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A038722 Take the sequence of natural numbers (A000027) and reverse successive subsequences of lengths 1,2,3,4,... . 28
1, 3, 2, 6, 5, 4, 10, 9, 8, 7, 15, 14, 13, 12, 11, 21, 20, 19, 18, 17, 16, 28, 27, 26, 25, 24, 23, 22, 36, 35, 34, 33, 32, 31, 30, 29, 45, 44, 43, 42, 41, 40, 39, 38, 37, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 78, 77, 76 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

The rectangular array having A038722 as antidiagonals is the transpose of the rectangular array given by A000217. Column 1 of array A038722 is A000124 (central polygonal numbers). Array A038722 is the dispersion of the complement of A000124. - Clark Kimberling, Apr 05 2003

a(n) is the smallest number not yet in the sequence such that n + a(n) is one more than a square. - Franklin T. Adams-Watters, Apr 06 2009

From Hieronymus Fischer, Apr 30 2012: (Start)

. A reordering of the natural numbers.

. The sequence is self-inverse in that a(a(n)) = n.

. Also: a(1) = 1, a(n) = m (where m is the least triangular number > a(k) for 1 <= k < n), if the minimal natural number not yet in the sequence is greater than a(n-1), otherwise a(n) = a(n-1)-1. (End)

REFERENCES

Suggested by correspondence with Michael Somos.

R. Honsberger, "Ingenuity in Mathematics", Table 10.4 on page 87.

LINKS

Hieronymus Fischer, Table of n, a(n) for n = 1..11401

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for sequences that are permutations of the natural numbers

FORMULA

a(n) = (sqrt(2n-1) - 1/2)*(sqrt(2n-1) + 3/2) - n + 2 = A061579(n-1) + 1. Seen as a square table by antidiagonals, T(n, k) = k + (n+k-1)*(n+k-2)/2, i.e., the transpose of A000027 as a square table.

G.f.: g(x) = (x/(1-x))*(psi(x) - x/(1-x) + 2*Sum_{k>=0} k*x^(k*(k+1)/2)) where psi(x) = Sum_{k>=0} x^(k*(k+1)/2) = (1/2)*x^(-1/8)*theta_2(0,x^(1/2) is a Ramanujan theta function. - Hieronymus Fischer, Aug 08 2007

a(n) = floor(sqrt(2*n) + 1/2)^2 - n + 1. - Clark Kimberling, Jun 05 2011; corrected by Paul D. Hanna, Jun 27 2011

From Hieronymus Fischer, Apr 30 2012: (Start)

. a(n) = a(n-1)-1, if a(n-1)-1 > 0 is not in the set {a(k)| 1<=k<n}, otherwise a(n) = m, where m is the least triangular number not yet in the sequence.

. a(n) = n for n = 2k(k+1)+1, k >= 0.

. a(n+1) = (m+2)(m+3)/2, if 8a(n)-7 is a square of an odd number, otherwise a(n+1) = a(n)-1, where m = (sqrt(8a(n)-7)-1)/2.

. a(n) = ceiling((sqrt(8n+1)-1)/2)^2 - n + 1. (End)

EXAMPLE

The rectangular array view is

   1    2    4    7   11   16   22   29   37   46

   3    5    8   12   17   23   30   38   47   57

   6    9   13   18   24   31   39   48   58   69

  10   14   19   25   32   40   49   59   70   82

  15   20   26   33   41   50   60   71   83   96

  21   27   34   42   51   61   72   84   97  111

  28   35   43   52   62   73   85   98  112  127

  36   44   53   63   74   86   99  113  128  144

  45   54   64   75   87  100  114  129  145  162

  55   65   76   88  101  115  130  146  163  181

MATHEMATICA

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

r=40; r1=12; c=40; c1=12; f[n_] := Floor[1/2+Sqrt[2n]]

  (* 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, r1}, {j, 1, c1}]]

(* A038722 array *)

Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]

(* A038722 sequence *) (* Clark Kimberling, Jun 06 2011 *)

Table[ n, {m, 12}, {n, m (m + 1)/2, m (m - 1)/2 + 1, -1}] // Flatten (* or *)

Table[ Ceiling[(Sqrt[8 n + 1] - 1)/2]^2 - n + 1, {n, 78}] (* Robert G. Wilson v, Jun 27 2014 *)

With[{nn=20}, Reverse/@TakeList[Range[(nn(1+nn))/2], Range[nn]]//Flatten] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Dec 14 2017 *)

PROG

(PARI) a(n)=local(t=floor(1/2+sqrt(2*n))); if(n<1, 0, t^2-n+1) /* Paul D. Hanna */

(Haskell)

a038722 n = a038722_list !! (n-1)

a038722_list = concat a038722_tabl

a038722_tabl = map reverse a000027_tabl

a038722_row n = a038722_tabl !! (n-1)

-- Reinhard Zumkeller, Nov 08 2013

CROSSREFS

A self-inverse permutation of the natural numbers.

Cf. A000027, A020703.

Cf. A132666, A132664, A132665, A132674.

Cf. A056011 (boustrophedon).

Sequence in context: A194878 A194910 A194909 * A277881 A145522 A283939

Adjacent sequences:  A038719 A038720 A038721 * A038723 A038724 A038725

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, May 02 2000

STATUS

approved

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Last modified November 12 09:29 EST 2019. Contains 329054 sequences. (Running on oeis4.)