A025581 -id:A025581 - cp's OEIS frontend

A075300 Array A read by antidiagonals upwards: A(n, k) = array A054582(n,k) - 1 = 2^n(2k+1) - 1 with n,k >= 0.

0, 1, 2, 3, 5, 4, 7, 11, 9, 6, 15, 23, 19, 13, 8, 31, 47, 39, 27, 17, 10, 63, 95, 79, 55, 35, 21, 12, 127, 191, 159, 111, 71, 43, 25, 14, 255, 383, 319, 223, 143, 87, 51, 29, 16, 511, 767, 639, 447, 287, 175, 103, 59, 33, 18, 1023, 1535, 1279, 895, 575, 351, 207, 119

Offset: 0

Antti Karttunen, Sep 12 2002

From Philippe Deléham, Feb 19 2014: (Start)
A(0,k)  = 2*k = A005843(k),
A(1,k)  = 4*k + 1 = A016813(k),
A(2,k)  = 8*k + 3 = A017101(k),
A(n,0)  = A000225(n),
A(n,1)  = A153893(n),
A(n,2)  = A153894(n),
A(n,3)  = A086224(n),
A(n,4)  = A052996(n+2),
A(n,5)  = A086225(n),
A(n,6)  = A198274(n),
A(n,7)  = A238087(n),
A(n,8)  = A198275(n),
A(n,9)  = A198276(n),
A(n,10) = A171389(n). (End)
A permutation of the nonnegative integers. - Alzhekeyev Ascar M, Jun 05 2016
The values in array row n, when expressed in binary, have n trailing 1-bits. - Ruud H.G. van Tol, Mar 18 2025

			The array A begins:
   0    2    4    6    8   10   12   14   16   18 ...
   1    5    9   13   17   21   25   29   33   37 ...
   3   11   19   27   35   43   51   59   67   75 ...
   7   23   39   55   71   87  103  119  135  151 ...
  15   47   79  111  143  175  207  239  271  303 ...
  31   95  159  223  287  351  415  479  543  607 ...
  ... - _Philippe Deléham_, Feb 19 2014
From _Wolfdieter Lang_, Jan 31 2019: (Start)
The triangle T begins:
   n\k   0    1    2   3   4   5   6   7  8  9 10 ...
   0:    0
   1:    1    2
   2:    3    5    4
   3:    7   11    9   6
   4:   15   23   19  13   8
   5    31   47   39  27  17  10
   6:   63   95   79  55  35  21  12
   7:  127  191  159 111  71  43  25  14
   8:  255  383  319 223 143  87  51  29 16
   9:  511  767  639 447 287 175 103  59 33 18
  10: 1023 1535 1279 895 575 351 207 119 67 37 20
  ...
T(3, 1) = 2^2*(2*1+1) - 1 = 12 - 1 = 11.  (End)

Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A075301. Transpose: A075302. The X-projection is given by A007814(n+1) and the Y-projection A025480.

Cf. A002262, A025581, A054582, A241957.

A075300bi := (x,y) -> (2^x * (2*y + 1))-1;
A075300 := n -> A075300bi(A025581(n), A002262(n));
A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);
A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1);

Table[(2^# (2 k + 1)) - 1 &[m - k], {m, 0, 10}, {k, 0, m}] (* Michael De Vlieger, Jun 05 2016 *)

From Wolfdieter Lang, Jan 31 2019: (Start)
Array A(n, k) = 2^n*(2*k+1) - 1, for n >= 0 and m >= 0.
The triangle is T(n, k) = A(n-k, k) = 2^(n-k)*(2*k+1) - 1, n >= 0, k=0..n.
See also A054582 after subtracting 1. (End)
From Ruud H.G. van Tol, Mar 17 2025: (Start)
A(0, k) is even. For n > 0, A(n, k) is odd and (3 * A(n, k) + 1) / 2 = A(n-1, 3*k+1).
A(n, k) = 2^n - 1 (mod 2^(n+1)) (equivalent to the comment about trailing 1-bits). (End)

A214604 Odd numbers by transposing the right half of A176271, triangle read by rows: T(n,k) = A176271(n - 1 + k, n), 1 <= k <= n.

