A060030 -id:A060030 - cp's OEIS frontend

A060482 New record highs reached in A060030.

1, 2, 3, 5, 9, 13, 21, 29, 45, 61, 93, 125, 189, 253, 381, 509, 765, 1021, 1533, 2045, 3069, 4093, 6141, 8189, 12285, 16381, 24573, 32765, 49149, 65533, 98301, 131069, 196605, 262141, 393213, 524285, 786429, 1048573, 1572861, 2097149, 3145725, 4194301, 6291453

Offset: 1

Henry Bottomley, Mar 19 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Cf. A060013, A060030.

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

LinearRecurrence[{1,2,-2},{1,2,3,5,9},50] (* Harvey P. Dale, Sep 11 2016 *)

{ for (n=1, 1000, if (n%2==0, m=n/2; a=2^(m + 1) - 3, m=(n - 1)/2; a=3*2^m - 3); if (n<3, a=n); write("b060482.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 05 2009

a(n) = a(n-1) + 2^((n-1)/2) = 2*a(n-2) + 3 = a(n-1) + A016116(n-1) = A027383(n-1) - 1 = 2*A027383(n-3) + 1 = 4*A052955(n-4) + 1. a(2n) = 2^(n+1) - 3; a(2n+1) = 3*2^n - 3.
From Colin Barker, Jan 12 2013: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) for n > 5.
G.f.: x*(2*x^4-x^2+x+1) / ((x-1)*(2*x^2-1)). (End)
E.g.f.: 1 + x + x^2/2 - 3*cosh(x) + 2*cosh(sqrt(2)*x) - 3*sinh(x) + 3*sinh(sqrt(2)*x)/sqrt(2). - Stefano Spezia, Jul 25 2024

A060000 If the numbers a(1)...a(n) contain a hole, then a(n+1) is the smallest hole; otherwise a(n+1) = a(n-1) + a(n).

Original entry on oeis.org

1, 2, 3, 5, 4, 9, 6, 7, 8, 15, 10, 11, 12, 13, 14, 27, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 51, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 99, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Rainer Rosenthal, Mar 09 2001

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-1) + a(n).
Permutation of the natural numbers with inverse A099424.
A060013(n+1) = a(A060013(n)+1). - Reinhard Zumkeller, Mar 04 2008

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

Cf. A060013, A060030, A000045.

a060000 n = a060000_list !! (n-1)
a060000_list = 1 : 2 : f 1 2 2 [] where
   f x y m []     = z : f y z z [m+1..z-1] where z = x + y
   f x y m (h:hs) = h : f y h m hs
-- Reinhard Zumkeller, Sep 22 2011

h = {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 ], { n, 3, 100} ]; h
(* faster program *) h = {1, 2, 3, 5, 4, 9}; lastSum = 9; Do[AppendTo[h, If[ ++akt < lastSum, akt, ++akt; lastSum = 2*lastSum - 3]], {akt, 5, 100}]; h (* Alfred Heiligenbrunner, Jun 05 2004 *)

a(k)=k-1 if k>=7 and k <> 2^m * 3 + 4; a(k)=(k-1)*2-3 if k>=7 and k == 2^m * 3 + 4. - Alfred Heiligenbrunner, Jun 08 2004

More terms from Robert G. Wilson v and Larry Reeves (larryr(AT)acm.org), Mar 15 2001

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A060482 New record highs reached in A060030.

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