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 1-6 of 6 results.

A256997 Square array A(row,col) read by antidiagonals: A(1,col) = A055938(col), and for row > 1, A(row,col) = A005187(A(row-1,col)).

Original entry on oeis.org

2, 5, 3, 6, 8, 4, 9, 10, 15, 7, 12, 16, 18, 26, 11, 13, 22, 31, 34, 49, 19, 14, 23, 41, 57, 66, 95, 35, 17, 25, 42, 79, 110, 130, 184, 67, 20, 32, 47, 81, 153, 215, 258, 364, 131, 21, 38, 63, 89, 159, 302, 424, 514, 723, 259, 24, 39, 73, 120, 174, 312, 599, 844, 1026, 1440, 515, 27, 46, 74, 143, 236, 343, 620, 1192, 1683, 2050, 2876, 1027
Offset: 2

Views

Author

Antti Karttunen, Apr 14 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.
This is transpose of array A256995.
If we assume that a(1) = 1 (but which is not explicitly included here because outside of the array proper), then A256998 gives the inverse permutation.

Examples

			The top left corner of the array:
    2,    5,    6,    9,   12,   13,   14,   17,   20,   21,    24,    27
    3,    8,   10,   16,   22,   23,   25,   32,   38,   39,    46,    50
    4,   15,   18,   31,   41,   42,   47,   63,   73,   74,    88,    97
    7,   26,   34,   57,   79,   81,   89,  120,  143,  145,   173,   191
   11,   49,   66,  110,  153,  159,  174,  236,  281,  287,   341,   375
   19,   95,  130,  215,  302,  312,  343,  467,  558,  568,   677,   743
   35,  184,  258,  424,  599,  620,  680,  928, 1111, 1132,  1349,  1479
   67,  364,  514,  844, 1192, 1235, 1356, 1852, 2216, 2259,  2693,  2951
  131,  723, 1026, 1683, 2380, 2464, 2707, 3697, 4428, 4512,  5381,  5895
  259, 1440, 2050, 3360, 4755, 4924, 5408, 7387, 8851, 9020, 10757, 11783
  ...

Links

Crossrefs

Cf. A005187, A055938 (row 1), A256994 (column 1), A256989 (row index), A256990 (column index).
Inverse: A256998.
Transpose: A256995.
Cf. also A254107, A255557 (variants), A246278 (another thematically similar construction).

Programs

Formula

A(1,col) = A055938(col), and for row > 1, A(row,col) = A005187(A(row-1,col)).

A256989 One-based column index of n in array A256995.

Original entry on oeis.org

0, 1, 2, 3, 1, 1, 4, 2, 1, 2, 5, 1, 1, 1, 3, 2, 1, 3, 6, 1, 1, 2, 2, 1, 2, 4, 1, 1, 1, 1, 3, 2, 1, 4, 7, 1, 1, 2, 2, 1, 3, 3, 1, 1, 1, 2, 3, 1, 5, 2, 1, 1, 2, 2, 1, 2, 4, 1, 1, 1, 1, 1, 3, 2, 1, 5, 8, 1, 1, 2, 2, 1, 3, 3, 1, 1, 1, 2, 4, 1, 4, 2, 1, 1, 2, 2, 1, 3, 4, 1, 1, 1, 1, 2, 6, 1, 3, 2, 1, 1, 2, 3, 1, 3, 2, 1, 1, 1, 3, 5, 1, 2, 2, 1, 1, 2, 2, 1, 2, 4
Offset: 1

Views

Author

Antti Karttunen, Apr 14 2015

Keywords

Comments

Also one-based row index for array A256997.
a(1) = 0 by convention, as 1 is outside of the actual arrays A256995 & A256997.

Links

Crossrefs

Cf. A005187, A055938, A213714, A256995, A256997.
Cf. A256990 (corresponding row index), A255559.

Formula

a(1) = 0; for n > 1, if A213714(n) = 0 [i.e., if n is one of the terms of A055938], then a(n) = 1, otherwise a(n) = 1 + a(A213714(n)).
In other words, a(1) = 0, and for n > 1, if n = A005187(k) for some k, then a(n) = 1 + a(k), otherwise it must be that n is in A055938, in which case a(n) = 1.
Other observations. For all n >= 1 it holds that:
a(n) <= A256993(n).

A256995 Square array A(row,col) read by antidiagonals: A(row,1) = A055938(row), and for col > 1, A(row,col) = A005187(A(row,col-1)).

Original entry on oeis.org

2, 3, 5, 4, 8, 6, 7, 15, 10, 9, 11, 26, 18, 16, 12, 19, 49, 34, 31, 22, 13, 35, 95, 66, 57, 41, 23, 14, 67, 184, 130, 110, 79, 42, 25, 17, 131, 364, 258, 215, 153, 81, 47, 32, 20, 259, 723, 514, 424, 302, 159, 89, 63, 38, 21, 515, 1440, 1026, 844, 599, 312, 174, 120, 73, 39, 24, 1027, 2876, 2050, 1683, 1192, 620, 343, 236, 143, 74, 46, 27
Offset: 2

Views

Author

Antti Karttunen, Apr 14 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.
This is transpose of array A256997.
If we assume that a(1) = 1 (but which is not explicitly included here because outside of the array proper), then A256996 gives the inverse permutation.

