A119281 Number of counting rods to represent n in the ancient Chinese rod numeral system.

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

Offset: 0

Author

Rick L. Shepherd, May 12 2006

Keywords

• base
• easy
• nonn

Comments

Contrast with A092196, the number of letters to represent n in ancient Roman numerals. Negative numbers were represented by the same number of rods but usually of a different color (usually black rods with red rods for positive numbers). It's unclear to me whether 0 itself was ever formally considered represented by the absence of all counting rods, but it does seem reasonable that a(0)=0 from the example below.

Examples

			a(105) = 6 because 105 was represented on a counting board by placing one counting rod in the compartment for hundreds, no rods where those representing tens were normally placed and five rods in the units compartment.
		

Links

• Wikipedia, Chinese numerals
• Wikipedia, Counting rods

Crossrefs

Cf. A092196, A007953, A102677.

Programs

PARI

a(n)= tmp=abs(n); r=0; l=length(Str(tmp)); for(k=1,l, d=tmp-(tmp\10)*10; tmp=tmp\10; if(d<6, r=r+d, r=r+d-4)); r

Formula

a(n) = a(-n) = A007953(n) - 4*A102677(n) = A092196(n) + 4*(number of 5s in n).