A046688 Antidiagonals of square array in which k-th row (k>0) is an arithmetic progression of difference 2^(k-1).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 5, 1, 1, 5, 7, 9, 9, 1, 1, 6, 9, 13, 17, 17, 1, 1, 7, 11, 17, 25, 33, 33, 1, 1, 8, 13, 21, 33, 49, 65, 65, 1, 1, 9, 15, 25, 41, 65, 97, 129, 129, 1, 1, 10, 17, 29, 49, 81, 129, 193, 257, 257, 1, 1, 11, 19, 33, 57, 97, 161, 257, 385, 513, 513, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy

Examples

			From _Gus Wiseman_, May 08 2021: (Start):
Array A(m,n) = 1 + n*2^(m-1) begins:
       n=0: n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9:
  m=0:   1    1    1    1    1    1    1    1    1    1
  m=1:   1    2    3    5    9   17   33   65  129  257
  m=2:   1    3    5    9   17   33   65  129  257  513
  m=3:   1    4    7   13   25   49   97  193  385  769
  m=4:   1    5    9   17   33   65  129  257  513 1025
  m=5:   1    6   11   21   41   81  161  321  641 1281
  m=6:   1    7   13   25   49   97  193  385  769 1537
  m=7:   1    8   15   29   57  113  225  449  897 1793
  m=8:   1    9   17   33   65  129  257  513 1025 2049
  m=9:   1   10   19   37   73  145  289  577 1153 2305
Triangle T(n,k) = 1 + (n-k)*2^(k-1) begins:
   1
   1   1
   1   2   1
   1   3   3   1
   1   4   5   5   1
   1   5   7   9   9   1
   1   6   9  13  17  17   1
   1   7  11  17  25  33  33   1
   1   8  13  21  33  49  65  65   1
   1   9  15  25  41  65  97 129 129   1
   1  10  17  29  49  81 129 193 257 257   1
   1  11  19  33  57  97 161 257 385 513 513   1
(End)
		

References

• G. H. Hardy, A Theorem Concerning the Infinite Cardinal Numbers, Quart. J. Math., 35 (1904), p. 90 = Collected Papers, Vol. VII, p. 430.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325

Crossrefs

Row sums are A000079.

Diagonal n = m + 1 of the array is A002064.

Diagonal n = m of the array is A005183.

Column m = 1 of the array is A094373.

Diagonal n = m - 1 of the array is A131056.

A002109 gives hyperfactorials (sigma: A260146, omega: A303281).

A009998(k,n) = n^k.

A009999(n,k) = n^k.

A057156 = (2^n)^(2^n).

A062319 counts divisors of n^n.

Cf. A000169, A000272, A000312, A036289, A343656, A343658.

Programs

Mathematica

Table[If[k==0,1,n*2^(k-1)+1],{n,0,9},{k,0,9}] (* ARRAY, Gus Wiseman, May 08 2021 *)
Table[If[k==0,1,1+(n-k)*2^(k-1)],{n,0,10},{k,0,n}] (* TRIANGLE, Gus Wiseman, May 08 2021 *)

PARI

A(m,n)={if(m>0, 1+n*2^(m-1), 1)}
{ for(m=0, 10, for(n=0, 10, print1(A(m,n), ", ")); print) } \\ Andrew Howroyd, Mar 07 2020

Formula

A(m,n) = 1 + n*2^(m-1) for m > 1. - Andrew Howroyd, Mar 07 2020

As a triangle, T(n,k) = A(k,n-k) = 1 + (n-k)*2^(k-1). - Gus Wiseman, May 08 2021

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Apr 06 2000