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A038722 Take the sequence of natural numbers (A000027) and reverse successive subsequences of lengths 1,2,3,4,... . 27
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), else 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}, else 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, else 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 *)

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 19 11:04 EST 2017. Contains 294936 sequences.