A101220 -id:A101220 - cp's OEIS frontend

A109754 Matrix defined by: a(i,0) = 0, a(i,j) = i*Fibonacci(j-1) + Fibonacci(j), for j > 0; read by ascending antidiagonals.

0, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 3, 3, 0, 1, 4, 4, 5, 5, 0, 1, 5, 5, 7, 8, 8, 0, 1, 6, 6, 9, 11, 13, 13, 0, 1, 7, 7, 11, 14, 18, 21, 21, 0, 1, 8, 8, 13, 17, 23, 29, 34, 34, 0, 1, 9, 9, 15, 20, 28, 37, 47, 55, 55, 0, 1, 10, 10, 17, 23, 33, 45, 60, 76, 89, 89

Offset: 0

Ross La Haye, Aug 11 2005; corrected Apr 14 2006

Lower triangular version is at A117501. - Ross La Haye, Apr 12 2006

			Table starts:
[0] 0, 1,  1,  2,  3,  5,  8, 13,  21,  34, ...
[1] 0, 1,  2,  3,  5,  8, 13, 21,  34,  55, ...
[2] 0, 1,  3,  4,  7, 11, 18, 29,  47,  76, ...
[3] 0, 1,  4,  5,  9, 14, 23, 37,  60,  97, ...
[4] 0, 1,  5,  6, 11, 17, 28, 45,  73, 118, ...
[5] 0, 1,  6,  7, 13, 20, 33, 53,  86, 139, ...
[6] 0, 1,  7,  8, 15, 23, 38, 61,  99, 160, ...
[7] 0, 1,  8,  9, 17, 26, 43, 69, 112, 181, ...
[8] 0, 1,  9, 10, 19, 29, 48, 77, 125, 202, ...
[9] 0, 1, 10, 11, 21, 32, 53, 85, 138, 223, ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Rows: A000045(j); A000045(j+1), for j > 0; A000032(j), for j > 0; A000285(j-1), for j > 0; A022095(j-1), for j > 0; A022096(j-1), for j > 0; A022097(j-1), for j > 0. Diagonals: a(i, i) = A094588(i); a(i, i+1) = A007502(i+1); a(i, i+2) = A088209(i); Sum[a(i-j, j), {j=0...i}] = A104161(i). a(i, j) = A101220(i, 0, j).
Rows 7 - 19: A022098(j-1), for j > 0; A022099(j-1), for j > 0; A022100(j-1), for j > 0; A022101(j-1), for j > 0; A022102(j-1), for j > 0; A022103(j-1), for j > 0; A022104(j-1), for j > 0; A022106(j-1), for j > 0; A022107(j-1), for j > 0; A022108(j-1), for j > 0; A022109(j-1), for j > 0; A022110(j-1), for j > 0.
a(2^i-2, j+1) = A118654(i, j), for i > 0.
Cf. A117501.

A := (n, k) -> ifelse(k = 0, 0,
      n*combinat:-fibonacci(k-1) + combinat:-fibonacci(k)):
seq(seq(A(n - k, k), k = 0..n), n = 0..6); # Peter Luschny, May 28 2022

T[n_, 0]:= 0; T[n_, 1]:= 1; T[n_, 2]:= n - 1; T[n_, 3]:= n - 1; T[n_, n_]:= Fibonacci[n]; T[n_, k_]:= T[n, k] = T[n - 1, k - 1] + T[n - 2, k - 2]; Table[T[n, k], {n, 0, 15}, {k, 0, n}] (* G. C. Greubel, Jan 07 2017 *)

a(i, 0) = 0, a(i, j) = i*Fibonacci(j-1) + Fibonacci(j), for j > 0.
a(i, 0) = 0, a(i, 1) = 1, a(i, 2) = i+1, a(i, j) = a(i, j-1) + a(i, j-2), for j > 2.
G.f.: (x*(1 + ix))/(1 - x - x^2).
Sum_{j=0..i+1} a(i-j+1, j) - Sum_{j=0..i} a(i-j, j) = A001595(i). - Ross La Haye, Jun 03 2006

More terms from G. C. Greubel, Jan 07 2017

A027934 a(0)=0, a(1)=1, a(2)=2; for n > 2, a(n) = 3a(n-1) - a(n-2) - 2a(n-3).

