A360099 - OEIS

A360099

To get A(n,k), replace 0's in the binary expansion of n with (-1) and interpret the result in base k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 1, -1, 0, 1, 0, 1, 0, 1, 1, 2, -1, 0, 1, 2, 3, -1, 1, 0, 1, 3, 4, 1, 1, -1, 0, 1, 4, 5, 5, 3, 1, 1, 0, 1, 5, 6, 11, 7, 5, 3, -1, 0, 1, 6, 7, 19, 13, 11, 7, -2, 1, 0, 1, 7, 8, 29, 21, 19, 13, 1, 0, -1, 0, 1, 8, 9, 41, 31, 29, 21, 14, 3, 0, 1, 0, 1, 9, 10, 55, 43, 41, 31, 43, 16, 5, 2, -1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,14

COMMENTS

The empty bit string is used as binary expansion of 0, so A(0,k) = 0.

LINKS

Alois P. Heinz, Antidiagonals n = 0..200, flattened

FORMULA

G.f. for column k satisfies g_k(x) = k*(x+1)*g_k(x^2) + x/(1+x).

A(n,k) = k*A(floor(n/2),k)+2*(n mod 2)-1 for n>0, A(0,k)=0.

A(n,k) mod 2 = A057427(n) if k is even.

A(n,k) mod 2 = A030300(n) if k is odd and n>=1.

A(2^(n+1),1) + n = 0.

EXAMPLE

Square array A(n,k) begins:

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...

-1, -1, 1, 5, 11, 19, 29, 41, 55, 71, 89, ...

1, 1, 3, 7, 13, 21, 31, 43, 57, 73, 91, ...

-1, 1, 5, 11, 19, 29, 41, 55, 71, 89, 109, ...

1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, ...

-1, -2, 1, 14, 43, 94, 173, 286, 439, 638, 889, ...

1, 0, 3, 16, 45, 96, 175, 288, 441, 640, 891, ...

-1, 0, 5, 20, 51, 104, 185, 300, 455, 656, 909, ...

MAPLE

A:= proc(n, k) option remember; local m;

`if`(n=0, 0, k*A(iquo(n, 2, 'm'), k)+2*m-1)

end:

seq(seq(A(n, d-n), n=0..d), d=0..12);

# second Maple program:

A:= (n, k)-> (l-> add((2*l[i]-1)*k^(i-1), i=1..nops(l)))(Bits[Split](n)):

seq(seq(A(n, d-n), n=0..d), d=0..12);

CROSSREFS

Columns k=0-6, 10 give: A062157, A145037, A006257, A147991, A147992, A153777, A147993, A359925.

Rows n=0-10 give: A000004, A000012, A023443, A000027(k+1), A165900, A002061, A165900(k+1), A002061(k+1), A083074, A152618, A062158.

Main diagonal gives A360096.

Cf. A030300, A057427.

Sequence in context: A119339 A037835 A116433 * A106509 A324692 A228110

Adjacent sequences: A360096 A360097 A360098 * A360100 A360101 A360102

KEYWORD

sign,tabl,look,base

AUTHOR

Alois P. Heinz, Jan 25 2023

STATUS

approved