A256998 -id:A256998 - cp's OEIS frontend

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)).

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

Antti Karttunen, Apr 14 2015

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.

			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
  ...

Antti Karttunen, Table of n, a(n) for n = 2..10441; the first 144 antidiagonals of square array

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).

(define (A256997 n) (if (<= n 1) n (A256997bi (A002260 (- n 1)) (A004736 (- n 1)))))
(define (A256997bi row col) (if (= 1 row) (A055938 col) (A005187 (A256997bi (- row 1) col))))

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

A279336 Permutation of natural numbers: a(1) = 1; for n > 1, if A079559(n) = 0, a(n) = 2*A234016(n), otherwise a(n) = A003961(a(A213714(n))).

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 7, 9, 8, 15, 11, 10, 12, 14, 25, 27, 16, 35, 13, 18, 20, 21, 45, 22, 33, 49, 24, 26, 28, 30, 125, 81, 32, 77, 17, 34, 36, 75, 63, 38, 55, 175, 40, 42, 44, 39, 65, 46, 121, 135, 48, 50, 51, 99, 52, 105, 343, 54, 56, 58, 60, 62, 625, 243, 64, 143, 19, 66, 68, 57, 225, 70, 245, 275, 72, 74, 76, 69, 91, 78, 539, 189, 80

Offset: 1

Antti Karttunen, Dec 10 2016

nonn

For n > 1, a(n) = the number which is in the same position of array A246278 where n is located in array A256997.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A279337.
Cf. A003961, A005187, A079559, A213714, A234016, A246278, A256997, A256998.
Cf. also A278501, A279338 (a variant).

(definec (A279336 n) (cond ((= 1 n) n) ((zero? (A079559 n)) (* 2 (A234016 n))) (else (A003961 (A279336 (A213714 n))))))

a(1) = 1, for n > 1, if A079559(n) = 0 [when n is a term of A055938], a(n) = 2*A234016(n), otherwise a(n) = A003961(a(A213714(n))).
Other identities:
For all n >= 2, a(n) = A246278(A256998(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

Antti Karttunen, Apr 14 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

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

(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))))))

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).

cp's OEIS Frontend

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)).

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A279336 Permutation of natural numbers: a(1) = 1; for n > 1, if A079559(n) = 0, a(n) = 2*A234016(n), otherwise a(n) = A003961(a(A213714(n))).

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

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