A045925 -id:A045925 - cp's OEIS frontend

A317408 a(n) = n * Fibonacci(2n).

0, 1, 6, 24, 84, 275, 864, 2639, 7896, 23256, 67650, 194821, 556416, 1578109, 4449354, 12480600, 34852944, 96949079, 268746336, 742675211, 2046683100, 5626200216, 15430992126, 42235173769, 115380647424, 314656725625, 856733282574, 2329224424344, 6323840144076

Offset: 0

Rigoberto Florez, Jul 27 2018

nonn

Derivative of Morgan-Voyce Lucas-type evaluated at 1.

Colin Barker, Table of n, a(n) for n = 0..1000
Mahir Bilen Can and Nestor Diaz Morera, Nearly Toric Schubert Varieties and Dyck Paths, arXiv:2212.01234 [math.AG], 2022.
Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
Rigoberto Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
Rigoberto Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Cf. A000045, A001906, A045925.

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|6|-11|6>>^n. <<0, 1, 6, 24>>)[1$2]:
seq(a(n), n=1..35);  # Alois P. Heinz, Jul 27 2018

CoefficientList[Series[-(x - 1) (x + 1) x/(x^2 - 3 x + 1)^2, {x, 0, 28}], x] (* or *)
LinearRecurrence[{6, -11, 6, -1}, {0, 1, 6, 24}, 29] (* or *)
Array[# Fibonacci[2 #] &, 29, 0] (* Michael De Vlieger, Jul 27 2018 *)

a(n)=n*fibonacci(2*n) \\ Andrew Howroyd, Jul 27 2018

Vec(-(x-1)*(x+1)*x/(x^2-3*x+1)^2 + O(x^30)) \\ Andrew Howroyd, Jul 27 2018

G.f.: -(x-1)*(x+1)*x/(x^2-3*x+1)^2. - Alois P. Heinz, Jul 27 2018
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4) for n > 4. - Andrew Howroyd, Jul 27 2018
a(n) = (2^(-n)*((-(3-sqrt(5))^n + (3+sqrt(5))^n)*n))/sqrt(5). - Colin Barker, Jul 28 2018
a(n) = n*A001906(n). - Omar E. Pol, Jul 29 2018

A094588 a(n) = n*F(n-1) + F(n), where F = A000045.

Original entry on oeis.org

0, 1, 3, 5, 11, 20, 38, 69, 125, 223, 395, 694, 1212, 2105, 3639, 6265, 10747, 18376, 31330, 53277, 90385, 153011, 258523, 436010, 734136, 1234225, 2072043, 3474029, 5817515, 9730748, 16258910, 27139509, 45258917, 75408775, 125538539

Offset: 0

Paul Barry, May 13 2004

This is the transform of the Fibonacci numbers under the inverse of the signed permutations matrix (see A094587).

Vincenzo Librandi, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045, A007502, A045925, A088209, A354044.

a094588 n = a094588_list !! n
a094588_list = 0 : zipWith (+) (tail a000045_list)
                               (zipWith (*) [1..] a000045_list)
-- Reinhard Zumkeller, Mar 04 2012

# The function 'fibrec' is defined in A354044.
function A094588(n)
    n == 0 && return BigInt(0)
    a, b = fibrec(n - 1)
    a*n + b
end
println([A094588(n) for n in 0:34]) # Peter Luschny, May 16 2022

[n*Fibonacci(n-1)+Fibonacci(n): n in [0..60]]; // Vincenzo Librandi, Apr 23 2011

CoefficientList[Series[x (1+x-2x^2)/(1-x-x^2)^2,{x,0,40}],x]  (* Harvey P. Dale, Apr 16 2011 *)

Vec((1+x-2*x^2)/(1-x-x^2)^2+O(x^99)) \\ Charles R Greathouse IV, Mar 04 2012

G.f. : x*(1 + x - 2*x^2)/(1 - x - x^2)^2.
a(n) = A101220(n, 0, n) - Ross La Haye, Jan 28 2005
a(n) = A109754(n, n). - Ross La Haye, Aug 20 2005
a(n) = (sin(c*n)*i - n*sin(c*(n - 1)))/(i^n*sqrt(5/4)) where c = arccos(i/2). - Peter Luschny, May 16 2022

A136376 a(n) = nF(n) + (n-1)F(n-1).

