A191450 - OEIS

%I #85 Sep 09 2017 19:37:47

%S 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 149,59,26,10,1094,1823,851,446,176,77,29,12,3281,5468,2552,1337,527,

%U 230,86,35,13,9842,16403,7655,4010,1580,689,257,104,38,15,29525

%N Dispersion of (3n-1), read by antidiagonals.

%C 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 For other examples of such sequences, please see the Crossrefs section.

%H Antti Karttunen, <a href="/A191450/b191450.txt">Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array</a>

%H Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/intersp.html">Interspersions and Dispersions</a>.

%H 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 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F 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 a(n) = A254051(A038722(n)). [When both this and transposed array A254051 are interpreted as one-dimensional sequences.] - _Antti Karttunen_, Jan 22 2015

%e The northwest corner of the square array:

%e 1,  2,  5,  14,  41,  122,  365,  1094,  3281,   9842,  29525,   88574, ...

%e 3,  8, 23,  68, 203,  608, 1823,  5468, 16403,  49208, 147623,  442868, ...

%e 4, 11, 32,  95, 284,  851, 2552,  7655, 22964,  68891, 206672,  620015, ...

%e 6, 17, 50, 149, 446, 1337, 4010, 12029, 36086, 108257, 324770,  974309, ...

%e 7, 20, 59, 176, 527, 1580, 4739, 14216, 42647, 127940, 383819, 1151456, ...

%e 9, 26, 77, 230, 689, 2066, 6197, 18590, 55769, 167306, 501917, 1505750, ...

%e etc.

%e 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 := proc(r, c)

%p     option remember;

%p     if c = 1 then

%p         A032766(r) ;

%p     else

%p         A016789(procname(r, c-1)-1) ;

%p     end if;

%p end proc: # _R. J. Mathar_, Jan 25 2015

%t (* Program generates the dispersion array T of increasing sequence f[n] *)

%t r=40; r1=12; c=40; c1=12;

%t f[n_] :=3n-1 (* complement of column 1 *)

%t mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]

%t rows = {NestList[f, 1, c]};

%t Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];

%t t[i_, j_] := rows[[i, j]];

%t TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]

%t (* A191450 array *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191450 sequence *)

%t (* Program by _Peter J. C. Moses_, Jun 01 2011 *)

%o (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 (Scheme)

%o (define (A191450 n) (A191450bi (A002260 n) (A004736 n)))

%o (define (A191450bi row col) (if (= 1 col) (A032766 row) (A016789 (- (A191450bi row (- col 1)) 1))))

%o (define (A191450bi row col) (/ (+ 3 (* (A000244 col) (- (* 2 (A032766 row)) 1))) 6)) ;; Another implementation based on _L. Edson Jeffery_'s direct formula.

%o ;; _Antti Karttunen_, Jan 21 2015

%Y Inverse: A254047.

%Y Transpose: A254051.

%Y Column 1: A032766.

%Y Cf. A007051, A057198, A199109, A199113 (rows 1-4).

%Y Cf. A253887 (row index of n in this array) & A254046 (column index, see also A253786).

%Y Cf. A000244, A016789, A038722, A048673, A254053, A254103, A254104.

%Y Examples of other arrays of dispersions: A114537, A035513, A035506, A191449, A191426-A191455.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jun 05 2011

%E Example corrected and description clarified by _Antti Karttunen_, Jan 24 2015