A001611 -id:A001611 - cp's OEIS frontend

A212262 a(n) = 3^n + Fibonacci(n).

1, 4, 10, 29, 84, 248, 737, 2200, 6582, 19717, 59104, 177236, 531585, 1594556, 4783346, 14349517, 43047708, 129141760, 387423073, 1162265648, 3486791166, 10460364149, 31381077320, 94143207484, 282429582849, 847288684468, 2541865949722, 7625597681405

Offset: 0

Bruno Berselli, May 08 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2,-3).

Cf. A000045, A000244, A001611, A117591, A212323.

Magma
```
[3^n+Fibonacci(n): n in [0..27]];
```
Mathematica
```
Table[3^n + Fibonacci[n], {n, 0, 27}]
```

for(n=0, 27, print1(3^n+fibonacci(n)", "));

[3^n +fibonacci(n) for n in (0..30)] # G. C. Greubel, Jul 05 2021

G.f.: (1-2*x)*(1+2*x)/((1-3*x)*(1-x-x^2)).

A226271 Index of 1/n in the Fibonacci (or rabbit) ordering of the positive rationals.

Original entry on oeis.org

1, 4, 6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766, 10947, 17712, 28658, 46369, 75026, 121394, 196419, 317812, 514230, 832041, 1346270, 2178310, 3524579, 5702888, 9227466, 14930353, 24157818, 39088170, 63245987, 102334156, 165580142

Offset: 1

M. F. Hasler, Jun 01 2013

The Fibonacci ordering of the rationals (cf. A226080) is the sequence of rationals produced from the initial vector [1] by appending iteratively the new rationals obtained by applying the map t-> (t+1, 1/t) to the vector (cf. example).

Apart from initial terms, the same as A001611=(1, 2, 2, 3, 4, 6,...), A020706=(4,6,9,...), A048577=(3, 4, 6, ...), A000381=(2, 3, 4, ...).

			Starting from the vector [1] and applying the map t->(1+t,1/t), we get [2,1] (but ignore the number 1 which already occurred earlier), then [3,1/2], then [4,1/3,3/2,2] (where we ignore 2), etc. This yields the sequence (1,2,3,1/2,4,1/3,3/2,5,1/4,4/3,5/2,2/3,....) The unit fractions 1=1/1, 1/2, 1/3, ... occur at positions 1,4,6,9,...

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

LinearRecurrence[{2,0,-1},{1,4,6,9},40] (* Harvey P. Dale, Feb 04 2016 *)

PARI
```
A226271(n)=if(n>1,fibonacci(n+2))+1
```

{k=1;print1(s=1,",");U=Set(g=[1]);for(n=1,9,U=setunion(U,Set(g=select(f->!setsearch(U,f), concat(apply(t->[t+1,k/t],g))))); for(i=1,#g,numerator(g[i])==1&&print1(s+i","));s+=#g)} \\ for illustrative purpose

Vec(-x*(2*x^3+2*x^2-2*x-1)/((x-1)*(x^2+x-1)) + O(x^50)) \\ Colin Barker, May 11 2016

a(n) = 2*a(n-1)-a(n-3) for n>4. G.f.: -x*(2*x^3+2*x^2-2*x-1) / ((x-1)*(x^2+x-1)). - Colin Barker, Jun 03 2013
a(n) = 1+(2^(-1-n)*((1-sqrt(5))^n*(-3+sqrt(5))+(1+sqrt(5))^n*(3+sqrt(5))))/sqrt(5) for n>1. - Colin Barker, May 11 2016
E.g.f.: -2*(1 + x) + exp(x) + (3*sqrt(5)*sinh(sqrt(5)*x/2) + 5*cosh(sqrt(5)*x/2))*exp(x/2)/5. - Ilya Gutkovskiy, May 11 2016

