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 1-7 of 7 results.

A277696 Permutation of natural numbers: a(1) = 1; a(2n) = A277817(1+a(n)), a(2n+1) = A277816(a(n)).

Original entry on oeis.org

1, 2, 5, 3, 10, 7, 13, 4, 39, 15, 26, 9, 11, 18, 29, 6, 20, 92, 75, 24, 27, 49, 58, 14, 21, 16, 19, 31, 42, 62, 41, 8, 78, 33, 52, 270, 172, 196, 147, 47, 312, 56, 51, 126, 101, 143, 82, 23, 22, 34, 45, 28, 80, 32, 35, 64, 59, 96, 90, 153, 118, 95, 61, 12, 40, 224, 150, 66, 57, 129, 116, 1134, 534, 606, 316, 752, 404, 520, 207, 120, 55, 1400, 600
Offset: 1

Views

Author

Antti Karttunen, Nov 06 2016

Keywords

Comments

This sequence can be represented as a binary tree. Each left hand child is produced as A277817(1+n), and each right hand child as A277816(n), when the parent node contains n:
1
................../ \..................
2 5
3......../ \........10 7......../ \........13
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
4 39 15 26 9 11 18 29
6 20 92 75 24 27 49 58 14 21 16 19 31 42 62 41
etc.

Links

Crossrefs

Inverse: A277695.
Cf. A277710, A277816, A277817.
Cf. A277701 (the rightmost edge of the tree).

Formula

a(1) = 1; and then after, a(2n) = A277817(1+a(n)), a(2n+1) = A277816(a(n)).

A277710 Square array A(r,c), where each row r lists all numbers k for which A264977(k) = r, read by downwards antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

1, 5, 2, 13, 10, 3, 29, 26, 39, 4, 41, 58, 75, 20, 9, 61, 82, 147, 52, 21, 6, 85, 122, 207, 116, 45, 78, 7, 125, 170, 291, 164, 93, 150, 11, 8, 173, 250, 411, 244, 189, 294, 19, 40, 81, 209, 346, 579, 340, 381, 414, 35, 104, 105, 18, 253, 418, 819, 500, 657, 582, 67, 232, 165, 42, 23, 281, 506, 927, 692, 765, 822, 131, 328, 213, 90, 43, 12
Offset: 1

Views

Author

Antti Karttunen, Oct 29 2016

Keywords

Comments

Alternative description: Each row r lists the positions of A019565(r) in A277330.
Odd terms occur only on rows with odd index, and even terms only on rows with even index. Specifically: all terms k on row r are equal to r modulo 4, thus the first differences of each row are all multiples of 4.
All the terms on any particular row are either all multiples of two (or respectively: three, or six), or none of them are.

Examples

			The top left 12 x 12 corner of the array:
   1,   5,  13,  29,  41,   61,   85,  125,  173,  209,  253,  281
   2,  10,  26,  58,  82,  122,  170,  250,  346,  418,  506,  562
   3,  39,  75, 147, 207,  291,  411,  579,  819,  927, 1155, 1635
   4,  20,  52, 116, 164,  244,  340,  500,  692,  836, 1012, 1124
   9,  21,  45,  93, 189,  381,  657,  765,  873, 1317, 1533, 1749
   6,  78, 150, 294, 414,  582,  822, 1158, 1638, 1854, 2310, 3270
   7,  11,  19,  35,  67,  131,  259,  311,  359,  515,  619,  655
   8,  40, 104, 232, 328,  488,  680, 1000, 1384, 1672, 2024, 2248
  81, 105, 165, 213, 333,  429,  669,  861, 1341, 1725, 2685, 2721
  18,  42,  90, 186, 378,  762, 1314, 1530, 1746, 2634, 3066, 3498
  23,  43,  79,  83, 103,  155,  163,  203,  307,  323,  403,  611
  12, 156, 300, 588, 828, 1164, 1644, 2316, 3276, 3708, 4620, 6540

Links

Crossrefs

Transpose: A277709.
Column 1: A277711, sorted into ascending order: A277817.
Row 1: A277701, Row 2: A277712 (= 2*A277701), Row 3: A277713, Row 4: 4*A277701, Row 5: A277715, Row 6: 2*A277713. Row 8: 8*A277701, Row 10: 2*A277715.
Cf. A277824 (the index of the column where n is located in this array).
Cf. A019565, A264977, A277330, A277816 and permutation pair A277695 & A277696.

Formula

A(r,1) = A277711(r); for c > 1, A(r,c) = A277816(A(r,c-1)).
Other identities. For all r>=1, c>=1:
A(2*r,c) = 2*A(r,c).
A(r,c) modulo 4 = r modulo 4.

Extensions

The dispersion-style formula added by Antti Karttunen, Nov 06 2016

A277695 Permutation of natural numbers: a(1) = 1; for n > 1, if A277815(n) = 0, a(n) = 2*a(A277814(n)-1), otherwise a(n) = 1 + 2*a(A277815(n)).

Original entry on oeis.org

1, 2, 4, 8, 3, 16, 6, 32, 12, 5, 13, 64, 7, 24, 10, 26, 128, 14, 27, 17, 25, 49, 48, 20, 257, 11, 21, 52, 15, 256, 28, 54, 34, 50, 55, 98, 515, 99, 9, 65, 31, 29, 97, 105, 51, 96, 40, 514, 22, 101, 43, 35, 1031, 513, 81, 42, 69, 23, 57, 104, 63, 30, 512, 56, 108, 68, 111, 100, 139, 199, 163, 110, 196, 203, 19, 211, 2063, 33, 195, 53, 1030, 47
Offset: 1

Views

Author

Antti Karttunen, Nov 06 2016

Keywords

Links

Crossrefs

Inverse: A277696.
Cf. A264977, A277710, A277814, A277815, A277816, A277817.

