A132664 -id:A132664 - cp's OEIS frontend

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

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

Offset: 1

N. J. A. Sloane, May 02 2000

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)

			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

Suggested by correspondence with Michael Somos.
R. Honsberger, "Ingenuity in Mathematics", Table 10.4 on page 87.

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

A self-inverse permutation of the natural numbers.
Cf. A000027, A020703.
Cf. A132666, A132664, A132665, A132674.
Cf. A056011 (boustrophedon).
Cf. A000122, A000700, A010054, A121373.
Cf. A000124, A000217.
Cf. A061579.

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

(* Program generates dispersion array T of the increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] := Floor[n+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, corrected Jan 26 2025 *)
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 *)

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

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<=ka(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)
G.f. as rectangular array: x*y*(1 - (1 + x)*y + (1 - x + x^2)*y^2)/((1 - x)^3*(1 - y)^3). - Stefano Spezia, Dec 25 2022

A132666 a(1)=1, a(n) = 2*a(n-1) if the minimal positive integer not yet in the sequence is greater than a(n-1), else a(n) = a(n-1)-1.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 10, 9, 8, 7, 14, 13, 12, 11, 22, 21, 20, 19, 18, 17, 16, 15, 30, 29, 28, 27, 26, 25, 24, 23, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 94, 93, 92, 91, 90, 89, 88, 87, 86, 85

Offset: 1

Hieronymus Fischer, Aug 24 2007, Sep 15 2007

Also: a(1)=1, a(n) = maximal positive number < a(n-1) not yet in the sequence, if it exists, else a(n) = 2*a(n-1).
Also: a(1)=1, a(n) = a(n-1)-1, if a(n-1) - 1 > 0 and has not been encountered so far, else a(n) = 2*a(n-1).
A reordering of the natural numbers. The sequence is self-inverse in that a(a(n)) = n.
Almost certainly a duplicate of A132340. - R. J. Mathar, Jun 12 2008

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

For formulas concerning a general parameter p (with respect to the recurrence rule ... a(n)=p*a(n-1) ...) see A132674.
For p=3 to p=10 see A132667 through A132674.
For a similar recurrence rule concerning Fibonacci (A000045) and Lucas numbers (A000032) see A132664 and A132665.
Cf. A027383.

import Data.List (delete)
a132666 n = a132666_list !! (n-1)
a132666_list = 1 : f 1 [2..] where
   f z xs  = y : f y (delete y xs) where
     y | head xs > z = 2 * z
       | otherwise   = z - 1
-- Reinhard Zumkeller, Sep 17 2001

max = 72; f[x_] := Sum[x^(2^k), {k, 0, Ceiling[ Log[2, max]]}]; g[x_] = (x (1 - 2x)/(1 - x) + 2x^2*f'[x^3] + 3/4*(f'[x] - 2x - 1))/(1 - x); Drop[ CoefficientList[ Series[ g[x], {x, 0, max}], x], 1] (* Jean-François Alcover, Dec 01 2011 *)

G.f.: g(x) = (x(1-2x)/(1-x) + 2x^2*f'(x^3) + 3/4*(f'(x)-2x-1))/(1-x) where f(x) = Sum_{k>=0} x^(2^k) and f'(z) = derivative of f(x) at x = z.
a(n) = 5*2^(r/2) - 3 - n, if both r and s are even, else a(n) = 7*2^((s-1)/2) - 3 - n, where r = ceiling(2*log_2((n+2)/3)) and s = ceiling(2*log_2((n+2)/2) - 1).
a(n) = 2^floor(1 + (k+1)/2) + 3*2^floor(k/2) - 3 - n, where k=r, if r is even, else k=s (with respect to r and s above; formally, k = ((r+s) + (r-s)*(-1)^r)/2).
a(n) = A027383(m) + A027383(m+1) + 1 - n, where m:=max{ k | A027383(k) < n }.
a(A027383(n) + 1) = A027383(n+1).
a(A027383(n)) = A027383(n-1) + 1 for n > 0.

A132674 a(1)=1, a(n) = 10*a(n-1) if the minimal positive integer not yet in the sequence is greater than a(n-1), else a(n) = a(n-1) - 1.

Original entry on oeis.org

1, 10, 9, 8, 7, 6, 5, 4, 3, 2, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 110, 109, 108, 107, 106, 105, 104, 103, 102, 101, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90, 89, 88, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62

Offset: 1

Hieronymus Fischer, Aug 24 2007, Sep 15 2007

nonn

Also: a(1)=1, a(n) = maximal positive integer < a(n-1) not yet in the sequence, if it exists, else a(n) = 10*a(n-1).
Also: a(1)=1, a(n) = a(n-1) - 1, if a(n-1) - 1 > 0 and has not been encountered so far, else a(n) = 10*a(n-1).
A permutation of the positive integers. The sequence is self-inverse, in that a(a(n)) = n.

For parameters p=2 to p=9 see A132666 - A132673.

For a similar recurrence rule concerning Fibonacci and Lucas numbers see A132664 and A132665.

