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A376102
Array read by ascending antidiagonals: A(n,k) = k*2^(n+1) + 1.
0
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
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987.
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. A000012 (k=0), A000051, A000337, A004119, A005408 (n=0), A016813 (n=1), A017077 (n=2), A158057 (n=3).
Sequence in context: A111125 A209159 A182397 * A343510 A344725 A209560
KEYWORD
nonn,easy,tabl
AUTHOR
Stefano Spezia, Sep 14 2024
STATUS
approved