Original entry on oeis.org

0, 1, 2, 5, 11, 24, 51, 107, 222, 457, 935, 1904, 3863, 7815, 15774, 31781, 63939, 128488, 257963, 517523, 1037630, 2079441, 4165647, 8342240, 16702191, 33433039, 66912446, 133899917, 267921227, 536038872, 1072395555, 2145305339

Offset: 0

Clark Kimberling

Number of compositions of n with at least one even part (offset 2). - Vladeta Jovovic, Dec 29 2004
First differences of A008466. a(n) = A008466(n+2) - A008466(n+1). - Alexander Adamchuk, Apr 06 2006
Starting with "1" = eigensequence of a triangle with the Fibonacci series as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010
An elephant sequence, see A175654. For the corner squares 24 A[5] vectors, with decimal values between 11 and 416, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A099036 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010
a(n) = Sum_{k=1..n} A108617(n,k) / 2. - Reinhard Zumkeller, Oct 07 2012
a(n) is the number of binary strings that contain the substring 11 or end in 1. a(3) = 5 because we have: 001, 011, 101, 110, 111. - Geoffrey Critzer, Jan 04 2014
a(n-1), n >= 1, is the number of nonexisting (due to the maturation delay) "[male-female] pairs of Fibonacci rabbits" at the beginning of the n-th month. - Daniel Forgues, May 06 2015
a(n-1) is the number of subsets of {1,2,..,n} that contain n that have at least one pair of consecutive integers. For example, for n=5, a(4) = 11 and the 11 subsets are {4,5}, {1,2,5}, {1,4,5}, {2,3,5}, {2,4,5}, {3,4,5}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}, {1,2,3,4,5}.  Note that A008466(n) is the number of all subsets of {1,2,..,n} that have at least one pair of consecutive integers. - Enrique Navarrete, Aug 15 2020

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Patrick Letendre, Polynomials with integer roots, arXiv:1911.00480 [math.NT], 2019. See p. 4.
Toufik Mansour and Mark Shattuck, Pattern avoidance in flattened derangements, Disc. Math. Lett. (2025) Vol. 15, 67-74. See p. 74.
Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
OEIS Wiki, Fibonacci rabbits
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Row sums of triangle A131767. - Gary W. Adamson, Jul 13 2007
a(n) = A101220(1, 2, n+1).
T(n, n) + T(n, n+1) + ... + T(n, 2n), T given by A027926.
Diagonal sums of A055248.
Cf. A000045, A000079, A008466, A059570, A099036, A047967, A000032.

List([0..35], n-> 2^n - Fibonacci(n+1) ); # G. C. Greubel, Sep 27 2019

a027934 n = a027934_list !! n
a027934_list = 0 : 1 : 2 : zipWith3 (\x y z -> 3 * x - y - 2 * z)
               (drop 2 a027934_list) (tail a027934_list) a027934_list
-- Reinhard Zumkeller, Oct 07 2012

[2^n - Fibonacci(n+1): n in [0..35]]; // G. C. Greubel, Sep 27 2019

A027934:= proc(n) local K; K:= Matrix ([[2,0,0], [0,1,1], [0,1,0]])^n; K[1,1]-K[2,2] end: seq (A027934(n), n=0..31); # Alois P. Heinz, Jul 28 2008
a := n -> 2^n - combinat:-fibonacci(n+1): seq(a(n),n=0..31); # Peter Luschny, May 09 2015

nn=31; a:=1/(1-x-x^2); b:=1/(1-2x); CoefficientList[Series[a*x*(1+x*b), {x,0,nn}], x] (* Geoffrey Critzer, Jan 04 2014 *)
LinearRecurrence[{3,-1,-2}, {0,1,2}, 32] (* Jean-François Alcover, Jan 09 2016 *)
nxt[{a_,b_,c_}]:={b,c,3c-b-2a}; NestList[nxt,{0,1,2},40][[;;,1]] (* Harvey P. Dale, Feb 02 2025 *)