The following formulas are given for a general parameter p > 2 considering the recurrence rule above (i.e., a(n) = p*a(n-1)...; p=10 for this sequence).
G.f.: g(x) = (x(1-2x)/(1-x) + px^2*f'(x^((2p-1)/(p-1))) + ((2p-1)/p^2)*(f'(x^(1/(p-1))) - px - 1)/(1-x) where f(x) = Sum_{k>=0} x^(p^k) and f'(z) = derivative of f(x) at x = z.
a(n) = ((3p-1)*p^(r/2) - p - 1)/(p-1) - n if both r and s are even, else a(n) = ((p^2 + 2p - 1)*p^((s-1)/2) - p - 1)/(p-1) - n, where r = ceiling(2*log_p(((p-1)n + p)/(2p-1))) and s = ceiling(2*log_p(((p-1)n + p)/p) - 1).
a(n) = (p^floor(1 + (k+1)/2) + (2p-1)*p^floor(k/2) - p - 1)/(p-1) - n, where k=r if r is odd, else k=s (with respect to r and s above; formally, k = ((r+s) - (r-s)*(-1)^r)/2).

A020703 Take the sequence of natural numbers (A000027) and reverse successive subsequences of lengths 1,3,5,7,...

Original entry on oeis.org

1, 4, 3, 2, 9, 8, 7, 6, 5, 16, 15, 14, 13, 12, 11, 10, 25, 24, 23, 22, 21, 20, 19, 18, 17, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 81, 80, 79, 78, 77

Offset: 1

N. J. A. Sloane, May 02 2000

nonn

Arrange A000027, the natural numbers, into a (square) spiral, say clockwise as shown in A068225. Read the numbers from the resulting counterclockwise spiral of the same shape that also begins with 1 and this sequence results. - Rick L. Shepherd, Aug 04 2006
Contribution from Hieronymus Fischer, Apr 30 2012: (Start)
The sequence may also be defined as follows: a(1)=1, a(n)=m^2 (where m^2 is the least square > a(k) for 1<=kA reordering of the natural numbers.
The sequence is self-inverse in that a(a(n))=n.
(End)

			a(2)=4=2^2, since 2^2 is the least square >2=a(1) and the minimal number not yet in the sequence is 2>1=a(1);
a(8)=6=a(7)-1, since the minimal number not yet in the sequence (=5) is <=7=a(7).

R. Honsberger, "Ingenuity in Mathematics", Table 10.4 on page 87.
Suggested by correspondence with Michael Somos.

Hieronymus Fischer, Table of n, a(n) for n = 1..11131
Index entries for sequences that are permutations of the natural numbers

A self-inverse permutation of the natural numbers.
Cf. A000027, A038722.
Cf. A132666, A132664, A132665, A132674.

Flatten[Table[Range[n^2,(n-1)^2+1,-1],{n,10}]] (* Harvey P. Dale, Jan 10 2016 *)
With[{nn=20},Flatten[Reverse/@TakeList[Range[nn^2],Range[1,nn,2]]]] (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Jan 28 2019 *)

a(n)=local(t); if(n<1,0,t=sqrtint(n-1); 2*(t^2+t+1)-n)

Contribution 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

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

a(n+1)=(sqrt(a(n)-1)+2)^2, if a(n)-1 is a square, a(n+1)=a(n)-1, else.

a(n)=2*(floor(sqrt(n-1))+1)*floor(sqrt(n-1))-n+2. (End)

A132665 a(1)=1, a(2)=3, a(n) = a(n-1) + n if the minimal positive integer not yet in the sequencer is greater than a(n-1), else a(n) = a(n-1)-1.

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 11, 10, 9, 8, 7, 19, 18, 17, 16, 15, 14, 13, 12, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69

Offset: 1

Hieronymus Fischer, Sep 15 2007

nonn

Also: a(1)=1, a(2)=3, a(n) = maximal positive number < a(n-1) not yet in the sequence, if it exists, else a(n) = a(n-1) + n.
Also: a(1)=1, a(2)=3, a(n) = a(n-1) - 1, if a(n-1) - 1 > 0 and has not been encountered so far, else a(n) = a(n-1) + n.
A permutation of the positive integers. The sequence is self-inverse, in that a(a(n)) = n.

Cf. A000045, A104152, A130233.
For an analog concerning Lucas numbers see A132664.
See A132666-A132674 for sequences with a similar recurrence rule.

G.f.: g(x) = (F'(x) - x^2 - 1/(1-x))/(1-x) where F(x) = Sum_{k>=0} x^Fibonacci(k). F(x) is the g.f. of the Fibonacci indicator sequence (see A104162) and F'(x) = derivative of F(x).
a(n) = Fibonacci(Fibonacci_inverse(n+1) + 2) - n - 3 = A000045(A130233(n+1) + 2) - n - 3.
a(n) = A000032(floor(log_phi(sqrt(5)*(n+1) + 1) + 2)) - n - 3, where phi = (1 + sqrt(5))/2 is the golden ratio.
a(n) = A000032(floor(log_phi(sqrt(5)*n + 2*phi) + 2)) - n - 3.

