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A064686 a(n) = number of n-digit base-3 biquams. 5
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 (list; graph; refs; listen; history; text; internal format)



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


Table of n, a(n) for n=1..26.

Index entries for linear recurrences with constant coefficients, signature (5,-6).


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



print([0]+[3**n - 2**(n-1) for n in range(1, 29)])

# Alex Ratushnyak, Jul 02 2012


Essentially the same as A083313.

Cf. A053152 (partial sums).

Sequence in context: A295136 A192906 A217664 * A083313 A077832 A030282

Adjacent sequences:  A064683 A064684 A064685 * A064687 A064688 A064689




David W. Wilson, Oct 10 2001



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Last modified June 22 09:57 EDT 2021. Contains 345375 sequences. (Running on oeis4.)