A160536 -id:A160536 - cp's OEIS frontend

A001611 a(n) = Fibonacci(n) + 1.

1, 2, 2, 3, 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

Offset: 0

N. J. A. Sloane

a(0) = 1, a(1) = 2 then the largest number such that a triangle is constructible with three successive terms as sides. - Amarnath Murthy, Jun 03 2003
a(n+2) = A^(n)B(1), n>=0, with compositions of Wythoff's complementary A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link under A135817 for the Wythoff representation of numbers (with A as 1 and B as 0 and the argument 1 omitted). E.g., 2=`0`, 3=`10`, 4=`110`, 6=`1110`, ..., in Wythoff code.
The first-difference sequence is the Fibonacci sequence (A000045). - Roland Schroeder (florola(AT)gmx.de), Aug 05 2010
2 and 3 are the only primes in this sequence.
a(n) is the number of 1 X n nonogram puzzles which can be solved uniquely. See A242876 for puzzle definition. - Lior Manor, Jan 23 2022

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..250
Andrei Asinowski, Cyril Banderier and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
K.-W. Chen, Greatest Common Divisors in Shifted Fibonacci Sequences, J. Int. Seq. 14 (2011) # 11.4.7.
Massimiliano Fasi and Gian Maria Negri Porzio, Determinants of Normalized Bohemian Upper Hessemberg Matrices, University of Manchester (England, 2019).
Martin Griffiths, On a Matrix Arising from a Family of Iterated Self-Compositions, Journal of Integer Sequences, 18 (2015), #15.11.8.
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Fumio Hazama, Spectra of graphs attached to the space of melodies, Discr. Math., 311 (2011), 2368-2383. See Table 5.1.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 402
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 97.
N. S. Mendelsohn, Permutations with confined displacement, Canad. Math. Bull., 4 (1961), 29-38.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A097280, A097281.
Cf. A000071, A157725, A001911, A157726, A006327, A157727, A157728, A157729, A167616. [Added by N. J. A. Sloane, Jun 25 2010 in response to a comment from Aviezri S. Fraenkel]
Cf. A002062, A019863, A160536, A212272, A242876.

a001611 = (+ 1) . a000045
a001611_list = 1 : 2 : map (subtract 1)
                       (zipWith (+) a001611_list $ tail a001611_list)
-- Reinhard Zumkeller, Jul 30 2013

[Fibonacci(n)+1: n in [1..37]]; // Bruno Berselli, Jul 26 2011

A001611:=-(-1+2*z**2)/(z-1)/(z**2+z-1); # Simon Plouffe in his 1992 dissertation
with(combinat): seq((fibonacci(n)+1), n=0..35);

a[0] = 1; a[1] = 2; a[n_] := a[n] = a[n-2] + a[n-1] - 1; Table[ a[n], {n, 0, 40} ]
Fibonacci[Range[0,50]]+1  (* Harvey P. Dale, Mar 23 2011 *)

a(n)=fibonacci(n)+1 \\ Charles R Greathouse IV, Jul 25 2011

G.f.: (1-2*x^2)/(1-2*x+x^3).
a(n) = 2*a(n-1) - a(n-3). - Tanya Khovanova, Jul 13 2007
a(0) = 1, a(1) = 2, a(n) = a(n - 2) + a(n - 1) - 1.
F(4*n) + 1 = F(2*n-1)*L(2*n+1); F(4*n+1) + 1 = F(2*n+1)*L(2*n); F(4*n+2) + 1 = F(2*n+2)*L(2*n); F(4*n+3) + 1 = F(2*n+1)*L(2*n+2) where F(n)=Fibonacci(n) and L(n)=Lucas(n). - R. K. Guy, Feb 27 2003
a(1) = 2; a(n+1)=floor(a(n)*(sqrt(5)+1)/2). - Roland Schroeder (florola(AT)gmx.de), Aug 05 2010
a(n) = Sum_{k=0..n+1} Fibonacci(k-3). - Ehren Metcalfe, Apr 15 2019
Product_{n>=1} (1 - (-1)^n/a(n)) = sin(3*Pi/10) (A019863). - Amiram Eldar, Nov 28 2024

A002062 a(n) = Fibonacci(n) + n.

Original entry on oeis.org

0, 2, 3, 5, 7, 10, 14, 20, 29, 43, 65, 100, 156, 246, 391, 625, 1003, 1614, 2602, 4200, 6785, 10967, 17733, 28680, 46392, 75050, 121419, 196445, 317839, 514258, 832070, 1346300, 2178341, 3524611, 5702921, 9227500, 14930388, 24157854, 39088207, 63246025

Offset: 0

N. J. A. Sloane

Let A006355(n+4)_0%20-%20A066982(n+1)_1%20(conjecture);%20(a(n))%20=%20em%5BK*%20%5Dseq(%20.25'i%20-%20.25'j%20-%20.25'k%20-%20.25i'%20+%20.25j'%20-%20.75k'%20-%20.25'ii'%20-%20.25'jj'%20-%20.25'kk'%20-%20.25'ij'%20-%20.25'ik'%20-%20.75'ji'%20+%20.25'jk'%20-%20.25'ki'%20-%20.75'kj'%20+%20.75e),%20apart%20from%20initial%20term.%20-%20_Creighton%20Dement">x indicate the sequence offset. Then a(n+2)_0 = A006355(n+4)_0 - A066982(n+1)_1 (conjecture); (a(n)) = em[K* ]seq( .25'i - .25'j - .25'k - .25i' + .25j' - .75k' - .25'ii' - .25'jj' - .25'kk' - .25'ij' - .25'ik' - .75'ji' + .25'jk' - .25'ki' - .75'kj' + .75e), apart from initial term. - _Creighton Dement, Nov 19 2004

R. Honsberger, Ingenuity in Math., Random House, 1970, p. 96.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
Hung Viet Chu, A Note on the Fibonacci Sequence and Schreier-type Sets, arXiv:2205.14260 [math.CO], 2022.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A001611, A160536, A212272.

