OFFSET
0,3
COMMENTS
Also the first differences of A261692.
Number of cells turned ON at n-th stage of the cellular automaton of A261692.
This irregular triangle A (instead of T) appears also in the linearization of the following product of Chebyshev T polynomials (A053120): PrT(n) := Product_{j=1..n} T(2^j, x) = (1/2^(n-1))*Sum_{k=1..2^(n-1)} T(2*A(n, k), x), for n >= 1. Proof via 2*T(n, x)*T(m, x) = T(n+m, x) + T(|n-m|, x). - Wolfdieter Lang, Oct 26 2019
LINKS
FORMULA
T(n, k) = 2^n + 1 - 2*k, n >= 1, 1 <= k <= 2^(n-1), and T(0, 0) = 0.
As a sequence: a(n) = A262621(n)/4, n >= 1, and a(0) = 0.
EXAMPLE
With the terms written as an irregular triangle T in which row lengths are the terms of A011782 the sequence begins:
0;
1;
3, 1;
7, 5, 3, 1;
15, 13, 11, 9, 7, 5, 3, 1;
31, 29, 27, 25, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1;
...
-------------------------------------------------------------------------------
From Wolfdieter Lang, Oct 26 2019: (Start)
Chebyshev T(2^j)-products (the argument x is here omitted):
n = 1: T(2) = (2^0)*T(2*1),
n = 2: T(2)*T(4) = (1/2)*(T(2*3) + T(2*1)) = (T(6) + T(2))/2,
n = 3: T(2)*T(4)*T(8) = (1/2^2)*(T(2*7) + T(2*5) + T(2*3) + T(2*1))
= (T(14) + T(10) + T(6) + T(2))/4.
... (End)
MAPLE
A261693 := n -> Bits:-Nor(2*n, 2*n):
seq(A261693(n), n=0..81); # Peter Luschny, Sep 23 2019
MATHEMATICA
Table[Reverse[2 Range[2^(n - 1)] - 1], {n, 0, 7}] /. {} -> 0 // Flatten (* Michael De Vlieger, Oct 05 2015 *)
PROG
(PARI) tabf(nn) = {for (n=0, nn, print1(n, ":"); for (k=1, 2^(n-2), print1(2^(n-1) - 2*k + 1, ", "); ); print(); ); } \\ Michel Marcus, Oct 27 2015
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Omar E. Pol, Sep 25 2015
EXTENSIONS
Corrections by Wolfdieter Lang, Nov 15 2019
STATUS
approved