OFFSET
3,1
COMMENTS
This sequence T(n,k) can be re-indexed into the form of b(m) where m is positive integer sequence that can be calculated by parametric function m = m(n,k): m(n=2j+1,k) = j^2-j+k, where n is odd and 1<=k<=j.
m(n=2j,k) = j^2-2j+1+k, where n is even and 1<=k<=j.
Here n is the number of digits of an odd number in balanced ternary representation, and 2k+1 is the number of nonzero trits (1 or T) of the same number in balanced ternary representation.
LINKS
Lei Zhou, Table of n, a(n) for n = 3..10002
Definition of Balanced Ternary.
FORMULA
T(n,k) = 2^(2k)*Binomial(n-1, 2k)
EXAMPLE
Odd numbers that can be expressed in 3 trits balanced ternary (bt) form are 5 = 1TT, 7 = 1T1, 9 = 100, 11 = 11T, 13 = 111. Among these five numbers, four have 3 nonzero digits, so a(1) = 4.
Odd numbers in 4 trits bt form are 15 = 1TT0, 17 = 1T0T, 19 = 1T01, 21 = 1T10, 23 = 10TT, 25 = 10T1, 27 = 1000, 29 = 101T, 31 = 1011, 33 = 11T0, 35 = 110T, 37 = 1101, 39 = 1110. Among these 13 numbers, 12 have 3 nonzero digits, so a(2) = 12.
The irregular triangle begins:
k=1 2 3 4 5 6 7
n=3 4
n=4 12
n=5 24 16
n=6 40 80
n=7 60 240 64
n=8 84 560 448
n=9 112 1120 1792 256
n=10 144 2016 5376 2304
n=11 180 3360 13440 11520 1024
n=12 220 5280 29568 42240 11264
n=13 264 7920 59136 126720 67584 4096
n=14 312 11440 109824 329472 292864 53248
n=15 364 16016 192192 768768 1025024 372736 16384
These are the odd columns with the 1st column removed in the table in A013609.
MATHEMATICA
a = {}; Do[Do[ct = 2^(2k)*Binomial[n - 1, 2k]; AppendTo[a, ct], {k, 1, Floor[(n-1)/2]}], {n, 3, 15}]; a
CROSSREFS
KEYWORD
base,nonn,tabf
AUTHOR
Lei Zhou, Oct 18 2016
STATUS
approved