A117591 -id:A117591 - cp's OEIS frontend

A099036 a(n) = 2^n - Fibonacci(n).

1, 1, 3, 6, 13, 27, 56, 115, 235, 478, 969, 1959, 3952, 7959, 16007, 32158, 64549, 129475, 259560, 520107, 1041811, 2086206, 4176593, 8359951, 16730848, 33479407, 66987471, 134021310, 268117645, 536356683, 1072909784, 2146137379, 4292788987, 8586410014

Offset: 0

Paul Barry, Sep 23 2004

Binomial transform of (-1)^n*A000045(n) + 1 = (-1)^n*A008346(n).
Number of compositions of n+1 that contain 1 as a part. - Vladeta Jovovic, Sep 26 2004
Generated from iterates of M * [1,1,1,...], where M = a tridiagonal matrix with [0,1,1,1,...] as the main diagonal, [1,1,1,...] as the superdiagonal and [1,0,0,0,...] as the subdiagonal. - Gary W. Adamson, Jan 05 2009
Starting with offset 1, generated from iterates of M * [1,1,1,...], M*ANS -> M*ANS,...; where M = = a tridiagonal matrix with (0,1,1,1,...) in the main diagonal, (1,1,1,...) in the superdiagonal and (1,0,0,0,...) in the subdiagonal. - Gary W. Adamson, Jan 04 2009
An elephant sequence, see A175655. For the central square 24 A[5] vectors, with decimal values between 11 and 416, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A027934 (without the leading 0). - Johannes W. Meijer, Aug 15 2010
Number of fixed points in all compositions of n+1. - Alois P. Heinz, Jun 18 2020

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Cf. A000045, A000079, A003056, A008346, A027934, A101220, A117591, A175655, A212323, A238350.

a099036 n = a099036_list !! n
a099036_list = zipWith (-) a000079_list a000045_list
-- Reinhard Zumkeller, Aug 15 2013

[2^n-Fibonacci(n): n in [0..35]]; // Vincenzo Librandi, May 03 2011

Table[2^n-Fibonacci[n],{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, May 02 2011 *)

a(n)=2^n-fibonacci(n) \\ Charles R Greathouse IV, Sep 24 2015

def A099036(n): return 2**n -fibonacci(n) # G. C. Greubel, Jun 05 2025

G.f.: (1 - x)^2/((1 - 2*x)*(1 - x - x^2)).
a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3).
a(n) = A101220(1,2,n+1) - A101220(1,2,n). - Ross La Haye, Aug 05 2005
a(n) = A000079(n+1) - A117591(n) = A117591(n) - 2 * A000045(n). - Reinhard Zumkeller, Aug 15 2013
a(n) = Sum_{t_1+2*t_2+...+n*t_n = n} multinomial(1+t_1+t_2+...+t_n, 1+t_1, t_2, ..., t_n). - Mircea Merca, Oct 09 2013
a(n) = Sum_{k=0..A003056(n+1)} k * A238350(n+1,k). - Alois P. Heinz, Jun 18 2020
E.g.f.: cosh(2*x) + sinh(2*x) - 2*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, Jan 31 2023

More terms from Ross La Haye, Aug 05 2005

A074824 Numbers k such that 2^k + Fibonacci(k) is prime.

Original entry on oeis.org

1, 2, 4, 5, 8, 11, 16, 19, 44, 88, 224, 349, 520, 1187, 2044, 2636, 2645, 3994, 6851, 12016, 16304, 31388, 54296, 56608, 100700, 134387

Offset: 1

Robert G. Wilson v, Sep 06 2002

2^(3n) + Fibonacci(3n) is even if n>0.

Cf. A000045, A074716, A117591.

Do[ If[ PrimeQ[ 2^n + Fibonacci[n]], Print[n]], {n, 1, 5000}]

is(n)=ispseudoprime(fibonacci(n)+2^n) \\ Charles R Greathouse IV, May 22 2017

a(19)-a(21) from Ryan Propper, Aug 30 2005
a(22) from Amiram Eldar, May 10 2022
a(23)-a(24) from Michael S. Branicky, May 15 2023
a(25)-a(26) from Michael S. Branicky, Oct 23 2024

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

Original entry on oeis.org

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)).

A131375 A007318 + A046854 - A049310.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 3, 4, 1, 1, 6, 6, 5, 1, 2, 5, 13, 10, 6, 1, 1, 9, 15, 24, 15, 7, 1, 2, 7, 27, 35, 40, 21, 8, 1, 1, 12, 28, 66, 70, 62, 28, 9, 1, 2, 9, 46, 84, 141, 126, 91, 36, 10, 1

Offset: 0

Gary W. Adamson, Jul 04 2007

Row sums = A117591: (1, 3, 5, 10, 19, 37, 72,...).

			First few rows of the triangle are:
1;
2, 1;
1, 3, 1;
2, 3, 4, 1;
1, 6, 6, 5, 1;
2, 5, 13, 10, 6, 1;
1, 9, 15, 24, 15, 7, 1;
...

Cf. A046854, A007318, A049310, A117591.

A007318 + A046854 - A049310 as infinite lower triangular matrices.

A131376 Triangle read by rows: T(n,k) = A007318(n,k) + A065941(n,k) - A168561(n,k).

