

A343071


Irregular triangle read by rows: T(n,k) = 2^(2n+1) * L(n,k), where L(n,k) is the kth left endpoint after the nth step of removal in the construction of the SmithVolterraCantor set (SVC), 0 <= k <= 2^n  1.


2



0, 0, 5, 0, 7, 20, 27, 0, 11, 28, 39, 80, 91, 108, 119, 0, 19, 44, 63, 112, 131, 156, 175, 320, 339, 364, 383, 432, 451, 476, 495, 0, 35, 76, 111, 176, 211, 252, 287, 448, 483, 524, 559, 624, 659, 700, 735, 1280, 1315, 1356, 1391, 1456, 1491, 1532, 1567, 1728, 1763, 1804, 1839, 1904, 1939, 1980, 2015
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OFFSET

0,3


COMMENTS

The SVC (or fat Cantor set) is a subset of [0, 1] similar to the standard Cantor set， constructed as follows:
We start from C_0 = [0, 1], which is written as [0/2, 2/2] for convenience.
 Step 1: Remove the middle open interval of length 1/4 in C_0 (namely (3/8, 5/8)), leaving the union of 2 closed intervals C_1 = [0/8, 3/8] U [5/8, 8/8];
 Step 2: Remove the middle open interval of length 1/4^2 in each of the closed interval of C_1 (namely (5/32, 7/32) and (27/32, 29/32)), leaving the union of 4 closed intervals C_2 = [0/32, 5/32] U [7/32, 12/32] U [20/32, 27/32] U [29/32, 32/32].
...
 Step n: Remove the middle open interval of length 1/4^n in each of the closed interval of C_{n1}, leaving the union of 4 closed intervals C_n.
The SVC is thereby defined as the intersection of all {C_n} for n >= 0.
After the nth step, the kth left closed interval (from left to right) is [L(n,k), R(n,k)] = [T(n,k)/2^(2n+1), A343072(n,k)/2^(2n+1)].
Like the standard Cantor set, the SVC is perfect (i.e., closed and every point inside is an accumulation point of itself) and has empty interior. The difference between the SVC and the standard Cantor set is that the SVC has positive Lebesgue measure, namely 1  (1/4)  (1/4^2)*2  (1/4^3)*2^2  ... = 1/2.
The construction of the famous Volterra's function (a function differentiable everywhere on [0, 1] whose derivative is bounded yet not Riemann integratable) is based on the SVC.


LINKS

Jianing Song, Table of n, a(n) for n = 0..16382 (Rows n = 0..13).
Jianing Song, Construction of SVC after steps 0..13
Wikipedia, SmithVolterraCantor set
Wikipedia, Volterra's function


FORMULA

A343072(n,k)  T(n,k) = 2^n + 1, which corresponds to the fact that each closed interval of C_n is of length (2^n + 1)/2^(2n+1).
For n >= 0, 0 <= k <= 2^(n1)  1, T(n,2k) = 4*T(n1,k), T(n,2k+1) = 4*A343072(n1,k)  (2^n + 1) = 4*T(n1,k) + (2^n + 3).
For k = Sum_{i=0..n1} (b_i) * 2^i, b_i = 0 or 1, T(n,k) = Sum_{i=0..n1} (b_i) * 4^i * (2^(ni) + 3).
T(n,k) = 2^(2n+1)  A343072(n,(2^n1)k).


EXAMPLE

Table begins
0,
0, 5,
0, 7, 20, 27,
0, 11, 28, 39, 80, 91, 108, 119,
0, 19, 44, 63, 112, 131, 156, 175, 320, 339, 364, 383, 432, 451, 476, 495,
0, 35, 76, 111, 176, 211, 252, 287, 448, 483, 524, 559, 624, 659, 700, 735, 1280, 1315, 1356, 1391, 1456, 1491, 1532, 1567, 1728, 1763, 1804, 1839, 1904, 1939, 1980, 2015,
...
After the nth step of construction, we have
C_0 = [0/2, 2/2],
C_1 = [0/8, 3/8] U [5/8, 8/8],
C_2 = [0/32, 5/32] U [7/32, 12/32] U [20/32, 27/32] U [29/32, 32/32],
C_3 = [0/128, 9/128] U [11/128, 20/128] U [28/128, 37/128] U [39/128, 48/128] U [80/128, 89/128] U [91/128, 100/128] U [108/128, 117/128] U [119/128, 128/128],
...


PROG

(PARI) T(n, k) = my(t=0); for(i=0, n1, t+=(k%2)*4^i*(2^(ni)+3); k=k\2); t


CROSSREFS

Cf. A343072 (the right endpoints).
Sequence in context: A084248 A201417 A147666 * A215892 A200643 A200231
Adjacent sequences: A343068 A343069 A343070 * A343072 A343073 A343074


KEYWORD

nonn,tabf,easy


AUTHOR

Jianing Song, Apr 04 2021


STATUS

approved



