A322411 - OEIS

%I #18 Oct 06 2019 09:28:49

%S 12,36,56,80,93,117,137,161,185,205,229,242,266,286,310,330,354,367,

%T 391,411,435,459,479,503,516,540,560,584,597,621,641,665,689,709,733,

%U 746,770,790,814,834,858,871,895,915,939,963,983,1007,1020,1044,1064,1088,1112,1132,1156,1169,1193,1213,1237,1257,1281

%N Compound tribonacci sequence with a(n) = A278040(A278041(n)), for n >= 0.

%C The nine sequences A308199, A319967, A319968, A322410, A322409,  A322411,  A322413, A322412, A322414 are based on defining the tribonacci ternary word to start with index 0 (in contrast to the usual definition, in A080843 and A092782, which starts with index 1). As a result these nine sequences differ from the compound tribonacci sequences defined in  A278040, A278041, and A319966-A319972. - _N. J. A. Sloane_, Apr 05 2019

%H L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/10-1/carlitz3-a.pdf">Fibonacci representations of higher order</a>, Fib. Quart., 10 (1972), 43-69.

%F a(n) = A(C(n)) = A(C(n) + 1) - 2 = 4*A(n) + 3*B(n) + 2*n + 8, for n >= 0, with A = A278040 and C = A278041. For a proof see the W. Lang link in A278040, Proposition 9, eq. (50).

%F This formula already follows from Theorem 15 in the 1972 paper by Carlitz et al., which gives that b(c(n)) = a(n) + 2b(n) + 2c(n), where a, b and c are the classical positional sequences of the letters in the tribonacci word. The connection is made by using that c(n) = a(n) + b(n) + n, and by making the translation B(n) = a(n+1)-1, A(n) = b(n+1)-1, C(n) = c(n+1)-1. (Note the switching of A and B!). - _Michel Dekking_, Apr 07 2019

%F a(n+1) = A319969(n)-1 = A003145(A003146(n))-1, the corresponding classical compound tribonacci sequence. - _Michel Dekking_, Apr 04 2019

%Y Cf. A278040, A278041, A322410.

%K nonn,easy

%O 0,1

%A _Wolfdieter Lang_, Jan 02 2019