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A261693 Irregular triangle read by rows in which row n lists the positive odd numbers in decreasing order starting with 2^n - 1. T(0, 1) = 0 and T(n, k) for n >= 1, 1 <= k <= 2^(n-1).

%I A261693 #47 Nov 15 2019 13:55:42
%S A261693 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,
%T A261693 9,7,5,3,1,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,
%U A261693 25,23,21,19,17,15,13,11,9,7,5,3,1,127,125,123,121,119,117,115,113,111,109,107,105,103,101,99,97,95,93
%N A261693 Irregular triangle read by rows in which row n lists the positive odd numbers in decreasing order starting with 2^n - 1. T(0, 1) = 0 and T(n, k) for n >= 1, 1 <= k <= 2^(n-1).
%C A261693 Also the first differences of A261692.
%C A261693 Number of cells turned ON at n-th stage of the cellular automaton of A261692.
%C A261693 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
%H A261693 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A261693 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A261693 T(n, k) = 2^n + 1 - 2*k, n >= 1, 1 <= k <= 2^(n-1), and T(0, 0) = 0.
%F A261693 As a sequence: a(n) = A262621(n)/4, n >= 1, and a(0) = 0.
%e A261693 With the terms written as an irregular triangle T in which row lengths are the terms of A011782 the sequence begins:
%e A261693 0;
%e A261693 1;
%e A261693 3, 1;
%e A261693 7, 5, 3, 1;
%e A261693 15, 13, 11, 9, 7, 5, 3, 1;
%e A261693 31, 29, 27, 25, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1;
%e A261693 ...
%e A261693 -------------------------------------------------------------------------------
%e A261693 From _Wolfdieter Lang_, Oct 26 2019: (Start)
%e A261693 Chebyshev T(2^j)-products (the argument x is here omitted):
%e A261693 n = 1: T(2) = (2^0)*T(2*1),
%e A261693 n = 2: T(2)*T(4) = (1/2)*(T(2*3) + T(2*1)) = (T(6) + T(2))/2,
%e A261693 n = 3: T(2)*T(4)*T(8) =  (1/2^2)*(T(2*7) + T(2*5) + T(2*3) + T(2*1))
%e A261693        = (T(14) + T(10) + T(6) + T(2))/4.
%e A261693 ... (End)
%p A261693 A261693 := n -> Bits:-Nor(2*n, 2*n):
%p A261693 seq(A261693(n), n=0..81); # _Peter Luschny_, Sep 23 2019
%t A261693 Table[Reverse[2 Range[2^(n - 1)] - 1], {n, 0, 7}] /. {} -> 0 // Flatten (* _Michael De Vlieger_, Oct 05 2015 *)
%o A261693 (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
%Y A261693 Cf. A036563, A005408, A011782, A080079, A099375, A147582, A262617, A261692, A262621.
%Y A261693 Column 1 is A000225. Row sums give A000302, n >= 1.
%K A261693 nonn,easy,tabf
%O A261693 0,3
%A A261693 _Omar E. Pol_, Sep 25 2015
%E A261693 Corrections by _Wolfdieter Lang_, Nov 15 2019