A060013 -id:A060013 - 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

A147595 a(n) is the number whose binary representation is A138144(n).

Original entry on oeis.org

1, 3, 7, 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

Offset: 1

Omar E. Pol, Nov 08 2008

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

Cf. A138144, A145641, A147537, A147538, A147539, A147540, A147590, A147596, A147597.

[1,3,7] cat [3*(1+2^(n-2)): n in [4..40]]; // G. C. Greubel, Oct 25 2022

LinearRecurrence[{3,-2},{1,3,7,15,27},40] (* Harvey P. Dale, Nov 30 2020 *)

Vec(-x*(2*x^2-1)*(2*x^2+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 15 2013

[1,3,7]+[3*(1+2^(n-2)) for n in range(4,40)] # G. C. Greubel, Oct 25 2022

a(n) = A060013(n+2), n > 3. - R. J. Mathar, Feb 05 2010
a(n+4) = 3*(2^(n+2) + 1), n >= 0. - Brad Clardy, Apr 03 2013
From Colin Barker, Sep 15 2013: (Start)
a(n) = 3*(4 + 2^n)/4 for n>3.
a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: x*(1-2*x^2)*(1+2*x^2) / ((1-x)*(1-2*x)). (End)
E.g.f.: (3/4)*(4*exp(x) + exp(2*x)) - (15/4) - 7*x/2 - 3*x^2/2 - x^3/3. - G. C. Greubel, Oct 25 2022

Extended by R. J. Mathar, Feb 05 2010

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

A195116 a(n) = (2+3^n)*(3+2^n).

Original entry on oeis.org

12, 25, 77, 319, 1577, 8575, 48977, 286759, 1699817, 10137775, 60645377, 363332599, 2178384857, 13065493375, 78378545777, 470228096839, 2821239178697, 16927047127375, 101561119454177, 609363227843479, 3656168902513337, 21936982025631775

Offset: 0

Bruno Berselli, Sep 09 2011

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-47,72,-36).

Cf. A060013 ((1+2^n)*(2+1) with n>3).

Cf. A021029 (for the recurrence).

Magma
```
[(2+3^n)*(3+2^n): n in [0..21]];
```

Table[(2 + 3^n) (3 + 2^n), {n, 0, 30}] (* Vincenzo Librandi, Mar 26 2013 *)

for(n=0, 21, print1((2+3^n)*(3+2^n)", "));

def a(n): return (2+3**n)*(3+2**n)
print([a(n) for n in range(23)]) # Michael S. Branicky, Dec 25 2021

G.f.: (12-119*x+341*x^2-294*x^3)/((1-x)*(1-2*x)*(1-3*x)*(1-6*x)).

Sum_{i=0..n} a(i) = (1/10)*(12*6^n+45*3^n+40*2^n+60*n+23).

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A060482 New record highs reached in A060030.

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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).

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A147595 a(n) is the number whose binary representation is A138144(n).

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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).

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A195116 a(n) = (2+3^n)*(3+2^n).

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