Original entry on oeis.org

1, 5, 9, 11, 17, 25, 19, 27, 37, 49, 29, 39, 51, 65, 81, 41, 53, 67, 83, 101, 121, 55, 69, 85, 103, 123, 145, 169, 71, 87, 105, 125, 147, 171, 197, 225, 89, 107, 127, 149, 173, 199, 227, 257, 289, 109, 129, 151, 175, 201, 229, 259, 291, 325, 361, 131, 153, 177, 203, 231, 261, 293, 327, 363, 401, 441

Offset: 1

Reinhard Zumkeller, Jul 25 2012

			.     Take the first n elements of the n-th diagonal (northeast to
.     southwest) of the triangle on the left side
.     and write this as n-th row on the triangle of the right side.
. 1:                1                    1
. 2:              _   5                  5  9
. 3:            _   9  11               11 17 25
. 4:         __  __  17  19             19 27 37 49
. 5:       __  __  25  27  29           29 39 51 65 ..
. 6:     __  __  __  37  39  41         41 53 67 .. .. ..
. 7:   __  __  __  49  51  53  55       55 69 .. .. .. .. ..
. 8: __  __  __  __  65  67  69  71     71 .. .. .. .. .. .. .. .

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

Cf. A000290, A016754, A028387, A025581, A094727, A176271.

Cf. A214659 (row sums), A214660 (main diagonal), A214661.

import Data.List (transpose)
a214604 n k = a214604_tabl !! (n-1) !! (k-1)
a214604_row n = a214604_tabl !! (n-1)
a214604_tabl = zipWith take [1..] $ transpose a176271_tabl

[(n+k)^2-n-3*k+1: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 10 2024

Table[(n+k)^2-n-3*k+1, {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 10 2024 *)

flatten([[(n+k)^2-n-3*k+1 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 10 2024

T(n,k) = (n+k)^2 - n - 3*k + 1.
Sum_{k=1..n} T(n, k) = A214659(n).
T(2*n-1, n) = A214660(n) (main diagonal).
T(n, 1) = A028387(n-1).
T(n, n) = A016754(n-1).
T(n, k) = A214661(n,k) + 2*A025581(n,k).
T(n, k) = 2*A000290(A094727(n,k)) - A214661(n,k).

A214661 Odd numbers obtained by transposing the left half of A176271 into rows of a triangle: T(n,k) = A176271(n - 1 + k, k), 1 <= k <= n.

Original entry on oeis.org

1, 3, 9, 7, 15, 25, 13, 23, 35, 49, 21, 33, 47, 63, 81, 31, 45, 61, 79, 99, 121, 43, 59, 77, 97, 119, 143, 169, 57, 75, 95, 117, 141, 167, 195, 225, 73, 93, 115, 139, 165, 193, 223, 255, 289, 91, 113, 137, 163, 191, 221, 253, 287, 323, 361, 111, 135, 161, 189, 219, 251, 285, 321, 359, 399, 441

Offset: 1

Reinhard Zumkeller, Jul 25 2012

			.     Take the first n elements of the n-th diagonal (northwest to
.     southeast) of the triangle on the left side
.     and write this as n-th row on the triangle of the right side.
. 1:                1                    1
. 2:              3   _                  3  9
. 3:            7   9  __                7 15 25
. 4:         13  15  __  __             13 23 35 49
. 5:       21  23  25  __  __           21 33 47 63 ..
. 6:     31  33  35  __  __  __         31 45 61 .. .. ..
. 7:   43  45  47  49  __  __  __       43 59 .. .. .. .. ..
. 8: 57  59  61  63  __  __  __  __     57 .. .. .. .. .. .. .. .

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

Cf. A000290, A002061, A016754, A025581, A094727, A176271, A214604.

