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A191450 Dispersion of (3*n-1), read by antidiagonals.

%I A191450 #88 Oct 21 2024 00:51:35
%S A191450 1,2,3,5,8,4,14,23,11,6,41,68,32,17,7,122,203,95,50,20,9,365,608,284,
%T A191450 149,59,26,10,1094,1823,851,446,176,77,29,12,3281,5468,2552,1337,527,
%U A191450 230,86,35,13,9842,16403,7655,4010,1580,689,257,104,38,15,29525
%N A191450 Dispersion of (3*n-1), read by antidiagonals.
%C A191450 Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1. The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...). Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers. The sequence u given by u(n) = {index of the row of D that contains n} is a fractal sequence. In this case s(n) = A016789(n-1), t(n) = A032766(n) [from term A032766(1) onward] and u(n) = A253887(n). [Author's original comment edited by _Antti Karttunen_, Jan 24 2015]
%C A191450 For other examples of such sequences, please see the Crossrefs section.
%H A191450 Antti Karttunen, <a href="/A191450/b191450.txt">Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array</a>
%H A191450 Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/intersp.html">Interspersions and Dispersions</a>.
%H A191450 Clark Kimberling, <a href="http://dx.doi.org/10.1090/S0002-9939-1993-1111434-0">Interspersions and dispersions</a>, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
%H A191450 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A191450 Conjecture: A(n,k) = (3 + (2*A032766(n) - 1)*A000244(k))/6. - _L. Edson Jeffery_, with slight changes by _Antti Karttunen_, Jan 21 2015
%F A191450 a(n) = A254051(A038722(n)). [When both this and transposed array A254051 are interpreted as one-dimensional sequences.] - _Antti Karttunen_, Jan 22 2015
%e A191450 The northwest corner of the square array:
%e A191450   1,  2,  5,  14,  41,  122,  365,  1094,  3281,   9842,  29525,   88574, ...
%e A191450   3,  8, 23,  68, 203,  608, 1823,  5468, 16403,  49208, 147623,  442868, ...
%e A191450   4, 11, 32,  95, 284,  851, 2552,  7655, 22964,  68891, 206672,  620015, ...
%e A191450   6, 17, 50, 149, 446, 1337, 4010, 12029, 36086, 108257, 324770,  974309, ...
%e A191450   7, 20, 59, 176, 527, 1580, 4739, 14216, 42647, 127940, 383819, 1151456, ...
%e A191450   9, 26, 77, 230, 689, 2066, 6197, 18590, 55769, 167306, 501917, 1505750, ...
%e A191450   etc.
%e A191450 The leftmost column is A032766, and each successive column to the right of it is obtained by multiplying the left neighbor on that row by three and subtracting one, thus the second column is (3*1)-1, (3*3)-1, (3*4)-1, (3*6)-1, (3*7)-1, (3*9)-1, ... = 2, 8, 11, 17, 20, 26, ...
%p A191450 A191450 := proc(r, c)
%p A191450     option remember;
%p A191450     if c = 1 then
%p A191450         A032766(r) ;
%p A191450     else
%p A191450         A016789(procname(r, c-1)-1) ;
%p A191450     end if;
%p A191450 end proc: # _R. J. Mathar_, Jan 25 2015
%t A191450 (* Program generates the dispersion array T of increasing sequence f[n] *)
%t A191450 r=40; r1=12; c=40; c1=12;
%t A191450 f[n_] :=3n-1 (* complement of column 1 *)
%t A191450 mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
%t A191450 rows = {NestList[f, 1, c]};
%t A191450 Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
%t A191450 t[i_, j_] := rows[[i, j]];
%t A191450 TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
%t A191450 (* A191450 array *)
%t A191450 Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191450 sequence *)
%t A191450 (* Program by _Peter J. C. Moses_, Jun 01 2011 *)
%o A191450 (PARI) a(n,k)=3^(n-1)*(k*3\2*2-1)\2+1 \\ =3^(n-1)*(k*3\2-1/2)+1/2, but 30% faster. - _M. F. Hasler_, Jan 20 2015
%o A191450 (Scheme)
%o A191450 (define (A191450 n) (A191450bi (A002260 n) (A004736 n)))
%o A191450 (define (A191450bi row col) (if (= 1 col) (A032766 row) (A016789 (- (A191450bi row (- col 1)) 1))))
%o A191450 (define (A191450bi row col) (/ (+ 3 (* (A000244 col) (- (* 2 (A032766 row)) 1))) 6)) ;; Another implementation based on _L. Edson Jeffery_'s direct formula.
%o A191450 ;; _Antti Karttunen_, Jan 21 2015
%Y A191450 Inverse: A254047.
%Y A191450 Transpose: A254051.
%Y A191450 Column 1: A032766.
%Y A191450 Cf. A007051, A057198, A199109, A199113 (rows 1-4).
%Y A191450 Cf. A253887 (row index of n in this array) & A254046 (column index, see also A253786).
%Y A191450 Cf. A000244, A016789, A038722, A048673, A254053, A254103, A254104.
%Y A191450 Examples of other arrays of dispersions: A114537, A035513, A035506, A191449, A191426-A191455.
%K A191450 nonn,tabl
%O A191450 1,2
%A A191450 _Clark Kimberling_, Jun 05 2011
%E A191450 Example corrected and description clarified by _Antti Karttunen_, Jan 24 2015