A091023 -id:A091023 - cp's OEIS frontend

A091263 Inverse permutation to A091023.

1, 2, 4, 7, 11, 5, 13, 20, 10, 22, 8, 24, 3, 26, 40, 23, 42, 19, 44, 17, 46, 15, 48, 12, 50, 6, 52, 79, 49, 81, 45, 83, 41, 85, 38, 87, 36, 89, 34, 91, 32, 93, 30, 95, 28, 97, 25, 99, 18, 101, 14, 103, 155, 100, 157, 96, 159, 92, 161, 88, 163, 84, 165, 80, 167, 77, 169, 75

Offset: 1

N. J. A. Sloane, following a suggestion of Leroy Quet, Feb 23 2004

Let S_n be the set of all positive integers except a(1), a(2), ..., a(n-1) and a(n). Let k_n be the number of members of S_n that are less than a(n). (k_n = A091068(n+1).) If k_n >= n, then a(n+1) is the (1+k_n-n)th member of S_n; if k_n < n then a(n+1) is the (k_n+n)th member of S_n. - David Wasserman, Feb 23 2006

Charlie Neder, Table of n, a(n) for n = 1..1000

More terms from John W. Layman, Feb 25 2004

A091052 Record values in A091023.

Original entry on oeis.org

1, 2, 13, 26, 205, 410, 3277, 6554, 52429, 104858, 838861, 1677722, 13421773, 26843546, 214748365, 429496730, 3435973837, 6871947674, 54975581389, 109951162778, 879609302221, 1759218604442, 14073748835533, 28147497671066

Offset: 1

N. J. A. Sloane, Feb 23 2004

Indranil Ghosh, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,17,0,-16)

Cf. A091023, A091053.

LinearRecurrence[{0,17,0,-16},{1,2,13,26},24] (* Indranil Ghosh, Feb 22 2017 *)

a(n)=(3*4^n - (-4)^n + 2*(-1)^n + 6)/20 \\ Charles R Greathouse IV, Feb 22 2017

def A091052(n): return (3*4**n-(-4)**n+2*(-1)**n+6)/20 # Indranil Ghosh, Feb 22 2017

Add 1 to every term of A077854, then take the terms with indices 4k and 4k+3.
(1/20) [3*4^n - (-4)^n + 2*(-1)^n + 6]. - Ralf Stephan, Dec 02 2004
G.f. -x*(2*x-1)*(1+2*x)^2 / ( (x-1)*(4*x-1)*(4*x+1)*(1+x) ). - R. J. Mathar, Jun 10 2013

More terms from David Wasserman, Feb 23 2006

A091053 Where records occur in A091023.

Original entry on oeis.org

1, 2, 3, 6, 9, 21, 33, 78, 129, 309

Offset: 1

N. J. A. Sloane, Feb 23 2004

Cf. A091023, A091052.

A110080 a(1) = 1; skipping over integers occurring earlier in the sequence, count down p(n) (p(n) = n-th prime) from a(n) to get a(n+1). If this is <= 0, instead count up from a(n) p(n) positions (skipping already occurring integers) to get a(n+1).

Original entry on oeis.org

1, 3, 6, 11, 2, 16, 29, 10, 32, 4, 39, 70, 31, 75, 27, 80, 20, 87, 17, 94, 9, 97, 176, 91, 183, 81, 188, 77, 193, 73, 198, 57, 203, 50, 206, 38, 209, 28, 216, 22, 223, 12, 226, 417, 222, 422, 219, 435, 202, 440, 199, 445, 190, 448, 177, 455, 169, 462, 166, 469, 161, 472

Offset: 1

Leroy Quet, Oct 12 2005

If we did not skip earlier occurring integers when counting, we would instead have Cald's sequence (A006509).

			The first 5 terms of the sequence can be plotted on the number line as:
1,2,3,*,*,6,*,*,*,*,11,*,*,*,*,*.
a(5) is 2. Counting p(5) = 11 down from 2 gets a negative integer. So we instead count up 11 positions, skipping the 3, 6 and 11 as we count, to arrive at 16 (which is at the rightmost * of the number line above).
Here is the calculation of the first 6 terms in more detail:
integers i : 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
i at n = ... : 1 5 2 . . 3 . . . .. .4 .. .. .. .. .6 ...
prime p used : - 7 2 . . 3 . . . .. .5 .. .. .. .. 11 ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A091023, A091263, A006509, A111187 (inverse).

import Data.Set (singleton, member, insert)
a110080 n = a110080_list !! (n-1)
a110080_list = 1 : f 1 a000040_list (singleton 1) where
   f x (p:ps) m = y : f y ps (insert y m) where
     y = g x p
     g 0 _ = h x p
     g u 0 = u
     g u v = g (u - 1) (if member (u - 1) m then v else v - 1)
     h u 0 = u
     h u v = h (u + 1) (if member (u + 1) m then v else v - 1)
-- Reinhard Zumkeller, Sep 02 2014

More terms from Klaus Brockhaus and Hans Havermann, Oct 17 2005

A091068 a(0) = 0; for n>0, a(n) = a(n-1) - n if that is >= 0, else a(n) = a(n-1) + n - 1.