a(n)=2^n-fibonacci(n+1) \\ Charles R Greathouse IV, Jun 11 2015

[2^n - fibonacci(n+1) for n in (0..35)] # G. C. Greubel, Sep 27 2019

a(n) = Sum_{j=0..floor(n/2)} Sum_{k=0..n-2*j} binomial(n-j, n-2*j-k). - Paul Barry, Feb 07 2003
From Paul Barry, Jan 23 2004: (Start)
Row sums of A105809.
G.f.: x*(1-x)/((1-2*x)*(1-x-x^2)).
a(n) = 2^n - Fibonacci(n+1). (End) - corrected Apr 06 2006 and Oct 05 2012
a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(n-k, k+j). - Paul Barry, Aug 29 2004
a(n) = (Sum of (n+1)-th row of the triangle in A108617) / 2. - Reinhard Zumkeller, Jun 12 2005
a(n) = term (1,1) - term (2,2) in the 3 X 3 matrix [2,0,0; 0,1,1; 0,1,0]^n. - Alois P. Heinz, Jul 28 2008
a(n) = 2^n - A000045(n+1). - Geoffrey Critzer, Jan 04 2014
a(n) ~ 2^n. - Daniel Forgues, May 06 2015
From Bob Selcoe, Mar 29 2016: (Start)
a(n) = 2*a(n-1) + A000045(n-2).
a(n) = 4*a(n-2) + A000032(n-2). (End)
a(n) = 2^(n-1) - ( ((1+sqrt(5))/2)^n - ((1-sqrt(5))/2)^n)/sqrt(5). - Haider Ali Abdel-Abbas, Aug 17 2019

Simpler definition from Miklos Kristof, Nov 24 2003
Initial zero added by N. J. A. Sloane, Feb 13 2008
Definition fixed by Reinhard Zumkeller, Oct 07 2012

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

Original entry on oeis.org

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

A022096 Fibonacci sequence beginning 1, 6.

Original entry on oeis.org

1, 6, 7, 13, 20, 33, 53, 86, 139, 225, 364, 589, 953, 1542, 2495, 4037, 6532, 10569, 17101, 27670, 44771, 72441, 117212, 189653, 306865, 496518, 803383, 1299901, 2103284, 3403185, 5506469, 8909654, 14416123, 23325777, 37741900, 61067677, 98809577, 159877254

Offset: 0

N. J. A. Sloane

a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(6;n-1-k,k), n>=1, with a(-1)=5. These are the sums of the SW-NE diagonals in P(6;n,k), the (6,1) Pascal triangle A093563. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs. Also sums of SW-NE diagonals in (1,5)-Pascal triangle A096940.
Subsequence of primes: 7, 13, 53, 139, 953, 44771, 189653, 1494692464747, ... - R. J. Mathar, Aug 09 2012
a(n) is the sum of seven consecutive Fibonacci numbers. a(n) = F(n-4) + F(n-3) + F(n-2) + F(n-1) + F(n) + F(n+1) + F(n+2), where F(n)=A000045(n), extended so that F(-1)=1, F(-2)=-1, F(-3)=2, and F(-4)=-3. - Graeme McRae, Apr 24 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jia Huang, Hecke algebras of simply-laced type with independent parameters, arXiv:1902.11139 [math.RT], 2019.
Tanya Khovanova, Recursive Sequences
José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368 [cs.DM], 2012-2014.
J. L. Ramírez and G. N. Rubiano, Properties and Generalizations of the Fibonacci Word Fractal, The Mathematica Journal, Vol. 16 (2014).
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000045, A093563, A096940, A101220, A109754.

a0:=1; a1:=6; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // Bruno Berselli, Feb 12 2013