Cf. A051673 (row sums), A214675 (main diagonal).

import Data.List (transpose)
a214661 n k = a214661_tabl !! (n-1) !! (k-1)
a214661_row n = a214661_tabl !! (n-1)
a214661_tabl = zipWith take [1..] $ transpose $ map reverse a176271_tabl

[(n+k)^2-3*n-k+1: k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 10 2024

Table[(n+k)^2-3*n-k+1, {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 10 2024 *)

flatten([[(n+k)^2-3*n-k+1 for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 10 2024

T(n, k) = (n+k)^2 - 3*n - k + 1.
T(n,k) = A176271(n+k-1, k).
T(n, k) = A214604(n,k) - 2*A025581(n,k).
T(n, k) = 2*A000290(A094727(n,k)) - A214604(n,k).
T(2*n-1, n) = A214675() (main diagonal).
T(n,1) = A002061(n).
T(n,n) = A016754(n-1).
Sum_{k=1..n} T(n, k) = A051673(n) (row sums).

A227189 Square array A(n>=0,k>=0) where A(n,k) gives the (k+1)-th part of the unordered partition which has been encoded in the binary expansion of n, as explained in A227183. The array is scanned antidiagonally as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), etc.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2

Offset: 0

Antti Karttunen, Jul 06 2013

Discarding the trailing zero terms, on each row n there is a unique partition of integer A227183(n). All possible partitions of finite natural numbers eventually occur. The first partition that sums to n occurs at row A227368(n).

Irregular table A227739 lists only the nonzero terms.

			The top-left corner of the array:
row #  row starts as
    0  0, 0, 0, 0, 0, ...
    1  1, 0, 0, 0, 0, ...
    2  1, 1, 0, 0, 0, ...
    3  2, 0, 0, 0, 0, ...
    4  2, 2, 0, 0, 0, ...
    5  1, 1, 1, 0, 0, ...
    6  1, 2, 0, 0, 0, ...
    7  3, 0, 0, 0, 0, ...
    8  3, 3, 0, 0, 0, ...
    9  1, 2, 2, 0, 0, ...
   10  1, 1, 1, 1, 0, ...
   11  2, 2, 2, 0, 0, ...
   12  2, 3, 0, 0, 0, ...
   13  1, 1, 2, 0, 0, ...
   14  1, 3, 0, 0, 0, ...
   15  4, 0, 0, 0, 0, ...
   16  4, 4, 0, 0, 0, ...
   17  1, 3, 3, 0, 0, ...
etc.
8 has binary expansion "1000", whose runlengths are [3,1] (the length of the run in the least significant end comes first) which maps to nonordered partition {3+3} as explained in A227183, thus row 8 begins as 3, 3, 0, 0, ...
17 has binary expansion "10001", whose runlengths are [1,3,1] which maps to nonordered partition {1,3,3}, thus row 17 begins as 1, 3, 3, ...

Antti Karttunen, The first 141 antidiagonals of the table, flattened
Index entries for sequences related to partitions

Only nonzero terms: A227739. Row sums: A227183. The product of nonzero terms on row n>0 is A227184(n). Number of nonzero terms on each row: A005811. The leftmost column, after n>0: A136480. The rightmost nonzero term: A227185.

Cf. A227368 and also arrays A227186 and A227188.

(define (A227189 n) (A227189bi (A002262 n) (A025581 n)))
(define (A227189bi n k) (cond ((< (A005811 n) (+ 1 k)) 0) ((zero? k) (A136480 n)) (else (+ (- (A136480 n) 1) (A227189bi (A163575 n) (- k 1))))))

A268830 Square array A(r,c): A(0,c) = c, A(r,0) = 0, A(r>=1,c>=1) = 1+A(r-1,A268718(c)-1) = 1 + A(r-1, A003188(A006068(c)-1)), read by descending antidiagonals.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 4, 2, 3, 1, 0, 5, 6, 2, 3, 1, 0, 6, 8, 9, 2, 3, 1, 0, 7, 3, 8, 9, 2, 3, 1, 0, 8, 7, 5, 5, 6, 2, 3, 1, 0, 9, 10, 4, 4, 7, 8, 2, 3, 1, 0, 10, 12, 13, 6, 4, 6, 7, 2, 3, 1, 0, 11, 15, 12, 13, 5, 4, 6, 7, 2, 3, 1, 0, 12, 11, 17, 17, 18, 5, 4, 6, 7, 2, 3, 1, 0, 13, 5, 16, 16, 19, 20, 5, 4, 6, 7, 2, 3, 1, 0, 14, 13, 7, 18, 16, 18, 19, 5, 4, 6, 7, 2, 3, 1, 0