Formula

a(1) = 1; for n > 1, if A277815(n) = 0 [when n is in A277817], a(n) = 2*a(A277814(n)-1), otherwise a(n) = 1 + 2*a(A277815(n)).
a(1) = 1, a(A277817(1+n)) = 2*a(n), a(A277816(n)) = 1 + 2*a(n). [Implicit form.]

A277815 a(n) = the largest k < n for which A264977(k) = A264977(n), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 7, 0, 5, 0, 0, 0, 0, 0, 11, 4, 9, 14, 0, 0, 17, 10, 15, 0, 13, 0, 0, 0, 0, 0, 19, 0, 25, 22, 3, 8, 29, 18, 23, 28, 21, 0, 0, 0, 0, 34, 27, 20, 37, 30, 47, 0, 33, 26, 31, 0, 41, 0, 0, 0, 0, 0, 35, 0, 57, 38, 55, 0, 0, 50, 39, 44, 53, 6, 43, 16, 0, 58, 79, 36, 61, 46, 0, 56, 73, 42, 71, 0, 45, 0, 0, 0, 0, 0, 51
Offset: 0

Views

Author

Antti Karttunen, Nov 06 2016

Keywords

Links

Crossrefs

Cf. A264977, A277695, A277814, A277817 (the positions of zeros).
Left inverse of A277816.

Programs

  • Scheme
    (define (A277815 n) (if (zero? n) n (let ((v (A264977 n))) (let loop ((k (- n 1))) (cond ((zero? k) 0) ((= v (A264977 k)) k) (else (loop (- k 1))))))))

Formula

For all n >= 0, a(A277816(n)) = n.

A277814 a(n) = number of zeros in A277815[0..n].

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 5, 6, 7, 8, 8, 8, 9, 9, 10, 11, 12, 13, 14, 14, 14, 14, 14, 15, 16, 16, 16, 16, 17, 17, 18, 19, 20, 21, 22, 22, 23, 23, 23, 23, 23, 23, 23, 23, 23, 23, 24, 25, 26, 27, 27, 27, 27, 27, 27, 27, 28, 28, 28, 28, 29, 29, 30, 31, 32, 33, 34, 34, 35, 35, 35, 35, 36, 37, 37, 37, 37, 37, 37, 37, 37, 38, 38, 38, 38, 38, 38, 39
Offset: 0

Views

Author

Antti Karttunen, Nov 06 2016

Keywords

Links

Crossrefs

Cf. A000007, A277815.
Left inverse of A277817.

Programs

A277824 a(0) = 0; for n >= 1, a(n) = 1 + a(A277815(n)); index of the column where n is located in array A277710.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 1, 1, 1, 3, 2, 2, 2, 1, 1, 2, 3, 2, 1, 4, 1, 1, 1, 1, 1, 4, 1, 3, 3, 2, 2, 5, 2, 2, 2, 3, 1, 1, 1, 1, 2, 3, 3, 4, 2, 2, 1, 2, 4, 2, 1, 6, 1, 1, 1, 1, 1, 5, 1, 3, 4, 3, 1, 1, 3, 3, 3, 5, 2, 3, 2, 1, 5, 4, 2, 7, 2, 1, 2, 2, 3, 4, 1, 4, 1, 1, 1, 1, 1, 4, 2, 2, 3, 5, 3, 2, 4, 5, 2, 6, 2, 1, 1, 2, 2, 3, 4, 4, 2, 2, 1
Offset: 0

Views

Author

Antti Karttunen, Nov 06 2016

Keywords

Comments

For n >= 1, a(n) = index of the column where n is located in array A277710.
Ordinal transform of A277826 and A277884.

Links

Crossrefs

Cf. A277815, A277710.
Cf. A277817 (gives the positions of ones in this sequence).
Cf. A277826, A277884 and their scatter-plots.

Programs

Formula

a(0) = 0; for n >= 1, a(n) = 1 + a(A277815(n)).

A277884 a(n) = A277814(A277826(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 5, 6, 7, 8, 2, 6, 9, 1, 10, 11, 12, 13, 14, 6, 4, 8, 10, 15, 16, 13, 2, 11, 17, 1, 18, 19, 20, 21, 22, 6, 23, 13, 10, 3, 7, 1, 14, 15, 17, 8, 24, 25, 26, 27, 22, 11, 4, 13, 18, 25, 28, 21, 2, 19, 29, 1, 30, 31, 32, 33, 34, 6, 35, 21, 10, 25, 36, 37, 22, 3, 17, 13, 5, 15, 12, 38, 2, 15, 23, 1, 24, 39, 28, 37, 14, 25, 40, 8, 41
Offset: 0

Views

Author

Antti Karttunen, Nov 06 2016

Keywords

Links

Crossrefs

Cf. A264977, A277814.
Left inverse of A277817.
Cf. A277701, A277712, A277713, A277715 (positions of 1, 2, 3 and 8 in this sequence).
Cf. A277824, A277826 and their scatter-plots.

Programs

Formula

a(n) = A277814(A277826(n)).
Other identities. For all n >= 0:
a(A277817(n)) = n.
Showing 1-7 of 7 results.

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