A268292 a(n) is the total number of isolated 1's at the boundary between n-th and (n-1)-th iterations in the pattern of A267489.

0, 0, 0, 0, 0, 0, 0, 1, 3, 5, 7, 9, 11, 14, 18, 22, 26, 30, 34, 39, 45, 51, 57, 63, 69, 76, 84, 92, 100, 108, 116, 125, 135, 145, 155, 165, 175, 186, 198, 210, 222, 234, 246, 259, 273, 287, 301, 315, 329, 344, 360, 376, 392, 408, 424, 441

Offset: 0

Author

Kival Ngaokrajang, Jan 31 2016

Keywords

• base
• nonn

Comments

Refer to pattern of A267489, The total number of isolated 1's is a(n) and A112421 when consider at the boundary between n-th and (n-1)-th iterations and at the boundary in the same iterations concatenate on horizontal respectively. See illustrations in the links.

Empirically, a(n+4) gives the number of solutions m where 0 < m < 2^n and A014682^n(m) < 3 and A014682^n(m+2^n) = A014682^n(m)+9. - Thomas Scheuerle, Apr 25 2021

Links

• Kival Ngaokrajang, Illustration of initial terms, Continuously concatenate pattern

Crossrefs

Cf. A112421, A267489, A014682.

Programs

PARI

a = 3; d1 = 2; print1("0, 0, 0, 0, 0, 0, 0, 1, 3, ");
for (n = 3,100, d2 = 0; if (Mod(n,6)==1 || Mod(n,6)==2, d2 = 1); d1 = d1 + d2; a = a + d1; print1(a, ", "))

Formula

Empirical g.f.: x^7 / ((1-x)^3*(1-x+x^2)*(1+x+x^2)). - Colin Barker, Jan 31 2016

For n>0: a(n) = floor((n-3)^2/12) + floor((n-4)^2/12). - Hoang Xuan Thanh, Jun 02 2025