Offset: 0

Antti Karttunen, Feb 14 2016

			The top left [0 .. 16] x [0 .. 19] section of the array:
0, 1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19
0, 1, 4, 2, 6, 8, 3, 7, 10, 12, 15, 11,  5, 13, 16, 14, 18, 20, 23, 19
0, 1, 3, 2, 9, 8, 5, 4, 13, 12, 17, 16,  7,  6, 15, 14, 21, 20, 25, 24
0, 1, 3, 2, 9, 5, 4, 6, 13, 17, 16, 18, 10,  8, 15,  7, 21, 25, 24, 26
0, 1, 3, 2, 6, 7, 4, 5, 18, 19, 16, 17, 10, 11,  8,  9, 26, 27, 24, 25
0, 1, 3, 2, 8, 6, 4, 5, 20, 18,  9, 17,  7, 11, 10, 12, 28, 26, 33, 25
0, 1, 3, 2, 7, 6, 4, 5, 19, 18, 11, 10,  9,  8, 13, 12, 27, 26, 35, 34
0, 1, 3, 2, 7, 6, 4, 5, 19, 11, 14, 12,  8, 10, 13,  9, 27, 35, 38, 36
0, 1, 3, 2, 7, 6, 4, 5, 12, 13, 14, 15,  8,  9, 10, 11, 36, 37, 38, 39
0, 1, 3, 2, 7, 6, 4, 5, 14, 16, 11, 15,  8,  9, 12, 10, 38, 40, 35, 39
0, 1, 3, 2, 7, 6, 4, 5, 17, 16, 13, 12,  8,  9, 11, 10, 41, 40, 37, 36
0, 1, 3, 2, 7, 6, 4, 5, 17, 13, 12, 14,  8,  9, 11, 10, 41, 37, 36, 38
0, 1, 3, 2, 7, 6, 4, 5, 14, 15, 12, 13,  8,  9, 11, 10, 38, 39, 36, 37
0, 1, 3, 2, 7, 6, 4, 5, 16, 14, 12, 13,  8,  9, 11, 10, 40, 38, 21, 37
0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13,  8,  9, 11, 10, 39, 38, 23, 22
0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13,  8,  9, 11, 10, 39, 23, 26, 24
0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13,  8,  9, 11, 10, 24, 25, 26, 27

Antti Karttunen, Table of n, a(n) for n = 0..32895; the first 256 antidiagonals of array

Cf. A003188, A006068.
Inverses of these permutations can be found in table A268820.
Row 0: A001477, Row 1: A268718, Row 2: A268822, Row 3: A268824, Row 4: A268826, Row 5: A268828, Row 6: A268832, Row 7: A268934.
Rows converge towards A006068.

def a003188(n): return n^(n>>1)
def a006068(n):
    s=1
    while True:
        ns=n>>s
        if ns==0: break
        n=n^ns
        s<<=1
    return n
def a278618(n): return 0 if n==0 else 1 + a003188(a006068(n) - 1)
def A(r, c): return c if r==0 else 0 if c==0 else 1 + A(r - 1, a278618(c) - 1)
for r in range(21): print([A(c, r - c) for c in range(r + 1)]) # Indranil Ghosh, Jun 07 2017

(define (A268830 n) (A268830bi (A002262 n) (A025581 n))) ;; o=0: Square array of shifted powers of A268718.
(define (A268830bi row col) (cond ((zero? row) col) ((zero? col) 0) (else (+ 1 (A268830bi (- row 1) (- (A268718 col) 1))))))
(define (A268830bi row col) (cond ((zero? row) col) ((zero? col) 0) (else (+ 1 (A268830bi (- row 1) (A003188 (+ -1 (A006068 col))))))))

A032531 An inventory sequence: triangle read by rows, where T(n, k), 0 <= k <= n, records the number of k's thus far in the flattened sequence.

