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A347683
Triangle read by rows: T(n,k) (1<=k<=n) = f(n,k), where f(x,y) = x*red_inv(x,y) + y*red_inv(y,x) if gcd(x,y)=1, or 0 if gcd(x,y)>1, and red_inv is defined in the comments.
5
0, 1, 0, 1, 5, 0, 1, 0, 7, 0, 1, 9, 11, 9, 0, 1, 0, 0, 0, 11, 0, 1, 13, 13, 15, 29, 13, 0, 1, 0, 17, 0, 31, 0, 15, 0, 1, 17, 0, 17, 19, 0, 55, 17, 0, 1, 0, 19, 0, 0, 0, 41, 0, 19, 0, 1, 21, 23, 23, 21, 23, 43, 65, 89, 21, 0, 1, 0, 0, 0, 49, 0, 71, 0, 0, 0, 23, 0, 1, 25, 25, 25, 51, 25, 27, 79, 53, 79, 131, 25, 0
OFFSET
1,5
COMMENTS
If u, v are positive integers with gcd(u,v) = 1, the "reduced inverse" red_inv(u,v) of u mod v is u^(-1) mod v if u^(-1) mod v <= v/2, otherwise it is v - u^(-1) mod v.
That is, we map u to whichever of +-u has a representative mod v in the range 0 to v/2. Stated another way, red_inv(u,v) is a number r in the range 0 to v/2 such that r*u == +-1 mod v.
For example, red_inv(3,11) = 4, since 3^(-1) mod 11 = 4. But red_inv(2,11) = 5 = 11-6, since red_inv(2,11) = 6.
Arises in the study of A344005.
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 1..5050 [The first 100 rows, flattened]
EXAMPLE
Triangle begins:
0,
1, 0,
1, 5, 0,
1, 0, 7, 0,
1, 9, 11, 9, 0,
1, 0, 0, 0, 11, 0,
1, 13, 13, 15, 29, 13, 0,
1, 0, 17, 0, 31, 0, 15, 0,
1, 17, 0, 17, 19, 0, 55, 17, 0,
1, 0, 19, 0, 0, 0, 41, 0, 19, 0,
...
MAPLE
myfun1 := proc(A, B) local Ar, Br;
if igcd(A, B) > 1 then return(0); fi;
Ar:=(A)^(-1) mod B;
if 2*Ar > B then Ar:=B-Ar; fi;
Br:=(B)^(-1) mod A;
if 2*Br > A then Br:=A-Br; fi;
A*Ar+B*Br;
end;
for i from 1 to 20 do lprint([seq(myfun1(i, j), j=1..i)]); od:
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Sep 18 2021
STATUS
approved