CoefficientList[Series[(1+5 x)/(1-x-x^2), {x,0,40}], x] (* Vincenzo Librandi, Apr 25 2014 *)
LinearRecurrence[{1,1},{1,6},40] (* Harvey P. Dale, Aug 07 2023 *)

a(n)=([0,1; 1,1]^n*[1;6])[1,1] \\ Charles R Greathouse IV, Jan 29 2016

A022096=BinaryRecurrenceSequence(1,1,1,6)
print([A022096(n) for n in range(41)]) # G. C. Greubel, Jun 02 2025

a(n) = a(n-1) + a(n-2), n>=2, a(0)=1, a(1)=6.
G.f.: (1+5*x)/(1-x-x^2).
a(n) = A109754(5, n+1).
a(n) = 5*Fibonacci(n+2) - 4*Fibonacci(n+1). - Gary Detlefs, Dec 21 2010
a(n) = (2^(-1-n)*((1 - sqrt(5))^n*(-11 + sqrt(5)) + (1 + sqrt(5))^n*(11 + sqrt(5))))/sqrt(5). - Herbert Kociemba, Dec 18 2011
a(n) = Fibonacci(n+3) - Fibonacci(n-4). - Greg Dresden and Sam Neale, Mar 08 2022

Spelling correction by Jason G. Wurtzel, Aug 22 2010

A022097 Fibonacci sequence beginning 1, 7.

Original entry on oeis.org

1, 7, 8, 15, 23, 38, 61, 99, 160, 259, 419, 678, 1097, 1775, 2872, 4647, 7519, 12166, 19685, 31851, 51536, 83387, 134923, 218310, 353233, 571543, 924776, 1496319, 2421095, 3917414, 6338509, 10255923, 16594432, 26850355, 43444787, 70295142, 113739929

Offset: 0

N. J. A. Sloane

a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(7;n-1-k,k) with n>=1, a(-1)=6. These are the SW-NE diagonals in P(7;n,k), the (7,1) Pascal triangle A093564. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.
Pisano period lengths: 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 10, 24, 28, 48, 40, 24, 36, 24, 18, 60, ... (perhaps the same as A001175). - R. J. Mathar, Aug 10 2012
For n >= 1, a(n) is the number of edge covers of the tadpole graph T_{4,n-1} with T_{4,0} interpreted as just the cycle graph C_4. Example: If n=2, we have C_4 and path P_1 joined by a bridge. This is the cycle with a pendant and has 7 edge covers. - Feryal Alayont, Sep 22 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Tadpole Graph.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000045, A001175, A093564, A101220, A109754, A118654, A131778.

a0:=1; a1:=7; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // Bruno Berselli, Feb 12 2013

First /@ NestList[{Last@ #, Total@ #} &, {1, 7}, 36] (* or *)
CoefficientList[Series[(1 + 6 x)/(1 - x - x^2), {x, 0, 36}], x] (* Michael De Vlieger, Feb 20 2017 *)
LinearRecurrence[{1,1},{1,7},40] (* Harvey P. Dale, May 17 2018 *)

a(n)=([0,1; 1,1]^n*[1;7])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

A022097=BinaryRecurrenceSequence(1,1,1,7)
print([A022097(n) for n in range(41)]) # G. C. Greubel, Jun 03 2025

a(n) = a(n-1) + a(n-2) for n>=2, a(0)=1, a(1)=7, a(-1):=6.
G.f.: (1+6*x)/(1-x-x^2).
a(n) = A109754(6, n+1).
a(n) = A118654(3, n).
a(n) = (2^(-1-n)*((1 - sqrt(5))^n*(-13 + sqrt(5)) + (1 + sqrt(5))^n*(13 + sqrt(5))))/sqrt(5). - Herbert Kociemba
a(n) = 6*A000045(n) + A000045(n+1). - R. J. Mathar, Aug 10 2012
a(n) = 8*A000045(n) - A000045(n-2). - Bruno Berselli, Feb 20 2017
From Aamen Muharram, Aug 05 2022: (Start)
a(n) = F(n-4) + F(n-1) + F(n+4),
a(n) = F(n) + F(n+4) - F(n-3),
where F(n) = A000045(n) is the Fibonacci numbers. (End)

A104161 G.f.: x(1 - x + x^2)/((1-x)^2 (1 - x - x^2)).