Original entry on oeis.org

0, 1, 1, 1, 3, 0, 2, 3, 1, 2, 2, 4, 3, 3, 1, 2, 5, 4, 4, 3, 1, 2, 6, 5, 5, 3, 3, 1, 2, 7, 6, 7, 3, 3, 2, 2, 2, 7, 9, 9, 3, 3, 2, 3, 0, 3, 7, 10, 13, 3, 3, 2, 4, 0, 2, 4, 7, 12, 15, 5, 4, 2, 5, 0, 2, 1, 5, 8, 14, 15, 6, 6, 4, 5, 1, 2, 1, 0, 6, 10, 15, 15, 7, 7, 5, 7, 1, 2, 2, 0, 1, 7, 12, 17

Offset: 0

Dmitri Papichev (Dmitri.Papichev(AT)iname.com)

Old name: a(n) = number of a(i) for 0<=iA002262(n).
This sequence is a variation of the Inventory sequence A342585. The same rules apply except that in this variation each row ends after k terms, where k is the current row count which starts at 1. The behavior up to the first 1 million terms is similar to A342585 but beyond that the most common terms do not increase, likely due to the rows being cut off after k terms thus numbers such as 1 and 2 no longer make regular appearances. Larger number terms do increase and overtake the leading early terms, and it appears this pattern repeats as n increases. See the linked images. - Scott R. Shannon, Sep 13 2021
The complexity of this sequence derives from the totals being updated during the calculation of each row. If each row recorded an inventory of only the earlier rows, we would get the much simpler A025581. - Peter Munn, May 06 2023

			Triangle begins:
  0;
  1, 1;
  1, 3, 0;
  2, 3, 1, 2;
  2, 4, 3, 3, 1;
  2, 5, 4, 4, 3, 1;
  2, 6, 5, 5, 3, 3, 1;
  2, 7, 6, 7, 3, 3, 2, 2;
  2, 7, 9, 9, 3, 3, 2, 3, 0;
  ...

Scott R. Shannon, Table of n, a(n) for n = 0..20000.
Scott R. Shannon, Image of the first 10^6 terms.
Scott R. Shannon, Image of the first 10^7 terms.
Scott R. Shannon, Image of the first 10^8 terms.
Index entries for sequences related to inventory sequences

Cf. A002262, A025581, A342585.

A002262 := proc(n)
    n - binomial(floor(1/2+sqrt(2*(1+n))), 2);
end proc:
A032531 := proc(n)
    option remember;
    local a,piv,i ;
    a := 0 ;
    piv := A002262(n) ;
    for i from 0 to n-1  do
        if procname(i) = piv then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A032531(n),n=0..100) ; # R. J. Mathar, May 08 2020

A002262[n_] :=  n - Binomial[Floor[1/2 + Sqrt[2*(1 + n)]], 2];
A032531[n_] := A032531[n] = Module[{a, piv, i}, a = 0; piv = A002262[n]; For[i = 0, i <= n-1, i++, If[A032531[i] == piv, a++]]; a];
Table[A032531[n], {n, 0, 100}] (* Jean-François Alcover, Mar 25 2024, after R. J. Mathar *)

from math import comb, isqrt
from collections import Counter
def idx(n): return n - comb((1+isqrt(8+8*n))//2, 2)
def aupton(nn):
    num, alst, inventory = 0, [0], Counter([0])
    for n in range(1, nn+1):
        c = inventory[idx(n)]
        alst.append(c)
        inventory[c] += 1
    return alst
print(aupton(93)) # Michael S. Branicky, May 07 2023

New name from Peter Munn, May 06 2023

A071654 Inverse permutation to A071653.

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 5, 7, 19, 15, 4, 22, 16, 52, 14, 13, 20, 60, 43, 51, 41, 11, 18, 53, 178, 42, 153, 39, 10, 21, 47, 155, 177, 125, 151, 38, 12, 61, 56, 136, 154, 555, 123, 150, 40, 33, 55, 179, 164, 135, 479, 553, 122, 152, 117, 29, 17, 159, 557, 163, 417, 477, 552, 124

Offset: 0

Antti Karttunen, May 30 2002

A014137(n-1) = A071654(A072638(n)) for n>0 - Antti Karttunen, Jul 30 2012, based on Paul D. Hanna's similar observation in A071653.

Antti Karttunen, Rows n=0..66 of triangle, flattened
A. Karttunen, Alternative Catalan Orderings
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A071653. A071672 gives the corresponding parenthesizations (from the term 1 onward) encoded as binary numbers, i.e. A071672(n) = A063171(A071654(n)) for n >= 1.

a(n) = A057163(A071652(n)). Cf. also A014486, A002262, A025581, A071651, A071652.