Original entry on oeis.org

0, 0, 1, 3, 6, 1, 6, 12, 4, 12, 2, 12, 0, 12, 25, 10, 25, 8, 25, 6, 25, 4, 25, 2, 25, 0, 25, 51, 23, 51, 21, 51, 19, 51, 17, 51, 15, 51, 13, 51, 11, 51, 9, 51, 7, 51, 5, 51, 3, 51, 1, 51, 102, 49, 102, 47, 102, 45, 102, 43, 102, 41, 102, 39, 102, 37, 102, 35, 102, 33

Offset: 0

N. J. A. Sloane, Feb 23 2004

nonn

A sequence equivalent to A091023. Let b(k) = A091023(k) for all k. Suppose we have just assigned b(x) = n. Then a(n-1) is the number of b(k) for 1 <= k < x that are not yet assigned.

			a(4) = 6, 6-5 = 1 >= 0, so a(5) = 1. 1-6 < 0, so a(6) = 1 + 5 = 6.
When in A091023 we assign b(8) = 11, there are 2 unassigned b's to the left, namely b(3) and b(6) and indeed a(10) = 2.

Indranil Ghosh, Table of n, a(n) for n = 0..10000

Cf. A091023, A008344.

t={0};Do[AppendTo[t,If[t[[-1]]-n>=0,t[[-1]]-n,t[[-1]]+n-1]],{n,1,69}];t (* Indranil Ghosh, Feb 22 2017 *)
nxt[{n_,a_}]:={n+1,If[a-n-1>=0,a-n-1,a+n]}; NestList[nxt,{0,0},70][[;;,2]] (* Harvey P. Dale, Jun 14 2023 *)

print("0 0")
i=1
a=0
if a-i>=0:b=a-i
else:b=a+i-1
while i<=100:
    print(str(i)+" "+str(b))
    a=b
    i+=1
    if a-i>=0:b=a-i
    else:b=a+i-1 # Indranil Ghosh, Feb 22 2017

This is a concatenation of blocks: b(-2) = [0], b(-1) = [0], b(0) = [1], b(1) = [3], b(2) = [6 1 6], b(3) = [12 4 12 2 12 0 12], b(4) = [25 10 25 8 25 6 25 4 25 2 25 0 25], ...

Let M(k) be the k-th term of A077854. Then block b(k) for k >= 2 is [M(k), x, M(k), x-2, M(k), x-4, M(k), ..., M(k), 0 or 1, M(k)] where x = M(k-1) - 2. The length of the block is M(k+1) - 2 M(k) + M(k-1) (the second difference of A077854, shifted one place).

A378822 Inverse permutation to A378821.

Original entry on oeis.org

1, 2, 6, 3, 9, 12, 4, 15, 18, 7, 5, 21, 24, 27, 30, 10, 33, 8, 36, 39, 42, 13, 45, 48, 51, 11, 16, 54, 57, 60, 63, 19, 66, 69, 14, 72, 22, 75, 78, 81, 25, 17, 84, 87, 90, 93, 28, 96, 99, 102, 20, 105, 31, 108, 111, 114, 34, 23, 117, 120, 123, 37, 126, 129, 132, 26

Offset: 1

Neal Gersh Tolunsky, Dec 08 2024

nonn

			A378821(6) = 3, so a(3) = 6.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A378821, A081146, A091023.

cp's OEIS Frontend

A091263 Inverse permutation to A091023.

Views

Author

Keywords

Comments

Links

Extensions

A091052 Record values in A091023.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A091053 Where records occur in A091023.

Views

Author

Keywords

Crossrefs

A110080 a(1) = 1; skipping over integers occurring earlier in the sequence, count down p(n) (p(n) = n-th prime) from a(n) to get a(n+1). If this is <= 0, instead count up from a(n) p(n) positions (skipping already occurring integers) to get a(n+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Extensions

A091068 a(0) = 0; for n>0, a(n) = a(n-1) - n if that is >= 0, else a(n) = a(n-1) + n - 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A378822 Inverse permutation to A378821.

Views

Author

Keywords

Examples

Links

Crossrefs