A358090 Partial inventory of positions as an irregular table; rows 1 and 2 contain 1, for n > 2, row n contains the 1-based positions of 1's, followed by the positions of 2's, 3's, etc. in rows n-2 and n-1 flattened.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 3, 2, 4, 5, 1, 4, 2, 6, 3, 5, 7, 8, 1, 6, 3, 8, 2, 10, 4, 7, 5, 11, 9, 12, 13, 1, 9, 3, 13, 5, 11, 2, 15, 6, 17, 4, 10, 7, 16, 8, 12, 19, 14, 18, 20, 21, 1, 14, 5, 20, 3, 16, 7, 24, 9, 18, 2, 22, 8, 26, 4, 28, 11, 15, 6, 25, 10, 19, 12, 29, 13, 17, 31, 21, 27, 23, 32, 30, 33, 34

Offset: 1

Rémy Sigrist, Oct 30 2022

This sequence is a variant of A356784; here we consider two prior rows, there all prior rows, hence the term "partial" in the name.

The n-th row contains A000045(n) terms, and is a permutation of 1..A000045(n).

			Table begins:
    1,
    1,
    1, 2,
    1, 2, 3,
    1, 3, 2, 4, 5,
    1, 4, 2, 6, 3, 5, 7, 8,
    1, 6, 3, 8, 2, 10, 4, 7, 5, 11, 9, 12, 13,
    ...
For n = 7:
- the terms in rows 5 and 6 are: 1, 3, 2, 4, 5, 1, 4, 2, 6, 3, 5, 7, 8,
- positions of 1's are: 1, 6,
- positions of 2's are: 3, 8,
- positions of 3's are: 2, 10,
- positions of 4's are: 4, 7,
- positions of 5's are: 5, 11,
- positions of 6's are: 9,
- positions of 7's are: 12,
- positions of 8's are: 13,
- so row 7 is: 1, 6, 3, 8, 2, 10, 4, 7, 5, 11, 9, 12, 13.

Rémy Sigrist, Table of n, a(n) for n = 1..10945
Rémy Sigrist, Scatterplot of the first 832039 terms
Rémy Sigrist, PARI program

See A358120 for a similar sequence.

Cf. A000045, A001611, A342585, A356784, A358123.

PARI
```
See Links section.
```

T(n, 1) = 1.

T(n, 2) = A001611(n-2) for n > 2.

A358120 Partial inventory of positions as an irregular table; rows 1 and 2 contain 1, for n > 2, row n contains the 1-based positions of 1's, followed by the positions of 2's, 3's, etc. in rows n-1 and n-2 flattened.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 1, 4, 3, 5, 2, 1, 6, 5, 8, 3, 7, 2, 4, 1, 9, 7, 13, 5, 11, 8, 10, 3, 12, 2, 6, 4, 1, 14, 11, 20, 9, 18, 13, 21, 5, 16, 12, 15, 3, 19, 7, 17, 2, 8, 6, 10, 4, 1, 22, 17, 32, 13, 30, 21, 34, 9, 26, 19, 33, 15, 24, 18, 28, 5, 23, 20, 29, 3, 27, 11, 31, 7, 25, 2, 12, 10, 16, 6, 14, 4, 8

Offset: 1

Rémy Sigrist, Oct 30 2022

The n-th row contains A000045(n) terms, and is a permutation of 1..A000045(n).

			Table begins:
    1,
    1,
    1, 2,
    1, 3, 2,
    1, 4, 3, 5, 2,
    1, 6, 5, 8, 3, 7, 2, 4,
    1, 9, 7, 13, 5, 11, 8, 10, 3, 12, 2, 6, 4,
    ...
For n = 7:
- terms in rows 6 and 5 are: 1, 6, 5, 8, 3, 7, 2, 4, 1, 4, 3, 5, 2,
- positions of 1's are: 1, 9,
- positions of 2's are: 7, 13,
- positions of 3's are: 5, 11,
- positions of 4's are: 8, 10,
- positions of 5's are: 3, 12,
- positions of 6's are: 2,
- positions of 7's are: 6,
- positions of 8's are: 4,
- so row 7 is: 1, 9, 7, 13, 5, 11, 8, 10, 3, 12, 2, 6, 4.

Rémy Sigrist, Table of n, a(n) for n = 1..10945
Rémy Sigrist, PARI program
Rémy Sigrist, Scatterplot of the first 832039 terms

See A358090 for a similar sequence.

Cf. A000045, A001611, A342585, A356784, A358124.

PARI
```
See Links section.
```

T(n, 1) = 1.

T(n, 2) = A001611(n-1) for n > 2.

A166876 a(n) = a(n-1) + Fibonacci(n), a(1)=1983.

Original entry on oeis.org

1983, 1984, 1986, 1989, 1994, 2002, 2015, 2036, 2070, 2125, 2214, 2358, 2591, 2968, 3578, 4565, 6162, 8746, 12927, 19692, 30638, 48349, 77006, 123374, 198399, 319792, 516210, 834021, 1348250, 2180290, 3526559, 5704868, 9229446, 14932333, 24159798

Offset: 1

Geoff Ahiakwo, Oct 22 2009

Starting at some a(1)=s and creating further terms with the recurrence a(n)=a(n-1)+A000045(n) defines a family of sequences with recurrences a(n)= 2*a(n-1) -a(n-3).
The generating functions are x*( s+(1-s)*x+(1-s)*x^2 )/((1-x) * (1-x-x^2)).
The terms are a(n) = A000045(n+2)+s-2 = s + A001911(n-1) = (2*s+1+k)/2 where k=A166863(n-1), n>=1.
Examples: Up to offsets, s=1 yields A000071, s=2 yields A000045 shifted left thrice, s=3 yields A001611 shifted left thrice, s=4 yields A018910.
I appreciate the editing by R. J. Mathar. However I would like further analysis of the following formula. The sequence which I call GAP can have any integer as its first term, not just 1983. Thus a(1) can be 0, 1, 2, 3,... Then a(2) is always a(1)+ 1, while a(3) is a(1) + k(n)/2; where k(n) = k(n-2)+ k(n-1)+4 (This is a separate sequence submitted for consideration). [Geoff Ahiakwo, Nov 19 2009]

			For s=1983, n=3, we have k= A166863(2)= 5, a(3) = (2s+1+k)/2 = (2*1983+1+5)/2 = 1986.
For n=3, a(3)= a(1)+ k(3)/2 = a(1)+ [K(3-2)+ k(3-1)]/2 + 2 = a(1)+ 1 + 2 thus if a(1)is 0, a(3)= 3; if a(1)= 5, a(3)= 8; if a(1)=1983, a(3)= 1986, etc. [_Geoff Ahiakwo_, Nov 19 2009]

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1)

Cf. A001911, A000071

LinearRecurrence[{2, 0, -1}, {1983, 1984, 1986}, 100] (* G. C. Greubel, May 27 2016 *)

a(n) = 2*a(n-1) - a(n-3).
G.f.: x*(-1983 + 1982*x + 1982*x^2)/((1-x)*(x^2+x-1)).
Let a(n)= a(1)+ k(n)/2, then G.f.: k(n)= k(n-2)+ k(n-1) + 4. - Geoff Ahiakwo, Nov 19 2009

Definition and comments edited by R. J. Mathar, Oct 26 2009

A187893 a(0)=1, a(1)=4, a(n) = a(n-1) + a(n-2) - 1.

Original entry on oeis.org

1, 4, 4, 7, 10, 16, 25, 40, 64, 103, 166, 268, 433, 700, 1132, 1831, 2962, 4792, 7753, 12544, 20296, 32839, 53134, 85972, 139105, 225076, 364180, 589255, 953434, 1542688, 2496121, 4038808, 6534928, 10573735, 17108662, 27682396, 44791057, 72473452, 117264508

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 15 2011

Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A000071, A001611, A001612, A039834, A187890.