A072734 Simple triangle-stretching N X N -> N bijection, variant of A072732.

Original entry on oeis.org

0, 1, 2, 3, 12, 4, 7, 17, 18, 5, 6, 23, 40, 24, 8, 11, 31, 49, 50, 25, 9, 10, 30, 59, 84, 60, 32, 13, 16, 39, 71, 97, 98, 61, 33, 14, 15, 38, 70, 111, 144, 112, 72, 41, 19, 22, 48, 83, 127, 161, 162, 113, 73, 42, 20, 21, 47, 82, 126, 179, 220, 180, 128, 85, 51, 26, 29, 58

Offset: 0

Antti Karttunen, Jun 12 2002

Index entries for sequences that are permutations of the natural numbers

Inverse: A072735, projections: A072740 & A072741, variant of the same theme: A072732. Used to construct the global arithmetic ranking scheme of plane binary trees presented in A072787/A072788. Cf. also A001477 and its projections A025581 & A002262.

(define (A072734 n) (packA072734 (A025581 n) (A002262 n)))
(define (packA001477 x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))
(define (packA072734 x y) (let ((x-y (- x y))) (cond ((negative? x-y) (packA001477 (+ (* 2 x) (modulo (1+ x-y) 2)) (+ (* 2 x) (floor->exact (/ (+ (- x-y) (modulo x-y 2)) 2))))) ((< x-y 3) (packA001477 (+ (* 2 y) x-y) (* 2 y))) (else (packA001477 (+ (* 2 y) (floor->exact (/ (1+ x-y) 2)) (modulo (1+ x-y) 2)) (+ (* 2 y) (modulo x-y 2)))))))

A163233 Two-dimensional Binary Reflected Gray Code: a(i,j) = bits of binary expansion of A003188(i) interleaved with that of A003188(j).

Original entry on oeis.org

0, 1, 2, 5, 3, 10, 4, 7, 11, 8, 20, 6, 15, 9, 40, 21, 22, 14, 13, 41, 42, 17, 23, 30, 12, 45, 43, 34, 16, 19, 31, 28, 44, 47, 35, 32, 80, 18, 27, 29, 60, 46, 39, 33, 160, 81, 82, 26, 25, 61, 62, 38, 37, 161, 162, 85, 83, 90, 24, 57, 63, 54, 36, 165, 163, 170, 84, 87, 91

Offset: 0

Antti Karttunen, Jul 29 2009

The top left 8 X 8 corner of the array is
+0 +1 +5 +4 20 21 17 16
+2 +3 +7 +6 22 23 19 18
10 11 15 14 30 31 27 26
+8 +9 13 12 28 29 25 24
40 41 45 44 60 61 57 56
42 43 47 46 62 63 59 58
34 35 39 38 54 55 51 50
32 33 37 36 52 53 49 48
By taking the top left 2 X 2 corner, 2 X 4 rectangle ((0,1,5,4),(2,3,7,6)) or 4 X 4 corner one obtains Karnaugh map templates for 2, 3 or 4 variables respectively (although not the standard ones usually given in the textbooks).

Antti Karttunen, Table of n, a(n) for n = 0..2079
Wikipedia, Karnaugh map
Index entries for sequences that are permutations of the natural numbers

Inverse: A163234. a(n) = A057300(A163235(n)). Transpose: A163235. Row sums: A163242. Cf. A054238, A147995.

Table[Function[k, FromDigits[#, 2] &@ Apply[Function[{a, b}, Riffle @@ Map[PadLeft[#, Max[Length /@ {a, b}]] &, {a, b}]], Map[IntegerDigits[#, 2] &@ BitXor[#, Floor[#/2]] &, {k, j}]]][i - j], {i, 0, 11}, {j, i, 0, -1}] // Flatten (* Michael De Vlieger, Jun 25 2017 *)

def a000695(n):
    n=bin(n)[2:]
    x=len(n)
    return sum([int(n[i])*4**(x - 1 - i) for i in range(x)])
def a003188(n): return n^(n>>1)
def a(n, k): return a000695(a003188(n)) + 2*a000695(a003188(k))
for n in range(21): print([a(n - k, k) for k in range(n + 1)]) # Indranil Ghosh, Jun 25 2017

(define (A163233bi x y) (+ (A000695 (A003188 x)) (* 2 (A000695 (A003188 y)))))
(define (A163233 n) (A163233bi (A025581 n) (A002262 n)))

a(x,y) = A000695(A003188(x)) + 2*A000695(A003188(y)).