Original entry on oeis.org

1, 3, 8, 18, 37, 73, 139, 259, 474, 856, 1529, 2707, 4757, 8307, 14428, 24942, 42941, 73661, 125951, 214739, 365166, 619508, 1048753, 1771943, 2988457, 5031843, 8459504, 14201994, 23811349, 39873841, 66695539, 111440227, 186016962

Offset: 1

Gary W. Adamson, Dec 28 2007

For n>2, mod 2 = (0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, ...), i.e., two evens followed by four odds (repeating).

Inverse binomial transform of A117202: (1, 4, 15, 52, ...). - Gary W. Adamson, Sep 03 2008

			a(5) = 37 = a(n)*F(n) + (n-1)*F(n-1) = 5*5 + 4*3 = 25 + 12.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000032, A000045, A045925, A128064, A117202, A238344.

Table[n*Fibonacci[n] + (n - 1)*Fibonacci[n - 1], {n, 1, 50}] (* Stefan Steinerberger, Dec 28 2007 *)

a(n)=n*fibonacci(n)+(n-1)*fibonacci(n-1) \\ Charles R Greathouse IV, Oct 07 2015

Vec(x*(1+x)*(1+x^2)/(x^2+x-1)^2 + O(x^100)) \\ Altug Alkan, Oct 28 2015

a(n) = n*F(n) + (n-1)*F(n-1). Equals the matrix product A128064 (unsigned) * A000045.
From R. J. Mathar, Jul 13 2009: (Start)
a(n) = A045925(n) + A045925(n-1).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4).
G.f.: x*(1+x)*(1+x^2)/(x^2+x-1)^2. (End)
a(n) = A238344(3n-2,n-1). - Alois P. Heinz, Apr 11 2014
From Vladimir Reshetnikov, Oct 28 2015: (Start)
a(n) = ((n+1)*F(n)+(n-1)*L(n))/2, where L(n) are Lucas numbers (A000032).
E.g.f.: (exp(phi*x)*(phi^3*x-1)-exp(-x/phi)*(phi^3+x)/phi)/(sqrt(5)*phi)+1, where phi=(1+sqrt(5))/2.
(End)

More terms from Stefan Steinerberger, Dec 28 2007

A169630 a(n) = n times the square of Fibonacci(n).

Original entry on oeis.org

0, 1, 2, 12, 36, 125, 384, 1183, 3528, 10404, 30250, 87131, 248832, 705757, 1989806, 5581500, 15586704, 43356953, 120187008, 332134459, 915304500, 2516113236, 6900949462, 18888143927, 51599794176, 140718765625, 383142771674, 1041660829548, 2828107288188, 7668512468789

Offset: 0

R. J. Mathar, Mar 13 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. Baron, H. Prodinger, R. F. Tichy, F. T. Boesch, and J. F. Wang, The number of spanning trees in the square of a cycle, Fibonacci Quart. 23 (1985), no. 3, 258-264 [MR0806296]
Zbigniew R. Bogdanowicz, The number of spanning trees in a superprism, Discrete Math. Lett. 13 (2024) 66-73. See Theorem 1.2.
R. Guy, Q on papers by Kleitman, Baron et al., SeqFan list, Mar 2010
D. J. Kleitman and B. Golden, Counting trees in a certain class of graphs, Amer. Math. Monthly 82 (1975), 40-44.
Index entries for linear recurrences with constant coefficients, signature (4,0,-10,0,4,-1).

Cf. A000045, A007598, A045925, A282464 (partial sums).

a169630 n = a007598 n * n  -- Reinhard Zumkeller, Sep 01 2013

I:=[0,1,2,12,36,125]; [n le 6 select I[n] else 4*Self(n-1)-10*Self(n-3)+4*Self(n-5)-Self(n-6): n in [1..30]]; // Vincenzo Librandi, Dec 19 2012

A169630 := proc(n) n*(combinat[fibonacci](n))^2 ; end proc:

