A180969 Array read by antidiagonals: a(k,n) = natural numbers each repeated 2^k times.

0, 1, 0, 2, 0, 0, 3, 1, 0, 0, 4, 1, 0, 0, 0, 5, 2, 0, 0, 0, 0, 6, 2, 1, 0, 0, 0, 0, 7, 3, 1, 0, 0, 0, 0, 0, 8, 3, 1, 0, 0, 0, 0, 0, 0, 9, 4, 1, 0, 0, 0, 0, 0, 0, 0, 10, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 11, 5, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 12, 5, 2, 1, 0, 0, 0, 0

Offset

0,4

Comments

Generalization of P. Barry's (2003) formula in A004526.

Links

Danny Rorabaugh, Table of n, a(n) for n = 0..10000

Formula

a(k,n) = (n/2^k) + Sum_{j=1..k} ((-1)^a(j-1,n) - 1)/2^(k-j+2).

a(k,n) = floor(n/2^k). - Adriano Caroli, Sep 30 2019

Example

Sequence gives the antidiagonals of the infinite square array with rows indexed by k and columns indexed by n:
0  1  2  3  4  5  6  7  8  9 10 11 12 13...
0  0  1  1  2  2  3  3  4  4   5   5   6...
0  0  0  0  1  1  1  1  2  2   2   2   3...
0  0  0  0  0  0  0  0  1  1   1   1   1...
0  0  0  0  0  0  0  0  0  0   0   0   0...
...........................................

Mathematica

Table[Floor[#/2^k] &[n - k], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 30 2019 *)

Prog

(MATLAB) function v=A180969(k, n, q)
% n=vector of natural numbers 0, 1, ..., n
% v=vector in which each n is repeated k times
% q=q-th term of v from where to start
if k==0; v=n+q; return; end
v=A180969(k-1, n, q);
% calculate repetition only if v terms are not all zeros
if any(v); v=v/2+((-1).^v-1)/4; end
% Adriano Caroli, Nov 28 2010
(PARI) matrix(10, 20, k, n, k--; n--; floor(n/2^k)) \\ Michel Marcus, Sep 09 2019

Crossrefs

Cf. A001477, A004526, A002265, A132292.

Sequence in context: A194586 A288437 A287736 * A369312 A259479 A238343

Adjacent sequences: A180966 A180967 A180968 * A180970 A180971 A180972

Keyword

easy,nonn,tabl,less

Author

Adriano Caroli, Nov 17 2010

Status

approved