A254047 -id:A254047 - cp's OEIS frontend

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

1, 2, 3, 5, 8, 4, 14, 23, 11, 6, 41, 68, 32, 17, 7, 122, 203, 95, 50, 20, 9, 365, 608, 284, 149, 59, 26, 10, 1094, 1823, 851, 446, 176, 77, 29, 12, 3281, 5468, 2552, 1337, 527, 230, 86, 35, 13, 9842, 16403, 7655, 4010, 1580, 689, 257, 104, 38, 15, 29525

Offset: 1

Clark Kimberling, Jun 05 2011

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]

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

			The northwest corner of the square array:
  1,  2,  5,  14,  41,  122,  365,  1094,  3281,   9842,  29525,   88574, ...
  3,  8, 23,  68, 203,  608, 1823,  5468, 16403,  49208, 147623,  442868, ...
  4, 11, 32,  95, 284,  851, 2552,  7655, 22964,  68891, 206672,  620015, ...
  6, 17, 50, 149, 446, 1337, 4010, 12029, 36086, 108257, 324770,  974309, ...
  7, 20, 59, 176, 527, 1580, 4739, 14216, 42647, 127940, 383819, 1151456, ...
  9, 26, 77, 230, 689, 2066, 6197, 18590, 55769, 167306, 501917, 1505750, ...
  etc.
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, ...

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array
Clark Kimberling, Interspersions and Dispersions.
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Index entries for sequences that are permutations of the natural numbers

Inverse: A254047.
Transpose: A254051.
Column 1: A032766.
Cf. A007051, A057198, A199109, A199113 (rows 1-4).
Cf. A253887 (row index of n in this array) & A254046 (column index, see also A253786).
Cf. A000244, A016789, A038722, A048673, A254053, A254103, A254104.
Examples of other arrays of dispersions: A114537, A035513, A035506, A191449, A191426-A191455.

A191450 := proc(r, c)
    option remember;
    if c = 1 then
        A032766(r) ;
    else
        A016789(procname(r, c-1)-1) ;
    end if;
end proc: # R. J. Mathar, Jan 25 2015

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=3n-1 (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191450 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191450 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

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

(define (A191450 n) (A191450bi (A002260 n) (A004736 n)))
(define (A191450bi row col) (if (= 1 col) (A032766 row) (A016789 (- (A191450bi row (- col 1)) 1))))
(define (A191450bi row col) (/ (+ 3 (* (A000244 col) (- (* 2 (A032766 row)) 1))) 6)) ;; Another implementation based on L. Edson Jeffery's direct formula.
;; Antti Karttunen, Jan 21 2015

Conjecture: A(n,k) = (3 + (2*A032766(n) - 1)*A000244(k))/6. - L. Edson Jeffery, with slight changes by Antti Karttunen, Jan 21 2015

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

Example corrected and description clarified by Antti Karttunen, Jan 24 2015

A253887 Row index of n in A191450: a(3n) = 2n, a(3n+1) = 2n+1, a(3n+2) = a(n+1).

Original entry on oeis.org

1, 1, 2, 3, 1, 4, 5, 2, 6, 7, 3, 8, 9, 1, 10, 11, 4, 12, 13, 5, 14, 15, 2, 16, 17, 6, 18, 19, 7, 20, 21, 3, 22, 23, 8, 24, 25, 9, 26, 27, 1, 28, 29, 10, 30, 31, 11, 32, 33, 4, 34, 35, 12, 36, 37, 13, 38, 39, 5, 40, 41, 14, 42, 43, 15, 44, 45, 2, 46, 47, 16, 48, 49, 17, 50, 51, 6, 52, 53, 18, 54, 55, 19, 56, 57, 7

Offset: 1

Antti Karttunen, Jan 22 2015

nonn

a(n) gives the row index of n in square array A191450, or equally, the column index of n in A254051.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Odd bisection of A126760.
Cf. A254046 (the corresponding column index).
Cf. A003602, A032766, A064216, A253786, A253893, A249746.
Cf. A191450, A254051, A254047, A254052, A254104.

def a(n):
    if n%3==0: return 2*n//3
    elif n%3==1: return 2*(n - 1)//3 + 1
    else: return a((n - 2)//3 + 1)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

a(3n) = 2n, a(3n+1) = 2n+1, a(3n+2) = a(n+1).
a(n) = A126760(2n-1).
a(n) = A249746(A003602(A064216(n))). - Antti Karttunen, Feb 04 2015

A254046 Column index of n in A191450: a(3n) = 1, a(3n+1) = 1, a(3n+2) = 1 + a(n+1).

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 6, 1, 1, 2, 1, 1, 2

Offset: 1

Antti Karttunen, Jan 24 2015

Equally, the row index of n in A254051.

a(n) is the 3-adic valuation of A087289(n-1). - Fred Daniel Kline, Jan 11 2017

Antti Karttunen, Table of n, a(n) for n = 1..9842

One more than A253786.
Cf. A253887 (the corresponding row index).
Cf. A191450, A254047, A254051, A254052.
Odd bisection of A051064.
Cf. A007949, A087289.

With[{nmax=200},IntegerExponent[6Range[nmax]-3,3]] (* Paolo Xausa, Nov 10 2023 *)

a(3n) = 1, a(3n+1) = 1, a(3n+2) = 1 + a(n+1).
a(n) = A253786(n) + 1.
a(n) = A253786(3n-1). - Cyril Damamme, Aug 04 2015
a(n) = A051064(2n-1), i.e., the 3-adic valuation of 6n-3. - Cyril Damamme, Aug 04 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/2. - Amiram Eldar, Nov 16 2023

A254052 Inverse permutation to A254051.

Original entry on oeis.org

1, 3, 2, 4, 6, 7, 11, 5, 16, 22, 8, 29, 37, 10, 46, 56, 12, 67, 79, 17, 92, 106, 9, 121, 137, 23, 154, 172, 30, 191, 211, 13, 232, 254, 38, 277, 301, 47, 326, 352, 15, 379, 407, 57, 436, 466, 68, 497, 529, 18, 562, 596, 80, 631, 667, 93, 704, 742, 24, 781, 821, 107, 862, 904, 122, 947, 991, 14, 1036, 1082, 138, 1129, 1177, 155, 1226, 1276, 31

Offset: 1

Antti Karttunen, Jan 24 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A254051.
Cf. A253887, A254046, A254047.
Related permutations: A064216, A254054.

(define (A254052 n) (let ((x (A254046 n)) (y (A253887 n))) (* (/ 1 2) (- (expt (+ x y) 2) x y y y -2))))

As a composition of other permutations:

a(n) = A254054(A064216(n)).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Scheme

Formula

Extensions

A253887 Row index of n in A191450: a(3n) = 2n, a(3n+1) = 2n+1, a(3n+2) = a(n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Formula

A254046 Column index of n in A191450: a(3n) = 1, a(3n+1) = 1, a(3n+2) = 1 + a(n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A254052 Inverse permutation to A254051.

Views

Author

Keywords

Links

Crossrefs

Programs

Scheme

Formula