119, A130744 (without their initial zero-terms).
Column 4: A213167 (without the initial one).
Column 5: A052571 (without initial zeros).
Cf. A007623, A256450, A255411.
Cf. also permutations A255565 and A255566.
Thematically similar arrays: A083412, A135764, A246278.

Programs

Formula

A(1,col) = A256450(col-1), and for row > 1, A(row,col) = A255411(A(row-1,col)).

Extensions

Formula changed because of the changed starting offset of A256450 - Antti Karttunen, May 30 2016

A257505 Square array A(row,col): A(row,1) = A256450(row-1), and for col > 1, A(row,col) = A255411(A(row,col-1)); Dispersion of factorial base shift A255411.

Original entry on oeis.org

1, 4, 2, 18, 12, 3, 96, 72, 16, 5, 600, 480, 90, 22, 6, 4320, 3600, 576, 114, 48, 7, 35280, 30240, 4200, 696, 360, 52, 8, 322560, 282240, 34560, 4920, 2880, 378, 60, 9, 3265920, 2903040, 317520, 39600, 25200, 2976, 432, 64, 10, 36288000, 32659200, 3225600, 357840, 241920, 25800, 3360, 450, 66, 11, 439084800, 399168000, 35925120, 3588480, 2540160, 246240, 28800, 3456, 456, 70, 13
Offset: 1

Views

Author

Antti Karttunen, Apr 27 2015

Keywords

Comments

The array is read by downward antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
In Kimberling's terminology, this array is called the dispersion of sequence A255411 (when started from its first nonzero term, 4). The left column is the complement of that sequence, which is A256450.

Examples

			The top left corner of the array:
   1,   4,  18,   96,   600,   4320,   35280,   322560,   3265920
   2,  12,  72,  480,  3600,  30240,  282240,  2903040,  32659200
   3,  16,  90,  576,  4200,  34560,  317520,  3225600,  35925120
   5,  22, 114,  696,  4920,  39600,  357840,  3588480,  39553920
   6,  48, 360, 2880, 25200, 241920, 2540160, 29030400, 359251200
   7,  52, 378, 2976, 25800, 246240, 2575440, 29352960, 362517120
   8,  60, 432, 3360, 28800, 272160, 2822400, 31933440, 391910400
   9,  64, 450, 3456, 29400, 276480, 2857680, 32256000, 395176320
  10,  66, 456, 3480, 29520, 277200, 2862720, 32296320, 395539200
  11,  70, 474, 3576, 30120, 281520, 2898000, 32618880, 398805120
  13,  76, 498, 3696, 30840, 286560, 2938320, 32981760, 402433920
  14,  84, 552, 4080, 33840, 312480, 3185280, 35562240, 431827200
  15,  88, 570, 4176, 34440, 316800, 3220560, 35884800, 435093120
  17,  94, 594, 4296, 35160, 321840, 3260880, 36247680, 438721920
  19, 100, 618, 4416, 35880, 326880, 3301200, 36610560, 442350720
  20, 108, 672, 4800, 38880, 352800, 3548160, 39191040, 471744000
  21, 112, 690, 4896, 39480, 357120, 3583440, 39513600, 475009920
  23, 118, 714, 5016, 40200, 362160, 3623760, 39876480, 478638720
  ...

Links

Crossrefs

Transpose: A257503.
Inverse permutation: A257506.
Row index: A257681, Column index: A257679.
Columns 1-3: A256450, A257692, A257693.
Rows 1-3: A001563, A062119, A130744 (without their initial zero-terms).
Row 4: A213167 (without the initial one).
Row 5: A052571 (without initial zeros).
Cf. A007623, A256450, A255411.
Cf. also permutations A255565, A255566.
Thematically similar arrays: A035513, A054582, A246279.

Programs

Formula

A(row,1) = A256450(row-1), and for col > 1, A(row,col) = A255411(A(row,col-1)).

Extensions

Formula changed because of the changed starting offset of A256450 - Antti Karttunen, May 30 2016

A273668 Permutation of nonnegative integers: a(0) = 0, a(A255411(n)) = A153880(a(n)), a(A256450(n)) = A273670(a(n)).

