A277821 -id:A277821 - cp's OEIS frontend

A277820 Square array: A(r,1) = A065621(r); for c > 1, A(r,c) = A048724(A(r,c-1)), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

1, 3, 2, 5, 6, 7, 15, 10, 9, 4, 17, 30, 27, 12, 13, 51, 34, 45, 20, 23, 14, 85, 102, 119, 60, 57, 18, 11, 255, 170, 153, 68, 75, 54, 29, 8, 257, 510, 427, 204, 221, 90, 39, 24, 25, 771, 514, 765, 340, 359, 238, 105, 40, 43, 26, 1285, 1542, 1799, 1020, 937, 306, 187, 120, 125, 46, 31, 3855, 2570, 2313, 1028, 1275, 854, 461, 136, 135, 114, 33, 28

Offset: 1

Antti Karttunen, Nov 01 2016

For all n >= 1, A277818 (= A268389(n)+1) gives the (one-based) index of the column where n is located in this array, while A268671(n) gives the (one-based) index of the row where it is on.

This array is obtained when one selects from A277320 the columns 1, 3, 5, 15, 17, 51, ..., i.e., those with an index A001317(k).

			The top left corner of the array:
   1,  3,   5,  15,  17,   51,   85,  255,   257,   771,  1285,  3855
   2,  6,  10,  30,  34,  102,  170,  510,   514,  1542,  2570,  7710
   7,  9,  27,  45, 119,  153,  427,  765,  1799,  2313,  6939, 11565
   4, 12,  20,  60,  68,  204,  340, 1020,  1028,  3084,  5140, 15420
  13, 23,  57,  75, 221,  359,  937, 1275,  3341,  5911, 14649, 19275
  14, 18,  54,  90, 238,  306,  854, 1530,  3598,  4626, 13878, 23130
  11, 29,  39, 105, 187,  461,  599, 1785,  2827,  7453, 10023, 26985
   8, 24,  40, 120, 136,  408,  680, 2040,  2056,  6168, 10280, 30840
  25, 43, 125, 135, 393,  667, 1965, 2295,  6425, 11051, 32125, 34695
  26, 46, 114, 150, 442,  718, 1874, 2550,  6682, 11822, 29298, 38550
  31, 33,  99, 165, 495,  561, 1619, 2805,  7967,  8481, 25443, 42405
  28, 36, 108, 180, 476,  612, 1708, 3060,  7196,  9252, 27756, 46260
  21, 63,  65, 195, 325,  975, 1105, 3315,  5397, 16191, 16705, 50115
  22, 58,  78, 210, 374,  922, 1198, 3570,  5654, 14906, 20046, 53970
  19, 53,  95, 225, 291,  869, 1455, 3825,  4883, 13621, 24415, 57825
  16, 48,  80, 240, 272,  816, 1360, 4080,  4112, 12336, 20560, 61680
  49, 83, 245, 287, 801, 1379, 4005, 4335, 12593, 21331, 62965, 73247
  50, 86, 250, 270, 786, 1334, 3930, 4590, 12850, 22102, 64250, 69390
  55, 89, 235, 317, 839, 1481, 3675, 4845, 14135, 22873, 60395, 80957

Inverse permutation: A277821.
Transpose: A277819.
Cf. A048724, A065621.
Row 1: A001317.
Column 1: A065621, column 2: A277823, column 3: A277825.
Cf. A268389, A277818, A268671.
Cf. also A193231, A277320, A277810, A276618 (A099884), A248513.
Other related tables or permutations: A277880, A277901.

(define (A277820 n) (A277820bi (A002260 n) (A004736 n)))
(define (A277820bi row col) (if (= 1 col) (A065621 row) (A048724 (A277820bi row (- col 1)))))

A(r,1) = A065621(r); for c > 1, A(r,c) = A048724(A(r,c-1)).
A(r,c) = A048675(A277810(r,c)).
As a composition of other permutations:
a(n) = A277901(A277880(n)).

A277902 If A010060(n) = 1, a(n) = A000069(A268671(n)), otherwise a(n) = A001969(1+a(A006068(n)/2)).

Original entry on oeis.org

1, 2, 3, 7, 6, 5, 4, 14, 9, 10, 13, 15, 8, 11, 12, 31, 24, 23, 28, 30, 25, 26, 17, 29, 16, 19, 18, 22, 27, 20, 21, 62, 43, 40, 61, 45, 56, 59, 54, 58, 49, 50, 33, 55, 36, 39, 52, 63, 32, 35, 48, 38, 57, 46, 37, 47, 34, 53, 44, 60, 41, 42, 51, 127, 102, 85, 124, 120, 121, 122, 83, 95, 112, 115, 68, 118, 89, 106

Offset: 1

Antti Karttunen, Nov 03 2016

a(n) gives the number that is in the same position in array A277880 as where n is located in array A277820.

			The top left corner of array A277820 is:
   1,  3,  5, 15
   2,  6, 10, 30
   7,  9, 27, 45
   4, 12, 20, 60
  13, 23, 57, 75
while the top left corner of A277880 is:
   1,  3,  6, 12
   2,  5, 10, 20
   4,  9, 18, 36
   7, 15, 30, 60
   8, 17, 34, 68
thus for example, a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 7, a(5) = 6, a(6) = 5, a(7) = 4, a(9) = 9, a(12) = 15, a(13) = 8 and a(27) = 18.

Inverse: A277901.
Cf. A000069, A001969, A001317, A003945, A006068, A010060, A065621, A129771, A268671, A277823, A277825.
Related permutations and arrays: A277820, A277821, A277880.

If A010060(n) = 1 [when n is one of the odious numbers, A000069], then a(n) = A000069(A268671(n)), otherwise a(n) = A001969(1+a(A006068(n)/2)).
As a composition of other permutations:
a(n) = A277880(A277821(n)).
Other identities. For all n >= 1:
A010060(a(n)) = A010060(n). [Preserves the parity of binary weight.]
a(A001317(n)) = A003945(n).
a(A065621(n)) = A000069(n).
a(A277823(n)) = A129771(n).
a(A277825(n)) = 2*A129771(n).

cp's OEIS Frontend

A277820 Square array: A(r,1) = A065621(r); for c > 1, A(r,c) = A048724(A(r,c-1)), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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