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 2, 5, 1, 1, 5, 6, 6, 1, 2, 3, 14, 9, 8, 1, 1, 7, 14, 24, 16, 9, 1, 2, 4, 27, 30, 45, 21, 11, 1, 1, 9, 25, 62, 70, 66, 31, 12, 1, 2, 5, 44, 71, 147, 120, 104, 38, 14, 1, 1, 11, 39, 128, 203, 273, 217, 140, 51, 15, 1, 2, 6, 65, 139, 366, 434, 518, 329, 200, 60, 17, 1

Offset: 0

Gary W. Adamson, Jul 04 2007

The old definition was: A007318 + A065941 - A049310. - N. J. A. Sloane, Aug 09 2019

Row sums = A117591: (1, 3, 5, 10, 19, 37, 72, ...).

			First few rows of the triangle are:
  1;
  2, 1;
  1, 3, 1;
  2, 2, 5, 1;
  1, 5, 6, 6, 1;
  2, 3, 14, 9, 8, 1;
  1, 7, 14, 24, 16, 9, 1;
  ...

Cf. A007318, A065941, A049310, A117591 (row sums), A131375, A168561, A309213.

Table[Binomial[n, k] + Binomial[n - Floor[(k+1)/2], Floor[k/2]] - If[EvenQ[n+k], Binomial[(n+k)/2, k], 0], {n, 0, 11}, {k, 0, n}] // Flatten  (* Amiram Eldar, May 31 2025 *)

The old definition of A131376 did not match its data, as Michel Marcus pointed out. The definition has been corrected here, keeping the data. The old definition with corrected data is now A309213. - N. J. A. Sloane, Aug 09 2019

More terms from Amiram Eldar, May 31 2025

A309213 A007318 + A065941 - A049310.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 2, 6, 5, 1, 1, 5, 12, 6, 1, 2, 3, 14, 17, 8, 1, 3, 7, 14, 24, 26, 9, 1, 2, 12, 27, 30, 45, 33, 11, 1, 1, 9, 45, 62, 70, 66, 45, 12, 1, 2, 5, 44, 111, 147, 120, 104, 54, 14, 1, 3, 11, 39, 128, 273, 273, 217, 140, 69, 15, 1, 2, 18, 65, 139, 366, 546, 518, 329, 200, 80, 17, 1

Offset: 0

Gary W. Adamson, Jul 04 2007

Row sums = 1, 3, 7, 14, 25, 45, 85, ... (This is probably a new sequence and should be added to the OEIS.) - N. J. A. Sloane, Aug 09 2019

			First few rows of the triangle are:
1,
2, 1,
3, 3, 1,
2, 6, 5, 1,
1, 5, 12, 6, 1,
2, 3, 14, 17, 8, 1,
3, 7, 14, 24, 26, 9, 1,
...

Jinyuan Wang, Table of n, a(n) for n = 0..5150

Cf. A007318, A065941, A049310, A117591, A131375, A168561.

T007318(n, k) = binomial(n, k);
T065941(n, k) = binomial(n - (k+1)\2, k\2);
T049310(n, k) = if ((n+k)%2, 0, (-1)^((n+k)/2 + k) * binomial((n+k)/2, k));
T(n, k) = T007318(n, k) + T065941(n, k) - T049310(n, k); \\ Michel Marcus, Apr 28 2014

A007318 + A065941 - A168561 as infinite lower triangular matrices.

The old definition of A131376 did not match the data, as Michel Marcus pointed out. The definition there has been corrected, keeping the old data. The present sequence uses the old definition with corrected data from Michel Marcus. - N. J. A. Sloane, Aug 09 2019

More terms from Jinyuan Wang, Aug 29 2019

A101353 a(n) = Sum_{k=0..n} (2^k + Fibonacci(k)).

Original entry on oeis.org

1, 4, 9, 19, 38, 75, 147, 288, 565, 1111, 2190, 4327, 8567, 16992, 33753, 67131, 133654, 266323, 531051, 1059520, 2114861, 4222959, 8434974, 16852239, 33675823, 67305280, 134535537, 268949683, 537702950, 1075088091, 2149661955, 4298491872, 8595637477

Offset: 0

Jorge Coveiro, Dec 25 2004

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4,-1,2).

Cf. A117591 (first differences). - R. J. Mathar, Feb 06 2010

seq(sum(2^x+fibonacci(x),x=0..a),a=0..30);

Accumulate[Table[2^k+Fibonacci[k],{k,0,40}]] (* or *) LinearRecurrence[{4,-4,-1,2},{1,4,9,19},40] (* Harvey P. Dale, Aug 17 2025 *)

Vec((1-3*x^2)/((1-x)*(2*x-1)*(x^2+x-1)) + O(x^40)) \\ Colin Barker, Nov 03 2016

Fibonacci(n+2) + 2^(n+1) + 2. - Ralf Stephan, May 16 2007
a(n)= 4*a(n-1) -4*a(n-2) -a(n-3) +2*a(n-4). G.f.: (1-3*x^2)/((1-x) * (2*x-1) * (x^2+x-1)). - R. J. Mathar, Feb 06 2010
a(n) = (-2+2^(1+n)+(2^(-1-n)*((1-sqrt(5))^n*(-3+sqrt(5))+(1+sqrt(5))^n*(3+sqrt(5))))/sqrt(5)). - Colin Barker, Nov 03 2016

cp's OEIS Frontend

A099036 a(n) = 2^n - Fibonacci(n).

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A074824 Numbers k such that 2^k + Fibonacci(k) is prime.

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

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A131375 A007318 + A046854 - A049310.

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A131376 Triangle read by rows: T(n,k) = A007318(n,k) + A065941(n,k) - A168561(n,k).

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A309213 A007318 + A065941 - A049310.

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A101353 a(n) = Sum_{k=0..n} (2^k + Fibonacci(k)).

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