Original entry on oeis.org

0, 1, 2, 5, 10, 19, 34, 59, 100, 167, 276, 453, 740, 1205, 1958, 3177, 5150, 8343, 13510, 21871, 35400, 57291, 92712, 150025, 242760, 392809, 635594, 1028429, 1664050, 2692507, 4356586, 7049123

Offset: 0

Creighton Dement, Mar 10 2005

A floretion-generated sequence.
Floretion Algebra Multiplication Program, FAMP Code: 1vesrokseq[ (- .25'i - .25i' - .25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj' - .25e)('i + i' + 'ji' + 'ki' + e) ] RokType: Y[sqa.Findk()] = Y[sqa.Findk()] + p.
Partial sums of Leonardo numbers A001595. - Jonathan Vos Post, Jan 01 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000045, A006355, A001595, A117501, A145912.

List([0..40], n-> 2*Fibonacci(n+2) -(n+2)); # G. C. Greubel, Jul 09 2019

[2*Fibonacci(n+2) -(n+2): n in [0..40]]; // G. C. Greubel, Jul 09 2019

a=0;b=1;Table[c=b+a+n; a=b; b=c, {n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *)
CoefficientList[Series[x*(1-x+x^2)/((1-x)^2*(1-x-x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{3,-2,-1,1},{0,1,2,5},40] (* Harvey P. Dale, Sep 06 2012 *)

my(x='x+O('x^40)); concat(0, Vec(x*(1-x+x^2)/((1-x)^2*(1-x-x^2)))) \\ G. C. Greubel, Sep 26 2017

[2*fibonacci(n+2) -(n+2) for n in (0..40)] # G. C. Greubel, Jul 09 2019

Superseeker results (incomplete): a(2) - 2a(n+1) + a(n) = A006355(n+1) (Number of binary vectors of length n containing no singletons); a(n+1) - a(n) = A001595(n) (2-ranks of difference sets constructed from Segre hyperovals); a(n) + n + 1 = A001595(n+1).
A107909(a(n)) = A000975(n). - Reinhard Zumkeller, May 28 2005
From Ross La Haye, Aug 03 2005: (Start)
a(n) = 2*(Fibonacci(n+2) - 1) - n.
a(n) = Sum_{k=0..n} A101220(n-k, 0, k). (End)
From Gary W. Adamson, Apr 02 2006: (Start)
a(n) = a(n-1) + a(n-2) + n-1.
a(n) = row sums of A117501, starting (1, 2, 5, 10, ...). (End)
a(n) = Sum_{k=0..n} A109754(n-k,k). - Ross La Haye, Apr 12 2006
a(n) = (Sum_{k=0..n} (n-k)*Fibonacci(k-1) + Fibonacci(k)) - n. - Ross La Haye, May 31 2006
From R. J. Mathar, Apr 18 2008: (Start)
a(n) = -2 - n + (-A094214)^n*(1-A010499/5) + (1+A010499/5)/A094214^n.
a(n) = A006355(n+3) - n - 2. (End)
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4); a(0)=0, a(1)=1, a(2)=2, a(3)=5. - Harvey P. Dale, Sep 06 2012

A022098 Fibonacci sequence beginning 1, 8.

Original entry on oeis.org

1, 8, 9, 17, 26, 43, 69, 112, 181, 293, 474, 767, 1241, 2008, 3249, 5257, 8506, 13763, 22269, 36032, 58301, 94333, 152634, 246967, 399601, 646568, 1046169, 1692737, 2738906, 4431643, 7170549, 11602192, 18772741, 30374933, 49147674, 79522607, 128670281

Offset: 0

N. J. A. Sloane

a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(8; n-1-k, k) with n >= 1, a(-1) = 7. These are the SW-NE diagonals in P(8; n, k), the (8, 1) Pascal triangle A093565. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.
Pisano period lengths: 1, 3, 8, 6, 4, 24, 16, 12, 24, 12, 10, 24, 28, 48, 8, 24, 36, 24, 18, 12, ... (is this the same as A106291?). - R. J. Mathar, Aug 10 2012
Also the sum of five consecutive Lucas numbers starting with L(-3). - Alonso del Arte, Sep 26 2013
The Pisano period lengths of this sequence are exactly the same as those of the Lucas sequence (A000032), given in A106291. - Klaus Purath, Apr 20 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045.

a(n) = A109754(7, n+1) = A101220(7, 0, n+1).

a0:=1; a1:=8; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // Bruno Berselli, Feb 12 2013

LinearRecurrence[{1, 1}, {1, 8}, 40] (* Alonso del Arte, Sep 26 2013 *)
CoefficientList[Series[(1 + 7 x)/(1 - x - x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 27 2013 *)
Table[LucasL[n + 3] + LucasL[n - 3] - 3 LucasL[n], {n, 2, 40}] (* Bruno Berselli, Dec 30 2016 *)

a(n)=([0,1; 1,1]^n*[1;8])[1,1] \\ Charles R Greathouse IV, Oct 07 2016

a(n) = a(n-1) + a(n-2) for n>1, a(0)=1, a(1)=8, and a(-1):=7.
G.f.: (1 + 7*x)/(1 - x - x^2).
a(n) = ((1 + sqrt(5))^n - (1 - sqrt(5))^n)/(2^n*sqrt(5)) + 3.5*((1 + sqrt(5))^(n-1) - (1 - sqrt(5))^(n-1))/(2^(n-2)*sqrt(5)) for n>0. - Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009
a(n) = 2^(-1-n)*((1 - sqrt(5))^n*(-15 + sqrt(5)) + (1 + sqrt(5))^n*(15 + sqrt(5)))/sqrt(5). - Herbert Kociemba, Dec 18 2011
a(n) = 7*A000045(n) + A000045(n+1). - R. J. Mathar, Aug 10 2012
a(n) = 9*A000045(n) - A000045(n-2). - Bruno Berselli, Feb 20 2017
a(n) = Lucas(n+5) + Lucas(n-1) - 3*Lucas(n+2). - Bruno Berselli, Dec 29 2016, corrected by Greg Dresden, Mar 02 2022
a(n) = Lucas(n+3) - Lucas(n-2). - Greg Dresden and Michael Nyc, Mar 02 2022

A027961 a(n) = Lucas(n+2) - 3.

Original entry on oeis.org

1, 4, 8, 15, 26, 44, 73, 120, 196, 319, 518, 840, 1361, 2204, 3568, 5775, 9346, 15124, 24473, 39600, 64076, 103679, 167758, 271440, 439201, 710644, 1149848, 1860495, 3010346, 4870844, 7881193, 12752040, 20633236, 33385279, 54018518

Offset: 1

Clark Kimberling

Sum of the first n Lucas numbers, that is, A000204(1) to A000204(n). - T. D. Noe, Oct 10 2005

G. C. Greubel, Table of n, a(n) for n = 1..2500
E. S. Egge and T. Mansour, Restricted permutations, Fibonacci numbers and k-generalized Fibonacci numbers, arXiv:math/0203226 [math.CO], 2002.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

T(n, n+1), T given by A027960.

Cf. A000045, A000204, A023537, A101220.

List([1..40], n-> Lucas(1, -1, n+2)[2] -3 ); # G. C. Greubel, Jun 01 2019

[Lucas(n+2)-3: n in [1..40]]; // Vincenzo Librandi, Apr 16 2011

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+a[n-2]+3 od: seq(a[n],n=1..40); # Miklos Kristof, Mar 09 2005
with(combinat): seq(fibonacci(n)+fibonacci(n+2)-3, n=2..40); # Zerinvary Lajos, Jan 31 2008
g:=(1+z^2)/(1-z-z^2): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)-3, n=3..40); # Zerinvary Lajos, Jan 09 2009

LucasL[Range[3, 40]] - 3 (* Alonso del Arte, Sep 26 2013 *)

vector(40, n, fibonacci(n+3) +fibonacci(n+1) -3) \\ G. C. Greubel, Dec 18 2017

first(n) = Vec(x*(1+2*x)/((1-x)*(1-x-x^2)) + O(x^(n+1))) \\ Iain Fox, Dec 19 2017

[lucas_number2(n+2, 1, -1) -3 for n in (1..40)] # G. C. Greubel, Jun 01 2019

a(0) = 0, a(1) = 1, a(n) = a(n-1) + a(n-2) + 3.
a(n) = A000204(n+2) - 3 = A000045(2n+4)/A000045(n+2) - 3. - Benoit Cloitre, Jan 05 2003
G.f.: x*(1+2*x)/((1-x)*(1-x-x^2)). Differences of A023537. - Ralf Stephan, Feb 07 2004
a(n) = A101220(3, 1, n). - Ross La Haye, Jan 28 2005
a(n) = F(n) + F(n+2) - 3, n >= 2, where F(n) is the n-th Fibonacci number. - Zerinvary Lajos, Jan 31 2008
a(n) = Sum_{k=1..n} ((-1/phi)^k + (phi)^k) where phi = 1/2+1/2*sqrt(5). - Dimitri Papadopoulos, Jan 07 2016
a(n) = 2*a(n-1)-a(n-3) for n>3. - Wesley Ivan Hurt, Jan 07 2016

A118654 Square array T(n,k) read by antidiagonals: T(n,k) = 2^n*Fibonacci(k) - Fibonacci(k-2).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 3, 2, 1, 1, 7, 4, 3, 2, 1, 15, 8, 7, 5, 3, 1, 31, 16, 15, 11, 8, 5, 1, 63, 32, 31, 23, 18, 13, 8, 1, 127, 64, 63, 47, 38, 29, 21, 13, 1, 255, 128, 127, 95, 78, 61, 47, 34, 21, 1, 511, 256, 255, 191, 158, 125, 99, 76, 55, 34

Offset: 0

Ross La Haye, May 17 2006

Inverse binomial transform (by columns) of A090888.

			T(2,3) = 7 because 2^2(Fibonacci(3)) - Fibonacci(3-2) = 4*2 - 1 = 7.
{1};
{1,  0};
{1,  1,  1};
{1,  3,  2,  1};
{1,  7,  4,  3,  2};
{1, 15,  8,  7,  5,  3};
{1, 31, 16, 15, 11,  8,  5};
{1, 63, 32, 31, 23, 18, 13,  8};

Rows: T(0,k) = A000045(k-1), for k > 0; T(1,k) = A000045(k+1); T(2,k) = A000032(k+1); T(3,k) = A022097(k); T(4,k) = A022105(k); T(5,k) = A022401(k).

Columns: T(n,1) = A000225(n); T(n,2) = A000079(n); T(n,3) = A000225(n+1); T(n,4) = A055010(n+1); T(n,5) = A051633(n); a(T,6) = A036563(n+3).

T(n,k) = 2^n*Fibonacci(k) - Fibonacci(k-2).
T(n,k) = (2^n-2)*Fibonacci(k) + Fibonacci(k+1).
T(n,0) = 1; T(n,1) = 2^n - 1; T(n,k) = T(n,k-1) + T(n,k-2), for k > 1.
T(0,k) = Fibonacci(k-1); T(1,k) = Fibonacci(k+1); T(n,k) = 3T(n-1,k) - 2T(n-2,k), for n > 1.
T(n,k) = 2T(n-1,k) + Fibonacci(k-2), for n > 0.
T(n,k) = A109754(2^n-2, k+1) = A101220(2^n-2, 0, k+1), for n > 0.
O.g.f. (by rows) = (1+(-2+2^n)x)/(1-x-x^2).
Sum_{k=0..n} T(n-k,k) = A119587(n+1). - Ross La Haye, May 31 2006

A023537 a(n) = Lucas(n+4) - (3*n+7).

Original entry on oeis.org

1, 5, 13, 28, 54, 98, 171, 291, 487, 806, 1324, 2164, 3525, 5729, 9297, 15072, 24418, 39542, 64015, 103615, 167691, 271370, 439128, 710568, 1149769, 1860413, 3010261, 4870756, 7881102, 12751946, 20633139, 33385179, 54018415, 87403694, 141422212, 228826012

Offset: 1

Clark Kimberling

Define a triangle with T(n, 1) = n*(n-1) + 1 and T(n, n) = n for n = 1, 2, 3, ... The interior terms T(r, c) = T(r - 1, c) + T(r - 2, c - 1); this triangle will give the sum of terms in row(n) = a(n). The rows begin 1; 3 2; 7 3 3; 13 6 5 4; 21 13 8 7 5 having row(n) sums 1, 5, 13, 28, 54. - J. M. Bergot, Feb 17 2013

Wolfdieter Lang in "Applications of Fibonacci Numbers", Vol. 7, p. 235, eds.: G. E. Bergum et al, Kluwer, 1998.

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

Cf. A000032, A000045, A027960, A027961, A101220.

List([1..40], n-> Lucas(1, -1, n+4)[2] -(3*n+7) ); # G. C. Greubel, Jun 01 2019

[Lucas(n+4) -(3*n+7): n in [1..40]]; // Vincenzo Librandi, Apr 16 2011

with(combinat): L:=n->fibonacci(n+2)-fibonacci(n-2): seq(L(n),n=0..12): seq(L(n+4)-3*n-7,n=1..40); # Emeric Deutsch, Aug 08 2005

Table[LucasL[n + 4] - (3n + 7), {n, 40}] (* Alonso del Arte, Feb 17 2013 *)

Vec(x*(1+2*x)/((1-x-x^2)*(1-x)^2) + O(x^40)) \\ Colin Barker, Mar 11 2017

[lucas_number2(n+4, 1, -1) -(3*n+7) for n in (1..40)] # G. C. Greubel, Jun 01 2019

def lucas(n: BigInt): BigInt = {
  val zero = BigInt(0)
  def fibTail(n: BigInt, a: BigInt, b: BigInt): BigInt = n match {
    case `zero` => a
    case _ => fibTail(n - 1, b, a + b)
  }
  fibTail(n, 2, 1)
}
(1 to 40).map(n => lucas(n + 4) - (3 * n + 7)) // Alonso del Arte, Oct 20 2019

Convolution of natural numbers with Lucas numbers A000204.
a(n) = A027960(n+1, n+3).
From Wolfdieter Lang: (Start)
a(n) = 7*(F(n+1) - 1) + 4*F(n) - 3*n; F(n) = A000045 (Fibonacci);
G.f.: x*(1 + 2*x)/((1-x)^2*(1 - x - x^2)). (End)
a(n) - a(n-1) = A101220(3, 1, n). - Ross La Haye, May 31 2006
a(n+1) - a(n) = A027961(n+1). - R. J. Mathar, Feb 21 2013
From Colin Barker, Mar 11 2017: (Start)
a(n) = -4 + (2^(-1 - n)*((1 - sqrt(5))^n*(-15 + 7*sqrt(5)) + (1 + sqrt(5))^n*(15 + 7*sqrt(5)))) / sqrt(5) - 3*(1+n).
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n > 4. (End)
From G. C. Greubel, Jun 08 2025: (Start)
a(n) = a(n-1) + a(n-2) + 3*n - 2.
a(n) = Sum_{i=0..n-1} Sum_{j=0..i} A027960(i,j).
E.g.f.: exp(x/2)*( 7*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2) ) - (3*x+7)*exp(x). (End)

More terms from Emeric Deutsch, Aug 08 2005

cp's OEIS Frontend

A109754 Matrix defined by: a(i,0) = 0, a(i,j) = i*Fibonacci(j-1) + Fibonacci(j), for j > 0; read by ascending antidiagonals.

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A027934 a(0)=0, a(1)=1, a(2)=2; for n > 2, a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3).

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

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A022096 Fibonacci sequence beginning 1, 6.

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A022097 Fibonacci sequence beginning 1, 7.

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A104161 G.f.: x*(1 - x + x^2)/((1-x)^2 * (1 - x - x^2)).

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A027934 a(0)=0, a(1)=1, a(2)=2; for n > 2, a(n) = 3a(n-1) - a(n-2) - 2a(n-3).

A104161 G.f.: x(1 - x + x^2)/((1-x)^2 (1 - x - x^2)).