CoefficientList[Series[x*(1 - 2*x + 4*x^2 - 2*x^3 + x^4)/((1 + x)^2*(x^2 - 3*x + 1)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)
Table[n Fibonacci[n]^2,{n,0,30}] (* or *) LinearRecurrence[{4,0,-10,0,4,-1},{0,1,2,12,36,125},30] (* Harvey P. Dale, Jul 07 2017 *)

vector(40, n, n--; n*fibonacci(n)^2) \\ Michel Marcus, Jul 09 2015

a(n) = A045925(n)*A000045(n) = n*A007598(n) = n *(A000045(n))^2.
a(n) = 4*a(n-1) -10*a(n-3) +4*a(n-5) -a(n-6).
G.f.: x*(1-2*x+4*x^2-2*x^3+x^4)/((1+x)^2*(x^2-3*x+1)^2).
a(n) = n*(((3 + sqrt(5))/2)^n + ((3 - sqrt(5))/2)^n - 2*(-1)^n)/5 (Bogdanowicz). - Stefano Spezia, May 05 2024

A190062 a(n) = n*Fibonacci(n) - Sum_{i=0..n-1} Fibonacci(i).

Original entry on oeis.org

0, 1, 1, 4, 8, 18, 36, 71, 135, 252, 462, 836, 1496, 2653, 4669, 8164, 14196, 24566, 42332, 72675, 124355, 212156, 360986, 612744, 1037808, 1754233, 2959801, 4985476, 8384480, 14080602, 23614932, 39556031, 66181311, 110608188, 184670694

Offset: 0

Bruno Berselli, May 04 2011

Bruno Berselli, Table of n, a(n) for n = 0..1000
Carlos Alirio Rico Acevedo, Ana Paula Chaves, Double-Recurrence Fibonacci Numbers and Generalizations, arXiv:1903.07490 [math.NT], 2019.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,1).

Cf. A122491, A045925, A000071.

[0] cat [n*Fibonacci(n)-(&+[Fibonacci(k): k in [0..n-1]]): n in [1..34]];

CoefficientList[Series[x (1 - 2 x + 2 x^2) / ((1 - x) (1 - x - x^2)^2), {x, 0, 35}], x] (* Vincenzo Librandi, Aug 19 2013 *)

concat(0, Vec(x*(1-2*x+2*x^2)/((1-x)*(1-x-x^2)^2) + O(x^50))) \\ Altug Alkan, Nov 13 2015

G.f.:  x*(1-2*x+2*x^2)/((1-x)*(1-x-x^2)^2).
a(n) = A045925(n) - A000071(n+1).
a(n) = (n-1)*Fibonacci(n) - Fibonacci(n-1) + 1.
a(n) = (((2*n-1)*r-5)*(1+r)^n-((2*n-1)*r+5)*(1-r)^n)/(10*2^n)+1,  where r=sqrt(5).

A131410 A127647 * A000012.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 5, 5, 8, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 21, 34, 34, 34, 34, 34, 34, 34, 34, 34, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 144, 144, 144

Offset: 1

Gary W. Adamson, Jul 08 2007

Row sums = A045925, n*Fib(n): (1, 2, 6, 12, 25, 48,...).

A104763 = A000012 * A127647.

			First few rows of the triangle are:
1;
1, 1;
2, 2, 2;
3, 3, 3, 3;
5, 5, 5, 5, 5;
8, 8, 8, 8, 8, 8;
...

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A127647, A045925, A104763, A000045.

a131410 n k = a131410_tabl !! (n-1) !! (n-1)
a131410_row n = a131410_tabl !! (n-1)
a131410_tabl = zipWith replicate [1..] $ tail a000045_list
-- Reinhard Zumkeller, Oct 07 2012

Table[Fibonacci[n], {n, 15}, {n}] // Flatten (* Vincenzo Librandi, Jan 28 2017 *)

A127647 * A000012 as infinite lower triangular matrices.
Partial sums of A127647 starting from the right, read by rows.
By rows, F(n) occurs n times.

A146005 a(n) = n*Lucas(n).

