A064686 a(n) = number of n-digit base-3 biquams.

0, 2, 7, 23, 73, 227, 697, 2123, 6433, 19427, 58537, 176123, 529393, 1590227, 4774777, 14332523, 43013953, 129074627, 387289417, 1161999323, 3486260113, 10459304627, 31378962457, 94138984523, 282421147873, 847271832227

Offset: 1

Author

David W. Wilson, Oct 10 2001

Keywords

• base
• easy
• nonn

Comments

A biquam or biquanimous number (A064544) is a number whose digits can be split into two groups with equal sum.

This is the same as A083313 (apart from the initial term). Proof: Let sum(w) denote the sum of the digits of w. There are 2*3^(n-1) n-digit base-3 numbers: w = (w_1,w_2,...,w_n) with w_i in {0,1,2} for all i and w_1 != 0. Partition them into 4 classes: (i) sum(w) is odd, (ii) sum(w) is even, w contains no 1's and has an odd number of 2s, (iii) sum(w) is even, w contains no 1's and has an even number of 2s and (iv) sum(w) is even and w contains some 1's. Clearly, no biquams occur in cases (i) and (ii), case (iii) consists entirely of biquams and, we claim, so does case (iv). For case (iv) forces an even number, say 2k, of 1's. An even number of 2s clearly gives a biquam and an odd number 2m+1 of 2s does too because {m 2s, (k+1) 1's} and {(m+1) 2s, (k-1) 1's} is a biquam split. There are 3^(n-1) w's in case (i) and 2^(n-2) w's in case (ii) and hence 2*3^(n-1) - (3^(n-1) + 2^(n-2)) = 3^(n-1) - 2^(n-2) (A083313) biquams among n-digit base-3 numbers. - David Callan, Sep 15 2004

a(n) % 100 = 23 for n = 4*k-1, k>=1; a(n) % 100 = 27 for n = 4*k+1, k>=1. - Alex Ratushnyak, Jul 03 2012

The fraction of biquams for any base approaches 1/2 as the number of digits grows but only if you count leading zeros. Without counting leading zeros, the fraction appears to converge to (b-1)/2b where b is the base used. For base 3 this is 1/3 which fits the data in this sequence (see paper cited below for proofs and the OEIS data collated as fractions). - Timothy Varghese, Aug 08 2021

Links

• Timothy Varghese and George Varghese, The Stash-Repair Method applied to Partitionable Numbers,
• Index entries for linear recurrences with constant coefficients, signature (5,-6).

Crossrefs

Essentially the same as A083313.

Cf. A053152 (partial sums).

Programs

Python

print([0]+[3**n - 2**(n-1) for n in range(1,29)])
# Alex Ratushnyak, Jul 02 2012

Formula

a(1) = 0, a(n) = 3^(n-1)-2^(n-2) for n>=2. - Alex Ratushnyak, Jul 02 2012

a(n) = 5*a(n-1)-6*a(n-2) for n>3. G.f.: -x^2*(3*x-2) / ((2*x-1)*(3*x-1)). - Colin Barker, May 27 2013