Examples

			The top left corner of the array:
   2,  3,  4,   7,  11,  19,   35,   67,  131,  259,   515,  1027
   5,  8, 15,  26,  49,  95,  184,  364,  723, 1440,  2876,  5745
   6, 10, 18,  34,  66, 130,  258,  514, 1026, 2050,  4098,  8194
   9, 16, 31,  57, 110, 215,  424,  844, 1683, 3360,  6716, 13425
  12, 22, 41,  79, 153, 302,  599, 1192, 2380, 4755,  9504, 19004
  13, 23, 42,  81, 159, 312,  620, 1235, 2464, 4924,  9841, 19675
  14, 25, 47,  89, 174, 343,  680, 1356, 2707, 5408, 10812, 21617
  17, 32, 63, 120, 236, 467,  928, 1852, 3697, 7387, 14765, 29521
  20, 38, 73, 143, 281, 558, 1111, 2216, 4428, 8851, 17696, 35388
  21, 39, 74, 145, 287, 568, 1132, 2259, 4512, 9020, 18033, 36059
  ...

Links

Crossrefs

Inverse permutation: A256996.
Transpose: A256997.
Cf. A005187, A055938 (column 1), A256994 (row 1), A256989 (column index), A256990 (row index).
Cf. also A254105, A255555 (variants), A114537, A246279 (other thematically similar constructions).

Programs

Formula

A(row,1) = A055938(row), and for col > 1, A(row,col) = A005187(A(row,col-1)).

A255560 One-based row index of n in array A255555.

Original entry on oeis.org

1, 2, 1, 2, 3, 4, 1, 2, 5, 3, 4, 6, 7, 8, 1, 2, 9, 5, 3, 10, 11, 4, 6, 12, 7, 8, 13, 14, 15, 16, 1, 2, 17, 9, 5, 18, 19, 3, 10, 20, 11, 4, 21, 22, 23, 6, 12, 24, 7, 8, 25, 26, 13, 14, 27, 15, 16, 28, 29, 30, 31, 32, 1, 2, 33, 17, 9, 34, 35, 5, 18, 36, 19, 3, 37, 38, 39, 10, 20, 40, 11, 4, 41, 42, 21, 22, 43, 23, 6
Offset: 1

Views

Author

Antti Karttunen, Apr 14 2015

Keywords

Comments

Equally: One-based column index of n in array A255557.

Links

Crossrefs

Cf. A005187, A055938, A213714, A234017, A255555, A255557.
Cf. also A255559 (corresponding column index).
Cf. A254112, A256990.

Formula

a(1) = 1; for n > 1, if A213714(n) = 0 [i.e., if n is one of the terms of A055938], then a(n) = 1+A234017(n), otherwise a(n) = a(A213714(n)-1).
In other words, a(1) = 1, and for n > 1, if n = A055938(k) for some k, then a(n) = k+1, otherwise it must be that n = A005187(h) for some h, in which case a(n) = a(h-1).

A256998 Inverse to A256997 considered as a permutation of natural numbers, with the assumed fixed term a(1) = 1.

Original entry on oeis.org

1, 2, 4, 7, 3, 5, 11, 6, 8, 9, 16, 12, 17, 23, 10, 13, 30, 14, 22, 38, 47, 18, 24, 57, 31, 15, 68, 80, 93, 107, 19, 39, 122, 20, 29, 138, 155, 48, 58, 173, 25, 32, 192, 212, 233, 69, 40, 255, 21, 81, 278, 302, 94, 108, 327, 123, 26, 353, 380, 408, 437, 467, 49, 139, 498, 27, 37, 530, 563, 156, 174, 597, 59, 70, 632, 668, 705, 193, 33, 743, 41
Offset: 1

Views

Author

Antti Karttunen, Apr 14 2015

Keywords

Links

Crossrefs

Inverse: A256997.
Cf. A256989, A256990.
Cf. also A256996, A255558.

Programs

  • Scheme
    (define (A256998 n) (if (= 1 n) n (let ((row (A256989 n)) (col (A256990 n))) (+ 1 (* (/ 1 2) (- (expt (+ row col) 2) row col col col -2))))))

Formula

a(1) = 1, and for n > 1: a(n) = (1/2) * ((c+r)^2 - r - 3*c + 4), where r = A256989(n), and c = A256990(n).

A256996 Inverse to A256995 considered as a permutation of natural numbers, with assumed fixed initial term a(1) = 1.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 8, 6, 11, 10, 12, 16, 22, 29, 9, 15, 37, 14, 17, 46, 56, 21, 28, 67, 36, 13, 79, 92, 106, 121, 20, 45, 137, 19, 23, 154, 172, 55, 66, 191, 27, 35, 211, 232, 254, 78, 44, 277, 18, 91, 301, 326, 105, 120, 352, 136, 26, 379, 407, 436, 466, 497, 54, 153, 529, 25, 30, 562, 596, 171, 190, 631, 65, 77, 667, 704, 742, 210, 34, 781, 43
Offset: 1

Views

Author

Antti Karttunen, Apr 14 2015

Keywords

Links

Crossrefs

Inverse: A256995.
Cf. A256989, A256990.
Cf. also A256998, A255556.

Programs

  • Scheme
    (define (A256996 n) (if (= 1 n) n (let ((col (A256989 n)) (row (A256990 n))) (+ 1 (* (/ 1 2) (- (expt (+ row col) 2) row col col col -2))))))

Formula

a(1) = 1, and for n > 1: a(n) = (1/2) * ((c+r)^2 - r - 3*c + 4), where c = A256989(n), and r = A256990(n).
Showing 1-6 of 6 results.

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