A364027 a(0) = 0, a(1) = 0; for n > 1, a(n) is the number of pairs of consecutive terms that sum to the same value as a(n-2) + a(n-1).

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

Offset

0,6

Comments

The same number cannot occur four times in a row as the second pair in a triplet of the same numbers increments the appearance count of the first pair by one, so the fourth number is always one more than the previous three numbers.

The occurrences of three consecutive equal numbers is quite rare, only occurring thirteen times in the first 20 million terms. The last such triplet is a(3641208) = a(3641209) = a(3641210) = 1177. It is likely such triplets occur infinitely often although this is unknown.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Scott R. Shannon, Image of the first 20 million terms.

Example

a(2) = 1 as there is one pair that sums to a(0) + a(1) = 0, namely a(0) + a(1).
a(4) = 1 as a(2) + a(3) = 1 + 1 = 2, and there has been one previous pair that also sums to 2, namely a(2) + a(3).
a(5) = 2 as a(3) + a(4) = 1 + 1 = 2, and there has been two previous pairs that also sums to 2, namely a(2) + a(3) and a(3) + a(4).

Mathematica

nn = 120; c[_] := 0; a[0] = a[1] = i = j = 0; Do[Set[k, 1 + c[i + j]++]; i = j; j = a[n] = k, {n, 2, nn}]; Array[a, nn + 1, 0] (* Michael De Vlieger, Jul 02 2023 *)

Prog

(Python)
def A364027_gen(): # generator of terms
    a, b, ndict = 0, 0, {0:1}
    yield from (0, 0)
    while True:
        a, b = b, ndict[a+b]
        yield b
        ndict[a+b] = ndict.get(a+b, 0)+1 # Chai Wah Wu, Jul 02 2023

Crossrefs

Cf. A364036 (do not include previous pair), A342585, A347062.

Sequence in context: A374443 A362890 A306251 * A308967 A241844 A233864

Adjacent sequences: A364024 A364025 A364026 * A364028 A364029 A364030

Keyword

nonn,look

Author

Scott R. Shannon, Jul 01 2023

Status

approved