A163328 Square array A, where entry A(y,x) has the ternary digits of x interleaved with the ternary digits of y, converted back to decimal. Listed by antidiagonals: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

Original entry on oeis.org

0, 1, 3, 2, 4, 6, 9, 5, 7, 27, 10, 12, 8, 28, 30, 11, 13, 15, 29, 31, 33, 18, 14, 16, 36, 32, 34, 54, 19, 21, 17, 37, 39, 35, 55, 57, 20, 22, 24, 38, 40, 42, 56, 58, 60, 81, 23, 25, 45, 41, 43, 63, 59, 61, 243, 82, 84, 26, 46, 48, 44, 64, 66, 62, 244, 246, 83, 85, 87, 47, 49

Offset: 0

Antti Karttunen, Jul 29 2009

			From _Kevin Ryde_, Oct 06 2020: (Start)
Array A(y,x) read by downwards antidiagonals, so 0, 1,3, 2,4,6, etc.
        x=0   1   2   3   4   5   6   7   8
      +--------------------------------------
  y=0 |   0,  1,  2,  9, 10, 11, 18, 19, 20,
    1 |   3,  4,  5, 12, 13, 14, 21, 22,
    2 |   6,  7,  8, 15, 16, 17, 24,
    3 |  27, 28, 29, 36, 37, 38,
    4 |  30, 31, 32, 39, 40,
    5 |  33, 34, 35, 42,
    6 |  54, 55, 56,
    7 |  57, 58,
    8 |  60,
(End)

Antti Karttunen, Table of n, a(n) for n = 0..3320
Index entries for sequences that are permutations of the natural numbers

Inverse: A163329. Transpose: A163330. Cf. A037314 (row y=0), A208665 (column x=0)

Cf. A054238 is an analogous sequence for binary. Cf. A007089, A163327, A163332, A163334.

A(y,x) = 3*fromdigits(digits(y,3),9) + fromdigits(digits(x,3),9); \\ Kevin Ryde, Oct 06 2020

(define (A163328 n) (+ (A037314 (A025581 n)) (* 3 (A037314 (A002262 n)))))

a(n) = A037314(A025581(n)) + 3*A037314(A002262(n))

a(n) = A163327(A163330(n)).

Edited by Charles R Greathouse IV, Nov 01 2009

cp's OEIS Frontend

A075300 Array A read by antidiagonals upwards: A(n, k) = array A054582(n,k) - 1 = 2^n*(2*k+1) - 1 with n,k >= 0.

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A214604 Odd numbers by transposing the right half of A176271, triangle read by rows: T(n,k) = A176271(n - 1 + k, n), 1 <= k <= n.

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A214661 Odd numbers obtained by transposing the left half of A176271 into rows of a triangle: T(n,k) = A176271(n - 1 + k, k), 1 <= k <= n.

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A227189 Square array A(n>=0,k>=0) where A(n,k) gives the (k+1)-th part of the unordered partition which has been encoded in the binary expansion of n, as explained in A227183. The array is scanned antidiagonally as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), etc.

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A268830 Square array A(r,c): A(0,c) = c, A(r,0) = 0, A(r>=1,c>=1) = 1+A(r-1,A268718(c)-1) = 1 + A(r-1, A003188(A006068(c)-1)), read by descending antidiagonals.

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A032531 An inventory sequence: triangle read by rows, where T(n, k), 0 <= k <= n, records the number of k's thus far in the flattened sequence.

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A071654 Inverse permutation to A071653.

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A072734 Simple triangle-stretching N X N -> N bijection, variant of A072732.

A075300 Array A read by antidiagonals upwards: A(n, k) = array A054582(n,k) - 1 = 2^n(2k+1) - 1 with n,k >= 0.