A277513 Irregular triangle read by rows: T(n,k) is the number of integers greater than 4 such that they have n trits and 2k+1 (k>=1) nonzero trits in their balanced ternary representation, with n>=3 and 1<=k<=(j-1)/2.

4, 12, 24, 16, 40, 80, 60, 240, 64, 84, 560, 448, 112, 1120, 1792, 256, 144, 2016, 5376, 2304, 180, 3360, 13440, 11520, 1024, 220, 5280, 29568, 42240, 11264, 264, 7920, 59136, 126720, 67584, 4096, 312, 11440, 109824, 329472, 292864, 53248, 364, 16016, 192192

Offset

3,1

Comments

This is a subset of A013609 and A188440.

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

Cf. A013609, A188440, A134021, A134022, A134023.

Sequence in context: A008065 A321685 A156678 * A319887 A319868 A274187

Adjacent sequences: A277510 A277511 A277512 * A277514 A277515 A277516

Keyword

base,nonn,tabf

Author

Lei Zhou, Oct 18 2016

Status

approved