Original entry on oeis.org

0, 1, 3, 5, 2, 9, 4, 15, 7, 21, 11, 29, 8, 17, 41, 13, 14, 23, 6, 57, 19, 20, 32, 33, 10, 77, 27, 28, 44, 45, 16, 101, 39, 40, 61, 63, 22, 129, 53, 55, 83, 87, 31, 165, 71, 75, 107, 111, 12, 43, 213, 95, 56, 99, 137, 141, 18, 59, 269, 119, 26, 76, 125, 177, 80, 183, 38, 25, 81, 341, 134, 153, 30, 37, 100, 161, 62, 225, 104, 231, 52, 35
Offset: 0

Views

Author

Antti Karttunen, May 30 2016

Keywords

Links

Crossrefs

Inverse: A273667.
Cf. A153880, A255411, A256450, A257680, A257684, A273662, A273670.
Similar or related permutations: A255565, A273666.

Formula

a(0) = 0; for n >= 1: if A257680(n) = 0 [when n is one of the terms of A255411] then a(n) = A153880(a(A257684(n))), otherwise [when n is one of the terms of A256450], a(n) = A273670(a(A273662(n))).
As a composition of other permutations:
a(n) = A273666(A255565(n)).

A255566 a(0) = 0; after which, a(2n) = A255411(a(n)), a(2n+1) = A256450(a(n)).

Original entry on oeis.org

0, 1, 4, 2, 18, 6, 12, 3, 96, 24, 48, 8, 72, 15, 16, 5, 600, 120, 240, 30, 360, 56, 60, 10, 480, 87, 88, 20, 90, 21, 22, 7, 4320, 720, 1440, 144, 2160, 270, 288, 36, 2880, 416, 420, 67, 432, 73, 66, 13, 3600, 567, 568, 107, 570, 109, 108, 26, 576, 111, 112, 27, 114, 28, 52, 9, 35280, 5040, 10080, 840, 15120, 1584, 1680, 168
Offset: 0

Views

Author

Antti Karttunen, May 05 2015

Keywords

Comments

This sequence can be represented as a binary tree. Each left hand child is produced as A255411(n), and each right hand child as A256450(n), when parent contains n >= 1:
0
|
...................1...................
4 2
18......../ \........6 12......../ \........3
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
96 24 48 8 72 15 16 5
600 120 240 30 360 56 60 10 480 87 88 20 90 21 22 7
etc.
Because all terms of A255411 are even it means that odd terms can occur only in odd positions (together with some even terms, for each one of which there is a separate infinite cycle), while terms in even positions are all even.
After its initial 1, A255567 seems to give all the terms like 2, 3, 12, ... where the left hand child of the right hand child is one more than the right hand child of the left hand child (as for 2: 16 = 15+1, as for 3: 22 = 21+1, as for 12: 88 = 87+1).

Links

Crossrefs

Inverse: A255565.
Cf. A000142, A001511, A001563, A083318, A255411, A256450, A257679.
Cf. also A255567 and arrays A257503, A257505.
Related or similar permutations: A273666, A273667.

Formula

a(0) = 0; after which, a(2n) = A255411(a(n)), a(2n+1) = A256450(a(n)).
Other identities:
For all n >= 0, a(2^n) = A001563(n+1). [The leftmost branch of the binary tree is given by n*n!]
For all n >= 0, a(A083318(n)) = A000142(n+1). [And the next innermost vertices by (n+1)! This follows because A256450(n*n! - 1) = (n+1)! - 1.]
For all n >= 1, A257679(a(n)) = A001511(n).

Extensions

Formula changed because of the changed starting offset of A256450 - Antti Karttunen, May 30 2016

A273665 a(0) = 0; for n >= 1: if n = A153880(k) for some k, then a(n) = 2*a(k), otherwise, n = A273670(h) for some h, and a(n) = 1 + 2*a(h).

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 4, 11, 6, 15, 9, 23, 10, 13, 14, 31, 19, 47, 21, 27, 29, 63, 39, 95, 8, 43, 22, 55, 59, 127, 12, 79, 30, 191, 17, 87, 18, 45, 46, 111, 119, 255, 25, 159, 61, 383, 35, 175, 20, 37, 26, 91, 93, 223, 28, 239, 62, 511, 51, 319, 38, 123, 94, 767, 71, 351, 41, 75, 53, 183, 187, 447, 42, 57, 54, 479, 125, 1023, 58
Offset: 0

Views

Author

Antti Karttunen, May 30 2016

Keywords

Links

Crossrefs

Inverse: A273666.
Cf. A153880, A225901, A257680, A266193, A273663, A273670.
Related or similar permutations: A255565, A273667.

Formula

a(0) = 0; for n >= 1: if A257680(A225901(n)) = 0 [i.e., n is one of the terms of A153880], then a(n) = 2*a(A266193(n)), otherwise [when n is one of the terms of A273670], a(n) = 1 + 2*a(A273663(n)).
Showing 1-5 of 5 results.

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