A056196 -id:A056196 - cp's OEIS frontend

A003156 A self-generating sequence (see Comments for definition).

1, 4, 5, 6, 9, 12, 15, 16, 17, 20, 21, 22, 25, 26, 27, 30, 33, 36, 37, 38, 41, 44, 47, 48, 49, 52, 55, 58, 59, 60, 63, 64, 65, 68, 69, 70, 73, 76, 79, 80, 81, 84, 85, 86, 89, 90, 91, 94, 97, 100, 101, 102, 105, 106, 107, 110, 111, 112, 115, 118, 121, 122, 123, 126, 129, 132

Offset: 1

N. J. A. Sloane

nonn

From N. J. A. Sloane, Dec 26 2020: (Start)
The best definitions of the triple [this sequence, A003157, A003158] are as the rows a(n), b(n), c(n) of the table:
n: 1, 2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, ...
a: 1, 4,  5,  6,  9, 12, 15, 16, 17, 20, 21, 22, ...
b: 3, 8, 11, 14, 19, 24, 29, 32, 35, 40, 43, 46, ...
c: 2, 7, 10, 13, 18, 23, 28, 31, 34, 39, 42, 45, ...
where a(1)=1, b(1)=3, c(1)=2, and thereafter
a(n) = mex{a(i), b(i), c(i), ib(n) = a(n) + 2*n,
c(n) = b(n) - 1.
Then a,b,c form a partition of the positive integers.
Note that there is another triple of sequences (A003144, A003145, A003146) also called a, b, c and also a partition of the positive integers, in a different paper by the same authors (Carlitz-Scovelle-Hoggatt) in the same volume of the same journal.
(End)
a(n) is the number of ones before the n-th zero in the Feigenbaum sequence A035263. - Philippe Deléham, Mar 27 2004
Number of odd numbers before the n-th even number in A007413, A007913, A001511, A029883, A033485, A035263, A036585, A065882, A065883, A088172, A092412. - Philippe Deléham, Apr 03 2004
Indices of a in the sequence closed under a -> abc, b -> a, c -> a, starting with a(1) = a; see A092606 where a = 0, b = 2, c = 1. - Philippe Deléham, Apr 12 2004

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Jean-Paul Allouche, On the morphism 1 -> 121, 2 -> 12221, Preprint, 2024 [Local copy, with permission]. See also Optimization, Discrete Mathematics and Applications to Data Sciences, Springer Optimization and Its Applications, Springer, Cham, 2025, and author's site, CNRS France, 2024. See p. 6.
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations of higher order, Fib. Quart., 10 (1972), 43-69. [Here a, b, c are A003144, A003145, A003146.]
L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., Representations for a special sequence, Fibonacci Quarterly 10.5 (1972), 499-518, 550. [A003156, A003157, A003158 are on page 500.]
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See pp. 8, 17

Cf. A001511, A003157, A003158, A003159, A007413, A007913, A029883, A033485, A035263, A036554, A036585, A056196, A065882, A065883, A072939, A079523, A072939, A080426, A088172, A092412, A092606.

See also A003144, A003145, A003146.

following Deléham
a003156 n = a003156_list !! (n-1)
a003156_list = scanl1 (+) a080426_list
-- Reinhard Zumkeller, Oct 27 2014

a:= proc(n) global l; while nops(l) [1, 3$d, 1][], l) od; `if` (n=1, 1, a(n-1) +l[n]) end: l:= [1]: seq (a(n), n=1..80); # Alois P. Heinz, Oct 31 2009

Position[Nest[Flatten[# /. {0 -> {0, 2, 1}, 1 -> {0}, 2 -> {0}}]&, {0}, 7], 0] // Flatten (* Jean-François Alcover, Mar 14 2014 *)

def A003156(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, s = n+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    return bisection(f,n,n)-n # Chai Wah Wu, Jan 29 2025

a(n) = A079523(n) - n + 1 = A003157(n) - 2n = A003158(n) - 2n + 1. - Philippe Deléham, Feb 28 2004
a(n) = A036554(n) - n = A072939(n) - n - 1 = 2*A003159(n) - n. - Philippe Deléham, Apr 10 2004
a(n) = Sum_{k = 1..n} A080426(k). - Philippe Deléham, Apr 16 2004

More terms from Alois P. Heinz, Oct 31 2009

Incorrect equation removed from formula by Peter Munn, Dec 11 2020

A386987 For n >= 2, a(n) is the least r >= 1 such that T(n - r) + ... + T(n - 1) = T(n + 1) + ... + T(n + r) where T(i) is A010060(i).

Original entry on oeis.org

2, 1, 1, 2, 4, 3, 3, 4, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 4, 3, 3, 4, 2, 1, 1, 2, 4, 3, 3, 4, 2, 1, 1, 2, 4, 3, 3, 4, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 4, 3, 3, 4, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 4, 3, 3, 4, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 4, 3, 3

Offset: 2

Ctibor O. Zizka, Aug 12 2025

nonn

a(n) is from {1, 2, 3, 4}.

			For n = 6: T(6 - r) + ... + T(5) = T(7) + ... + T(6 + r) is true for the least r = 4  because A010060(2) + A010060(3) + A010060(4) + A010060(5) = A010060(7) + A010060(8) + A010060(9) + A010060(10), thus a(6) = 4.

Cf. A010060, A056196, A079523, A081706, A131323, A225822, A249034.

a[n_] := Module[{s = 0, r = 1}, While[r <= n && (r == 1 || s != 0), s += (ThueMorse[n - r] - ThueMorse[n + r]); r++]; r-1]; Array[a, 100, 2] (* Amiram Eldar, Aug 12 2025 *)

a(A081706(n) + 1) = 1.
a(2*A079523(n)) = 2.
a(A249034(n))= 2.
a(A225822(n)) = 3.
a(A056196(n)) = 3.
a(2*A131323(n)) = 4.
a(2*A249034(n) - 1) = 4.

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cp's OEIS Frontend

A003156 A self-generating sequence (see Comments for definition).

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