A306306 Number of binary words of length n such that in every prefix and in every suffix the difference between the number of 1's and the number of 0's is in the interval [-2,3].

1, 2, 4, 7, 12, 21, 35, 62, 102, 184, 299, 551, 882, 1666, 2615, 5085, 7782, 15658, 23219, 48603, 69402, 151945, 207695, 477987, 622062, 1511741, 1864139, 4803125, 5588322, 15319484, 16756775, 49018968, 50253942, 157270414, 150729059, 505697248, 452121642

Offset: 0

Author

Alois P. Heinz, Feb 05 2019

Keywords

• nonn
• easy

Examples

			a(3) = 7: 001, 010, 011, 100, 101, 110, 111.
a(4) = 12: 0010, 0011, 0100, 0101, 0110, 0111, 1001, 1010, 1011, 1100, 1101, 1110.
a(5) = 21: 00101, 00110, 00111, 01001, 01010, 01011, 01100, 01101, 01110, 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11001, 11010, 11011, 11100, 11101.
a(6) = 35: 001010, 001011, 001100, 001101, 001110, 010010, 010011, 010100, 010101, 010110, 010111, 011001, 011010, 011011, 011100, 011101, 011110, 100011, 100101, 100110, 100111, 101001, 101010, 101011, 101100, 101101, 101110, 110001, 110010, 110011, 110100, 110101, 110110, 111001, 111010.
a(7) = 62: 0010101, 0010110, 0010111, ..., 1110010, 1110011, 1110100, 1110101.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3912
• Index entries for linear recurrences with constant coefficients, signature (0,14,0,-81,0,250,0,-443,0,451,0,-249,0,65,0,-6)

Crossrefs

Cf. A306293, A306315.

Formula

G.f.: -(x^16 -10*x^14 +31*x^12 +54*x^11 -27*x^10 -153*x^9 -27*x^8 +165*x^7 +59*x^6 -85*x^5 -37*x^4 +21*x^3 +10*x^2 -2*x-1) / ((x-1) *(x+1) *(3*x^2-1) *(2*x^2-1) *(x^2+x-1) *(x^2-x-1) *(x^3-2*x^2-x+1) *(x^3+2*x^2-x-1)).

a(n) >= A306293(n).