A364027 - OEIS

%I #14 Jul 02 2023 20:38:48

%S 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,

%T 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,

%U 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

%N 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).

%C 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.

%C 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.

%H Michael De Vlieger, <a href="/A364027/b364027.txt">Table of n, a(n) for n = 0..10000</a>

%H Scott R. Shannon, <a href="/A364027/a364027_1.png">Image of the first 20 million terms</a>.

%e a(2) = 1 as there is one pair that sums to a(0) + a(1) = 0, namely a(0) + a(1).

%e 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).

%e 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).

%t 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 *)

%o (Python)

%o def A364027_gen(): # generator of terms

%o     a, b, ndict = 0, 0, {0:1}

%o     yield from (0,0)

%o     while True:

%o         a, b = b, ndict[a+b]

%o         yield b

%o         ndict[a+b] = ndict.get(a+b,0)+1 # _Chai Wah Wu_, Jul 02 2023

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

%K nonn,look

%O 0,6

%A _Scott R. Shannon_, Jul 01 2023