A182194 a(1)=2, a(n)=a(n-1)^2 if the minimal natural number > 1 not yet in the sequence is greater than a(n-1), else a(n)=a(n-1)-1.

Original entry on oeis.org

2, 4, 3, 9, 8, 7, 6, 5, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90, 89, 88, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54

Offset: 1

Hieronymus Fischer, Apr 30 2012

nonn

A reordering of the natural numbers > 1.

The sequence is quasi self-inverse in that a(a(n-1)-1)=n.

			a(2)=4=a(1)^2, since 3>2=a(1) is the minimal number not yet in the sequence (because of a(1)=2);
a(15)=19=a(14)-1, since the minimal number not yet in the sequence (=10) is <=a(14)=20.
a(10^4)=b(8)+b(7)-10^4-2=877.
a(10^6)=b(10)+b(9)-10^6-2= 103539133.

Hieronymus Fischer, Table of n, a(n) for n = 1..10200

Cf. A020703, A038722, A132666, A132664, A132665, A132674, A210882.

a(n)=a(n-1)-1, if a(n-1)-1 > 1 is not in the set {a(k)| 1<=k<=n-1}, else a(n)=a(n-1)^2.
a(a(n)-1)=n+1.
If we define b(1)=2, b(2)=3, b(k)=b(k-2)^2+1, we get the sequence 2, 3, 5, 10, 26, 101, 677, 10202, 458330, 104080805, …. The b(k) are those terms a(n) of the original sequence for which a(n+1)=a(n)^2.
With these b(k) we obtain for k>1:
a(b(k)-2)=b(k-1),
a(b(k)-1)=b(k-1)^2.
a(b(k))=b(k-1)^2 - 1.
a(n)=b(m)+b(m-1)-n-2, where m is the least index such that b(m)>n+1 (valid for n>=1).

A210882 a(1)=1, a(n)=a(n-1)-1 if a(n-1)-1 > 0 is not in the set {a(k)| 1<=k

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 11, 10, 9, 8, 13, 12, 17, 16, 15, 14, 19, 18, 23, 22, 21, 20, 29, 28, 27, 26, 25, 24, 31, 30, 37, 36, 35, 34, 33, 32, 41, 40, 39, 38, 43, 42, 47, 46, 45, 44, 53, 52, 51, 50, 49, 48, 59, 58, 57, 56, 55, 54, 61, 60, 67, 66, 65, 64, 63, 62, 71

Offset: 1

Hieronymus Fischer, Apr 30 2012

nonn

A reordering of the natural numbers.
The sequence is self-inverse in that a(a(n))=n.
If n is a prime, then a(n+1) is the next prime > n. Hence, the subsequence 2, a(2+1), a(a(2+1)+1), a(a(a(2+1)+1)+1), a(a(a(a(2+1)+1)+1)+1), ... generates the sequence of primes A000040.

			a(4)=5, since 5 is the least prime > a(1), a(2), a(3), and the minimal number not yet in the sequence (=4) is greater than 3=a(3).
a(5)=4, since 4 is not in the set {1,2,3,5}={a(k)| 1<=k<n}.
7=p(4)=a(p(3)+1)=a(a(p(2)+1)+1)= a(a(a(p(1)+1)+1)+1)= a(a(a(2+1)+1)+1).

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A020703, A000040, A038722, A132666, A132664, A132665, A132674, A182194.

a(1)=1, a(n)=p (where p is the least prime number > a(k) for 1<=k

a(n)<>n for all n>3.

p(n+1)=a(p(n)+1), where p(n) is the n-th prime.

a(n+1)=p(m+2), if a(n)-1 is the m-th prime, else a(n+1)=a(n)-1, for n>2.

a(n)=p(m)+p(m-1)-n+1, where m is the least index such that p(m)>n-1 (valid for n>2).

Showing 1-7 of 7 results.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A132666 a(1)=1, a(n) = 2*a(n-1) if the minimal positive integer not yet in the sequence is greater than a(n-1), else a(n) = a(n-1)-1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A132674 a(1)=1, a(n) = 10*a(n-1) if the minimal positive integer not yet in the sequence is greater than a(n-1), else a(n) = a(n-1) - 1.

Views

Author

Keywords

Comments

Crossrefs

Formula

A020703 Take the sequence of natural numbers (A000027) and reverse successive subsequences of lengths 1,3,5,7,...

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A132665 a(1)=1, a(2)=3, a(n) = a(n-1) + n if the minimal positive integer not yet in the sequencer is greater than a(n-1), else a(n) = a(n-1)-1.

Views

Author

Keywords

Comments

Crossrefs

Formula

A182194 a(1)=2, a(n)=a(n-1)^2 if the minimal natural number > 1 not yet in the sequence is greater than a(n-1), else a(n)=a(n-1)-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A210882 a(1)=1, a(n)=a(n-1)-1 if a(n-1)-1 > 0 is not in the set {a(k)| 1<=k

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula