A376102 Array read by ascending antidiagonals: A(n,k) = k*2^(n+1) + 1.

1, 1, 3, 1, 5, 5, 1, 9, 9, 7, 1, 17, 17, 13, 9, 1, 33, 33, 25, 17, 11, 1, 65, 65, 49, 33, 21, 13, 1, 129, 129, 97, 65, 41, 25, 15, 1, 257, 257, 193, 129, 81, 49, 29, 17, 1, 513, 513, 385, 257, 161, 97, 57, 33, 19, 1, 1025, 1025, 769, 513, 321, 193, 113, 65, 37, 21

Offset

0,3

Comments

In 1747, Euler showed that any factor of a Fermat number A000215(n) is of the form k*2^(n+1) + 1. See Wells at p. 148.

References

William Dunham, Journey Through Genius, Wiley, 1990, Chapter 10, p. 233.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 70-71, 237-242.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 136.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987.

Links

Table of n, a(n) for n=0..65.

Formula

G.f.: (1 - 2*x + y)/((1 - x)*(1 - 2*x)*(1 - y)^2).

E.g.f.: exp(x+y)*(1 + 2*exp(x)*y).

Sum_{0<=k<=n} A(n-k,k) = A000295(n+2).

A(n,1) = A000051(n+1).

A(n,3) = A004119(n+2).

A(n,n) = A000337(n+1).

Example

The array begins as:
  1,   3,   5,   7,   9,  11,  13, ...
  1,   5,   9,  13,  17,  21,  25, ...
  1,   9,  17,  25,  33,  41,  49, ...
  1,  17,  33,  49,  65,  81,  97, ...
  1,  33,  65,  97, 129, 161, 193, ...
  1,  65, 129, 193, 257, 321, 385, ...
  1, 129, 257, 385, 513, 641, 769, ...
  ...

Mathematica

A[n_, k_]:=k*2^(n+1)+1; Table[A[n-k, k], {n, 0, 10}, {k, 0, n}]//Flatten

Crossrefs

Cf. A000079, A000215, A000295.

Cf. A000012 (k=0), A000051, A000337, A004119, A005408 (n=0), A016813 (n=1), A017077 (n=2), A158057 (n=3).

Sequence in context: A209159 A182397 A388052 * A343510 A344725 A209560

Adjacent sequences: A376099 A376100 A376101 * A376103 A376104 A376105

Keyword

nonn,easy,tabl

Author

Stefano Spezia, Sep 14 2024

Status

approved