Join[{a=1,b=4},Table[c=a+b-1;a=b;b=c,{n,100}]]
LinearRecurrence[{2,0,-1},{1,4,4},40] (* Harvey P. Dale, Jun 06 2020 *)

a(n)=3*fibonacci(n)+1 \\ Charles R Greathouse IV, Oct 29 2016

G.f.: -x*(-1-2*x+4*x^2) / ( (x-1)*(x^2+x-1) ). - R. J. Mathar, Mar 15 2011

a(n) = 1+3*|A039834(n)| = 1+3*A000045(n). - R. J. Mathar, Mar 15 2011

A228078 a(n) = 2^n - Fibonacci(n) - 1.

Original entry on oeis.org

0, 0, 2, 5, 12, 26, 55, 114, 234, 477, 968, 1958, 3951, 7958, 16006, 32157, 64548, 129474, 259559, 520106, 1041810, 2086205, 4176592, 8359950, 16730847, 33479406, 66987470, 134021309, 268117644, 536356682, 1072909783, 2146137378, 4292788986, 8586410013

Offset: 0

Reinhard Zumkeller, Aug 15 2013

a(n+1) = sum of n-th row of the triangle in A228074.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Tamás Szakács, Linear recursive sequences and factorials, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See p. 30.
Index entries for linear recurrences with constant coefficients, signature (4,-4,-1,2).

Haskell
```
a228078 = subtract 1 . a099036
```

[2^n - Fibonacci(n) - 1: n in [0..40]]; // Vincenzo Librandi, Aug 16 2013

Table[(2^n - Fibonacci[n] - 1), {n, 0, 40}] (* Vincenzo Librandi, Aug 16 2013 *)

concat([0,0], Vec(x^2*(3*x-2)/((x-1)*(2*x-1)*(x^2+x-1)) + O(x^100))) \\ Colin Barker, Mar 20 2015

a(n) = A000079(n) - A000045(n) - 1 = A000225(n) - A000045(n) = A000079(n) - A001611(n) = A099036(n) - 1.
From Colin Barker, Mar 20 2015: (Start)
a(n) = 4*a(n-1)-4*a(n-2)-a(n-3)+2*a(n-4) for n>3.
G.f.: x^2*(3*x-2) / ((x-1)*(2*x-1)*(x^2+x-1)). (End)
a(n) = (-1+2^n+(((1-sqrt(5))/2)^n-((1+sqrt(5))/2)^n)/sqrt(5)). - Colin Barker, Nov 02 2016

A011369 a(n+1) = a(n) - F(n) if > 0, otherwise a(n) + F(n), where F() are Fibonacci numbers; a(0) = 0.

Original entry on oeis.org

0, 0, 1, 2, 4, 1, 6, 14, 1, 22, 56, 1, 90, 234, 1, 378, 988, 1, 1598, 4182, 1, 6766, 17712, 1, 28658, 75026, 1, 121394, 317812, 1, 514230, 1346270, 1, 2178310, 5702888, 1, 9227466, 24157818, 1, 39088170, 102334156, 1, 165580142, 433494438, 1, 701408734, 1836311904

Offset: 0

N. J. A. Sloane

Seiichi Manyama, Table of n, a(n) for n = 0..4000
Index entries for linear recurrences with constant coefficients, signature (1,0,4,-4,0,1,-1).

Cf. A000045, A001611, A014437.

Module[{n = 0, f}, NestList[If[(f = Fibonacci[n++]) < #, # - f, # + f] &, 0, 49]] (* Paolo Xausa, Nov 08 2024 *)
Flatten[Join[{0, 0}, Table[{1, Fibonacci[{k, k+2}] + 1}, {k, 2, 49, 3}]]] (* Paolo Xausa, Nov 08 2024 *)
LinearRecurrence[{1, 0, 4, -4, 0, 1, -1}, {0, 0, 1, 2, 4, 1, 6, 14, 1}, 50] (* Paolo Xausa, Nov 08 2024 *)

a(n) = if (n==0, 0, my(d=a(n-1)-fibonacci(n-1)); if (d>0, d, d+2*fibonacci(n-1))) \\ Michel Marcus, Dec 29 2018