List([0..50], n-> Fibonacci(n)+n); # G. C. Greubel, Jul 09 2019

a002062 n = a000045 n + toInteger n
a002062_list = 0 : 2 : 3 : (map (subtract 1) $
   zipWith (-) (map (* 2) $ drop 2 a002062_list) a002062_list)
-- Reinhard Zumkeller, Oct 03 2012

[Fibonacci(n)+n: n in [0..50]]; // G. C. Greubel, Jul 09 2019

a:= n-> combinat[fibonacci](n)+n: seq(a(n), n=0..50); # Zerinvary Lajos, Mar 20 2008

Table[Fibonacci[n]+n,{n,0,50}] (* Harvey P. Dale, Jul 27 2011 *)

numlib::fibonacci(n)+n $ n = 0..50; // Zerinvary Lajos, May 08 2008

a(n)=fibonacci(n) + n \\ Charles R Greathouse IV, Oct 03 2016

[fibonacci(n)+n for n in (0..50)] # G. C. Greubel, Jul 09 2019

G.f.: x*(-2+3*x) / ( (x^2+x-1)*(x-1)^2 ). - Simon Plouffe in his 1992 dissertation
From Wolfdieter Lang: (Start)
Convolution of natural numbers n >= 1 with Fibonacci numbers F(k), k >= -3, (F(-k)=(-1)^(k+1)*F(k));
G.f.: x*(2-3*x)/((1-x-x^2)*(1-x)^2). (End)
a(n) = 2*a(n-1) - a(n-3) - 1. - Kieren MacMillan, Nov 08 2008
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4). - Emmanuel Vantieghem, May 19 2016
E.g.f.: 2*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5) + x*exp(x). - Ilya Gutkovskiy, Apr 11 2017

A212272 a(n) = Fibonacci(n) + n^3.

Original entry on oeis.org

0, 2, 9, 29, 67, 130, 224, 356, 533, 763, 1055, 1420, 1872, 2430, 3121, 3985, 5083, 6510, 8416, 11040, 14765, 20207, 28359, 40824, 60192, 90650, 138969, 216101, 339763, 538618, 859040, 1376060, 2211077, 3560515, 5742191, 9270340, 14977008, 24208470, 39143041

Offset: 0

Bruno Berselli, May 09 2012

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,6,1,-3,1).

Cf. A000045, A000578; A001611, A002062, A160536.

Magma
```
[Fibonacci(n)+n^3: n in [0..38]];
```
Mathematica
```
Table[Fibonacci[n] + n^3, {n, 0, 38}]
```

for(n=0, 38, print1(fibonacci(n)+n^3", "));

G.f.: x*(2-x+2*x^2-9*x^3)/((1-x-x^2)*(1-x)^4).

A179992 a(n) = a(n-1) + a(n-2) + n^2 for n >= 3, a(1)=2, and a(2)=5.

Original entry on oeis.org

2, 5, 16, 37, 78, 151, 278, 493, 852, 1445, 2418, 4007, 6594, 10797, 17616, 28669, 46574, 75567, 122502, 198469, 321412, 520365, 842306, 1363247, 2206178, 3570101, 5777008, 9347893, 15125742, 24474535, 39601238, 64076797, 103679124

Offset: 1

Carmine Suriano, Aug 05 2010

Each term is the sum of the previous two plus the square of its index.

			a(5) = a(4)+a(3)+5^2 = 16+37+25 = 78.

Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A000045, A001622, A179991.

Cf. A160536, A163250. - Bruno Berselli, Aug 25 2010

a(n) = F(n-2) + (Sum_{i=1..n} i^2) + Sum_{k=0..n-2} F(k)*Sum_{j=0..n-k-1} j^2, where F(i) is the i-th Fibonacci number. [Corrected by Jason Yuen, Apr 09 2025]
G.f.: x*(x^4-4*x^3+6*x^2-3*x+2)/((1-x-x^2)*(1-x)^3). [Corrected by Bruno Berselli, Aug 25 2010 and R. J. Mathar, Oct 18 2010]
Limiting ratio a(n+1)/a(n) = Phi = 1.618038...
a(n) = 2*A022095(n+2)-6*n-13-n^2. - R. J. Mathar, Aug 06 2010
a(n)-4*a(n-1)+5*a(n-2)-a(n-3)-2*a(n-4)+a(n-5) = 0 with n>5. - Bruno Berselli, Aug 25 2010

cp's OEIS Frontend

A001611 a(n) = Fibonacci(n) + 1.

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A002062 a(n) = Fibonacci(n) + n.

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A212272 a(n) = Fibonacci(n) + n^3.

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A179992 a(n) = a(n-1) + a(n-2) + n^2 for n >= 3, a(1)=2, and a(2)=5.

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