A075300 - OEIS

%I #40 Feb 04 2019 11:21:52

%S 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,

%T 21,12,127,191,159,111,71,43,25,14,255,383,319,223,143,87,51,29,16,

%U 511,767,639,447,287,175,103,59,33,18,1023,1535,1279,895,575,351,207,119

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

%C From _Philippe Deléham_, Feb 19 2014: (Start)

%C A(0,k)  = 2*k = A005843(k),

%C A(1,k)  = 4*k + 1 = A016813(k),

%C A(2,k)  = 8*k + 3 = A017101(k),

%C A(n,0)  = A000225(n),

%C A(n,1)  = A153893(n),

%C A(n,2)  = A153894(n),

%C A(n,3)  = A086224(n),

%C A(n,4)  = A052996(n+2),

%C A(n,5)  = A086225(n),

%C A(n,6)  = A198274(n),

%C A(n,7)  = A238087(n),

%C A(n,8)  = A198275(n),

%C A(n,9)  = A198276(n),

%C A(n,10) = A171389(n). (End)

%C A permutation of the nonnegative integers. - _Alzhekeyev Ascar M_, Jun 05 2016

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F From _Wolfdieter Lang_, Jan 31 2019: (Start)

%F Array A(n, k) = 2^n*(2*k+1) - 1, for n >= 0 and m >= 0.

%F The triangle is T(n, k) = A(n-k, k) = 2^(n-k)*(2*k+1) - 1, n >= 0, k=0..n.

%F See also A054582 after subtracting 1.

%F (End)

%e The array A begins:

%e    0    2    4    6    8   10   12   14   16   18 ...

%e    1    5    9   13   17   21   25   29   33   37 ...

%e    3   11   19   27   35   43   51   59   67   75 ...

%e    7   23   39   55   71   87  103  119  135  151 ...

%e   15   47   79  111  143  175  207  239  271  303 ...

%e   31   95  159  223  287  351  415  479  543  607 ...

%e   ...

%e - _Philippe Deléham_, Feb 19 2014

%e From _Wolfdieter Lang_, Jan 31 2019: (Start)

%e The triangle T begins:

%e    n\k   0    1    2   3   4   5   6   7  8  9 10 ...

%e    0:    0

%e    1:    1    2

%e    2:    3    5    4

%e    3:    7   11    9   6

%e    4:   15   23   19  13   8

%e    5    31   47   39  27  17  10

%e    6:   63   95   79  55  35  21  12

%e    7:  127  191  159 111  71  43  25  14

%e    8:  255  383  319 223 143  87  51  29 16

%e    9:  511  767  639 447 287 175 103  59 33 18

%e   10: 1023 1535 1279 895 575 351 207 119 67 37 20

%e   ...

%e T(3, 1) = 2^2*(2*1+1) - 1 = 12 - 1 = 11.  (End)

%p A075300bi := (x,y) -> (2^x * (2*y + 1))-1;

%p A075300 := n -> A075300bi(A025581(n), A002262(n));

%p A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);

%p A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1);

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

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

%Y Cf. A002262, A025581, A054582, A241957.

%K nonn,tabl,easy

%O 0,3

%A _Antti Karttunen_, Sep 12 2002