a(n) = if (n<=1, 0, my(m=(n % 3)); if (m==0, fibonacci(n-1)+1, if (m==1, fibonacci(n)+1, 1))); \\ Michel Marcus, Dec 29 2018

a(n) = 0, if n <= 1; F(n-1)+1, if n == 0 (mod 3); F(n)+1, if n == 1 (mod 3); 1, if n == 2 (mod 3). - David W. Wilson; corrected by Michel Marcus, Dec 29 2018
For n>=1, a(n) = F(0)<+>F(1)<+>...<+>F(n-1), where operation <+> is defined in comment in A245618. - Vladimir Shevelev, Nov 05 2014
Empirical g.f.: -x^2*(2*x^6 - x^4 + 7*x^3 - 2*x^2 - x - 1) / ((x-1)*(x^2 + x - 1)*(x^4 - x^3 + 2*x^2 + x + 1)). - Colin Barker, Nov 06 2014

Name edited by Michel Marcus, Dec 29 2018

A052661 E.g.f. (2-3x)/((1-x)(1-x-x^2)).

Original entry on oeis.org

2, 1, 4, 12, 72, 480, 4320, 45360, 564480, 7983360, 127008000, 2235340800, 43110144000, 902918016000, 20399720140800, 494300911104000, 12783824621568000, 351419178958848000, 10230993181753344000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Harvey P. Dale, Table of n, a(n) for n = 0..417
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 608

spec := [S,{S=Union(Sequence(Z), Sequence(Prod(Z,Z, Sequence(Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[(2-3x)/((1-x)(1-x-x^2)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 25 2023 *)

E.g.f.: -(-2+3*x)/(-1+x+x^2)/(-1+x)
Recurrence: {a(1)=1, a(2)=4, a(0)=2, (n^3+6*n^2+11*n+6)*a(n)+(-2*n-6)*a(n+2)+a(n+3)=0}
(1+Sum(1/5*(-1+3*_alpha)*_alpha^(-1-n), _alpha =RootOf(-1+_Z+_Z^2)))*n!
a(n) = n!*A001611(n-1), n>0. - R. J. Mathar, Nov 27 2011

A128206 Inverse of number triangle A128207.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 0, 3, 1, 0, 0, 0, 0, 5, 1, 0, 0, 0, 0, 0, 8, 1, 0, 0, 0, 0, 0, 0, 13, 1, 0, 0, 0, 0, 0, 0, 0, 21, 1, 0, 0, 0, 0, 0, 0, 0, 0, 34, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 1

Offset: 0

Paul Barry, Feb 19 2007

Row sums are A001611. Subdiagonal is A000045(n+1).

			Triangle begins
1,
1, 1,
0, 1, 1,
0, 0, 2, 1,
0, 0, 0, 3, 1,
0, 0, 0, 0, 5, 1,
0, 0, 0, 0, 0, 8, 1,
0, 0, 0, 0, 0, 0, 13, 1,
0, 0, 0, 0, 0, 0, 0, 21, 1,
0, 0, 0, 0, 0, 0, 0, 0, 34, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 1

cp's OEIS Frontend

A212262 a(n) = 3^n + Fibonacci(n).

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A226271 Index of 1/n in the Fibonacci (or rabbit) ordering of the positive rationals.

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A358090 Partial inventory of positions as an irregular table; rows 1 and 2 contain 1, for n > 2, row n contains the 1-based positions of 1's, followed by the positions of 2's, 3's, etc. in rows n-2 and n-1 flattened.

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A358120 Partial inventory of positions as an irregular table; rows 1 and 2 contain 1, for n > 2, row n contains the 1-based positions of 1's, followed by the positions of 2's, 3's, etc. in rows n-1 and n-2 flattened.

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A166876 a(n) = a(n-1) + Fibonacci(n), a(1)=1983.

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A187893 a(0)=1, a(1)=4, a(n) = a(n-1) + a(n-2) - 1.

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A228078 a(n) = 2^n - Fibonacci(n) - 1.

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