A037827 Number of i such that d(i) >= d(i-1), where Sum_{i=0..m} d(i)*4^i is the base-4 representation of n.

0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2

Offset

1,16

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

From Robert Israel, Oct 16 2015: (Start)

G.f.: (1-x)^(-1) * Sum_{j>=0} x^(4^(j+1)*(1-x^(4^j))*(1+x^(4^j)+x^(4*4^j)+x^(5*4^j)+x^(6*4^j)+x^(8*4^j)+x^(9*4^j)+x^(10*4^j)+x^(11*4^j)+x^(12*4^j))/(1-x^(4^(j+2))).

For n >= 4, a(n) - a(floor(n/4)) = 0 if n == 1, 2, 3, 6, 7, or 11 (mod 16), 1 otherwise. (End)

Maple

A037827 := proc(n)
    a := 0 ;
    dgs := convert(n, base, 4);
    for i from 2 to nops(dgs) do
        if op(i, dgs)>=op(i-1, dgs) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 16 2015
S:= [1, 2, 3, 6, 7, 11]:
f:= proc(n) option remember;
procname(floor(n/4)) + `if`(has(S, n mod 16), 0, 1)
end proc:
f(0):= 0:
seq(f(n), n=1..100); # Robert Israel, Oct 16 2015

Prog

(PARI) a(n) = {my(d = Vecrev(digits(n, 4))); my(dd = vector(#d-1, k, d[k+1] - d[k])); #select(x->(x>=0), dd); } \\ Michel Marcus, Oct 16 2015
(Python)
def A037827(n):
    s = '0'*(n.bit_length()&1)+bin(n)[2:]
    return sum(1 for i in range(0, len(s)-2, 2) if s[i:i+2]>=s[i+2:i+4]) # Chai Wah Wu, Feb 02 2023

Crossrefs

Cf. A037811.

Sequence in context: A175940 A299236 A353970 * A086074 A180601 A331048

Adjacent sequences: A037824 A037825 A037826 * A037828 A037829 A037830

Keyword

nonn,base

Author

Clark Kimberling

Extensions

Sign in name corrected by R. J. Mathar, Oct 16 2015

Status

approved