A060000 -id:A060000 - cp's OEIS frontend

A060013 New record highs reached in A060000.

1, 2, 3, 5, 9, 15, 27, 51, 99, 195, 387, 771, 1539, 3075, 6147, 12291, 24579, 49155, 98307, 196611, 393219, 786435, 1572867, 3145731, 6291459, 12582915, 25165827, 50331651, 100663299, 201326595, 402653187, 805306371, 1610612739, 3221225475, 6442450947, 12884901891

Offset: 1

Robert G. Wilson v, Mar 15 2001

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000051, A060000.

h = f = {1, 2}; a = 1; b = 2; Do[ g = Sort[ h ]; If[ g[ [ -1 ] ] + 1 == n, c = a + b, k = 1; While[ g[ [ k ] ] == k, k++ ]; c = k ]; a = b; b = c; h = Append[ h, c ]; If[ c > g[ [ -1 ] ], f = Append[ f, c ] ], { n, 3, 10^4 } ]; f
LinearRecurrence[{3,-2},{1,2,3,5,9,15},40] (* Harvey P. Dale, Dec 12 2018 *)

For n>4: a(n) = 2*a(n-1)-3. For n>3: a(n) = 3*2^(n-3)+3 = 3*A000051(n-3) = A007283(n-3)+3.
a(n+1) = A060000(a(n)+1), a(1) = 1. - Reinhard Zumkeller, Mar 04 2008
G.f.: -x*(x^2-x+1)*(2*x^3+2*x^2-1) / ((x-1)*(2*x-1)). - Colin Barker, Jan 12 2013
E.g.f.: (144*exp(x) + 9*exp(2*x) - 153 - 114*x - 42*x^2 - 12*x^3 - 2*x^4)/48. - Stefano Spezia, Jul 25 2024

Formulae and more terms from Henry Bottomley and Larry Reeves (larryr(AT)acm.org), Mar 19 2001

A099424 Inverse of A060000.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 8, 9, 6, 11, 12, 13, 14, 15, 10, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 16, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 28, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Reinhard Zumkeller, Oct 15 2004

nonn

Permutation of the natural numbers.

Index entries for sequences that are permutations of the natural numbers

A060030 a(1) = 1, a(2) = 2; thereafter a "hole" is defined to be any positive number not in the sequence a(1)..a(n-1) and less than the largest term; if there exists at least one hole, then a(n) is the largest hole, otherwise a(n) = a(n-2) + a(n-1).

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 8, 7, 6, 13, 12, 11, 10, 21, 20, 19, 18, 17, 16, 15, 14, 29, 28, 27, 26, 25, 24, 23, 22, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 93, 92, 91, 90, 89, 88, 87, 86, 85, 84, 83

Offset: 1

William Nelles (wnelles(AT)flashmail.com), Mar 17 2001

A self-inverse permutation of the natural numbers: a(a(n)) = n and a(n) <> n for n > 3. [Reinhard Zumkeller, Apr 29 2012]

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

See A060482 for successive records, A027383 for the final hole-filling values, A016116 for the difference between top and bottom of downward subsequences, A052551 for number of terms in downward subsequences.

Cf. A060000, A000045.

import Data.List (delete)
a060030 n = a060030_list !! (n-1)
a060030_list = 1 : 2 : f 1 2 [3..] where
   f u v ws = y : f v y (delete y ws) where
     y = if null xs then u + v else last xs
     xs = takeWhile (< v) ws
-- Reinhard Zumkeller, Apr 29 2012

a[1] = 1; a[2] = 2;
a[n_] := a[n] = Module[{A, H}, A = Array[a, n-1]; H = Complement[ Range[a[n-1]], A]; If[H != {}, H[[-1]], a[n-2] + a[n-1]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Apr 23 2024 *)

Offset corrected by Reinhard Zumkeller, Apr 29 2012

Name made more explicit by Jean-François Alcover, Apr 23 2024

A138153 If the numbers a(1)...a(n) contain a hole, then a(n+1) is the smallest hole; otherwise a(n+1) = a(n-2) + a(n-1) + a(n).

Original entry on oeis.org

1, 2, 3, 6, 4, 5, 15, 7, 8, 9, 10, 11, 12, 13, 14, 39, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 111, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66

Offset: 1

Jonathan Vos Post, May 04 2008

This is to A060000 as tribonacci A000073 is to Fibonacci A000045. Let H be the set of positive numbers less than a(n) which are not equal to some a(i), i < n. This H is the 'set of holes so far'. If H is nonempty, then define a(n+1) = minimum(H). Otherwise define a(n+1) = a(n-2) + a(n-1) + a(n). Permutation of the natural numbers with inverse not yet in the OEIS.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000040, A000073, A060000, A060013.

a[n_] := a[n] = If[n < 4, n, Block[{s = Array[a, n-1]}, s = Complement[ Range[ Max@s], s]; If[s == {}, a[n - 1] + a[n - 2] + a[n - 3], First[s]]]]; Array[a, 80] (* Giovanni Resta, Jun 20 2016 *)

Data corrected by Giovanni Resta, Jun 20 2016

cp's OEIS Frontend

A060013 New record highs reached in A060000.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A099424 Inverse of A060000.

Views

Author

Keywords

Comments

Links

A060030 a(1) = 1, a(2) = 2; thereafter a "hole" is defined to be any positive number not in the sequence a(1)..a(n-1) and less than the largest term; if there exists at least one hole, then a(n) is the largest hole, otherwise a(n) = a(n-2) + a(n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A138153 If the numbers a(1)...a(n) contain a hole, then a(n+1) is the smallest hole; otherwise a(n+1) = a(n-2) + a(n-1) + a(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions