A055849 -id:A055849 - cp's OEIS frontend

A100545 Expansion of (7-2x) / (1-3x+x^2).

7, 19, 50, 131, 343, 898, 2351, 6155, 16114, 42187, 110447, 289154, 757015, 1981891, 5188658, 13584083, 35563591, 93106690, 243756479, 638162747, 1670731762, 4374032539, 11451365855, 29980065026, 78488829223, 205486422643, 537970438706, 1408424893475, 3687304241719, 9653487831682, 25273159253327

Offset: 0

Creighton Dement, Dec 31 2004

A Floretion integer sequence relating to Fibonacci numbers.

Inverse binomial transform of A013655; inversion of A097924.

Colin Barker, Table of n, a(n) for n = 0..1000
Mark W. Coffey, James L. Hindmarsh, Matthew C. Lettington, John Pryce, On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials, arXiv:1502.03085 [math.NT], 2015 (see p. 31).
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Cf. A000032, A000045, A001906, A005248, A013655, A097924.

List([0..30], n-> Fibonacci(2*n+4) +Lucas(1,-1,2*n+3)[2] ); # G. C. Greubel, Jan 17 2020

[Fibonacci(2*n+4) +Lucas(2*n+3): n in [0..30]]; // G. C. Greubel, Jan 17 2020

F := proc(n) combinat[fibonacci](n) ; end: A100545 := proc(n) 4*F(2*(n+1)) + F(2*n+1)+F(2*n+3) ; end: for n from 0 to 30 do printf("%d,",A100545(n)) ; od ; # R. J. Mathar, Oct 26 2006

Table[Fibonacci[2*(n+2)] + LucasL[2*n+3], {n,0,30}] (* G. C. Greubel, Jan 17 2020 *)

Vec((7-2*x)/(1-3*x+x^2) + O(x^30)) \\ Michel Marcus, Feb 11 2015

[fibonacci(2*n+4) +lucas_number2(2*n+3,1,-1) for n in (0..30)] # G. C. Greubel, Jan 17 2020

a(n-1) = 4*Fibonacci(2*n) + Fibonacci(2*n-1) + Fibonacci(2*n+1).
a(n) + a(n+1) = A055849(n+2).
a(n) = 3*a(n-1) - a(n-2) with a(0)=7 and a(1)=19. - Philippe Deléham, Nov 16 2008
a(n) = (2^(-1-n)*((3-sqrt(5))^n*(-17+7*sqrt(5)) + (3+sqrt(5))^n*(17+7*sqrt(5)))) / sqrt(5). - Colin Barker, Oct 14 2015
From G. C. Greubel, Jan 17 2020: (Start)
a(n) = Fibonacci(2*n+4) + Lucas(2*n+3).
E.g.f.: 2*exp(3*t/2)*(cosh(sqrt(5)*t/2) + (4/sqrt(5))*sinh(sqrt(5)*t/2)). (End)

Corrected and extended by T. D. Noe and R. J. Mathar, Oct 26 2006

A141751 Triangle, read by rows, where T(n,k) = [T(n-1,k-1)T(n-1,k) + 1]/T(n-2,k-1) for 0=0 and T(n,0) = Fibonacci(2n-1) for n>=1.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 13, 13, 8, 4, 1, 34, 34, 21, 11, 5, 1, 89, 89, 55, 29, 14, 6, 1, 233, 233, 144, 76, 37, 17, 7, 1, 610, 610, 377, 199, 97, 45, 20, 8, 1, 1597, 1597, 987, 521, 254, 118, 53, 23, 9, 1, 4181, 4181, 2584, 1364, 665, 309, 139, 61, 26, 10, 1

Offset: 0

Paul D. Hanna, Jul 04 2008

			Generating rule.
Given nonzero elements W, X, Y, Z, relatively arranged like so:
.. W .....
.. X Y ...
.... Z ...
then Z = (X*Y + 1)/W.
Triangle begins:
1;
1, 1;
2, 2, 1;
5, 5, 3, 1;
13, 13, 8, 4, 1;
34, 34, 21, 11, 5, 1;
89, 89, 55, 29, 14, 6, 1;
233, 233, 144, 76, 37, 17, 7, 1;
610, 610, 377, 199, 97, 45, 20, 8, 1;
1597, 1597, 987, 521, 254, 118, 53, 23, 9, 1;
4181, 4181, 2584, 1364, 665, 309, 139, 61, 26, 10, 1; ...

Paul D. Hanna, Table of n, a(n) for n = 0..1080, as a flattened table of rows 0..45

Cf. row sums: A141752; columns: A001519, A001906, A002878, A054486, A054492, A055267, A055273, A055849.

PARI
```
T(n,k)=if(n
				
```

T(n,k)=fibonacci(2*(n-k))*k+fibonacci(2*(n-k)-1)
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

T(n,k) = Fibonacci(2*(n-k)-1) + k*Fibonacci(2*(n-k)) for 0<=k<=n.

A102714 Expansion of (x+2) / ((x+1)(x^2-3x+1)).

Original entry on oeis.org

2, 5, 14, 36, 95, 248, 650, 1701, 4454, 11660, 30527, 79920, 209234, 547781, 1434110, 3754548, 9829535, 25734056, 67372634, 176383845, 461778902, 1208952860, 3165079679, 8286286176, 21693778850, 56795050373, 148691372270, 389279066436, 1019145827039

Offset: 0

Creighton Dement, Feb 06 2005

A floretion-generated sequence relating Fibonacci numbers.

Floretion Algebra Multiplication Program, FAMP code: (a(n)) = 2dia[I]forseq[ + .5'i + .5'ii' + .5'ij' + .5'ik' ], 2dia[J]forseq = 2dia[K]forseq = A001654, mixforseq = A001519, tesforseq = A099016, vesforseq = A000004. Identity used: dia[I] + dia[J] + dia[K] + mix + tes = ves

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

Cf. A001519, A099016, A001654, A100545, A055849, A000004.

CoefficientList[Series[(x+2)/((x+1)(x^2-3x+1)),{x,0,30}],x] (* or *) LinearRecurrence[{2,2,-1},{2,5,14},30] (* Harvey P. Dale, Apr 22 2012 *)

a(n) = round((2^(-1-n)*((-1)^n*2^(1+n)+(9-5*sqrt(5))*(3-sqrt(5))^n+(3+sqrt(5))^n*(9+5*sqrt(5))))/5) \\ Colin Barker, Oct 01 2016

Vec((x+2)/((x+1)*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Oct 01 2016

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3), a(0) = 2, a(1) = 5, a(2) = 14.
a(n) + a(n+1) = A100545(n).
a(n) + 2*a(n+1) + a(n+2) = A055849(n+2).
a(n) + 2*A001654(n) - A099016(n+2) + 2*A001519(n) = 0.
a(n) = (2^(-1-n)*((-1)^n*2^(1+n)+(9-5*sqrt(5))*(3-sqrt(5))^n+(3+sqrt(5))^n*(9+5*sqrt(5))))/5. - Colin Barker, Oct 01 2016
a(n) = (-1)^n +9*A001906(n+1) -A001906(n) . - R. J. Mathar, Sep 11 2019

Corrected by T. D. Noe, Nov 02 2006, Nov 07 2006

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A100545 Expansion of (7-2*x) / (1-3*x+x^2).

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A141751 Triangle, read by rows, where T(n,k) = [T(n-1,k-1)*T(n-1,k) + 1]/T(n-2,k-1) for 0=0 and T(n,0) = Fibonacci(2*n-1) for n>=1.

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A102714 Expansion of (x+2) / ((x+1)*(x^2-3*x+1)).

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Mathematica

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PARI

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A100545 Expansion of (7-2x) / (1-3x+x^2).

A141751 Triangle, read by rows, where T(n,k) = [T(n-1,k-1)T(n-1,k) + 1]/T(n-2,k-1) for 0=0 and T(n,0) = Fibonacci(2n-1) for n>=1.

A102714 Expansion of (x+2) / ((x+1)(x^2-3x+1)).