Original entry on oeis.org

0, 1, 6, 12, 28, 55, 108, 203, 376, 684, 1230, 2189, 3864, 6773, 11802, 20460, 35312, 60707, 104004, 177631, 302540, 513996, 871266, 1473817, 2488368, 4194025, 7057518, 11858508, 19898116, 33345679, 55814940, 93320819, 155867104

Offset: 0

R. J. Mathar, Oct 26 2008

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

I:=[0, 1, 6, 12]; [n le 4 select I[n] else 2*Self(n-1) + Self(n-2) - 2*Self(n-3) - Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 13 2012

Table[LucasL[n, 1]*n, {n, 0, 36}] (* Zerinvary Lajos, Jul 09 2009 *)
CoefficientList[Series[x * (1 + 4*x - x^2)/(1 - x - x^2)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 13 2012 *)
LinearRecurrence[{2,1,-2,-1},{0,1,6,12},40] (* Harvey P. Dale, Apr 03 2013 *)

a(n) = n*A000032(n).
G.f.: x(1+4x-x^2)/(1-x-x^2)^2.
a(n) = 2*a(n-1)+a(n-2)-2*a(n-3)-a(n-4).
a(n) = A000045(n)-5*A000045(n+1)+5*A010049(n+1).
a(n) = A045925(n)+2*A099920(n-1).
E.g.f.: x*exp(x/2)*(cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2)). - G. C. Greubel, Jan 30 2016

A086804 a(0)=0; for n > 0, a(n) = (n+1)^(n-2)*2^(n^2).

Original entry on oeis.org

0, 1, 16, 2048, 1638400, 7247757312, 164995463643136, 18446744073709551616, 9803356117276277820358656, 24178516392292583494123520000000, 271732164163901599116133024293512544256

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 05 2003

Discriminant of Chebyshev polynomial U_n (x) of second kind.
Chebyshev second kind polynomials are defined by U(0)=0, U(1)=1 and U(n) = 2xU(n-1) - U(n-2) for n > 1.
The absolute value of the discriminant of Pell polynomials is a(n-1).
Pell polynomials are defined by P(0)=0, P(1)=1 and P(n) = 2x P(n-1) + P(n-2) if n > 1. - Rigoberto Florez, Sep 01 2018

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990; p. 219, 5.1.2.

Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the Second Kind
Eric Weisstein's World of Mathematics, Pell Polynomial
Index entries for sequences related to Chebyshev polynomials.

Cf. A007701, A127670 (discriminant for S-polynomials), A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.

[0] cat [(n+1)^(n-2)*2^(n^2): n in [1..10]]; // G. C. Greubel, Nov 11 2018

Join[{0},Table[(n+1)^(n-2) 2^n^2,{n,10}]] (* Harvey P. Dale, May 01 2015 *)

PARI
```
a(n)=if(n<1,0,(n+1)^(n-2)*2^(n^2))
```

a(n)=if(n<1,0,n++; poldisc(poltchebi(n)'/n))

a(n) = ((n+1)^(n-2))*2^(n^2), n >= 1, a(0):=0.
a(n) = ((2^(2*(n-1)))*Det(Vn(xn[1],...,xn[n])))^2, n >= 1, with the determinant of the Vandermonde matrix Vn with elements (Vn)i,j:= xn[i]^j, i=1..n, j=0..n-1 and xn[i]:=cos(Pi*i/(n+1)), i=1..n, are the zeros of the Chebyshev U(n,x) polynomials.
a(n) = ((-1)^(n*(n-1)/2))*(2^(n*(n-2)))*Product_{i=1..n}((d/dx)U(n,x)|_{x=xn[i]}), n >= 1, with the zeros xn[i], i=1..n, given above.

Formula and more terms from Vladeta Jovovic, Aug 07 2003

A117202 Binomial transform of n*F(n).

Original entry on oeis.org

0, 1, 4, 15, 52, 170, 534, 1631, 4880, 14373, 41810, 120406, 343884, 975325, 2749852, 7713435, 21540304, 59917826, 166094370, 458998523, 1264919720, 3477182961, 9536877614, 26102772910, 71309161752, 194468551225, 529490287924

Offset: 0

Paul Barry, Mar 02 2006

Binomial transform of A045925.
Number of acyclic subgraphs of the wheel graph W_n (on n+1 vertices) with exactly n-1 edges. - Emil R. Vaughan, Jun 12 2007
Equivalently, number of two-component spanning forests of the wheel graph W_n (on n+1 vertices). - Harry Richman, Jul 31 2023
Starting (1, 4, 15, 52, ...) = binomial transform of A136376. - Gary W. Adamson, Sep 03 2008

Indranil Ghosh, Table of n, a(n) for n = 0..1000
Harry Richman, Farbod Shokrieh, and Chenxi Wu, Counting two-forests and random cut size via potential theory, arXiv:2308.03859 [math.CO], 2023. See p. 18.
J. Salas and A. D. Sokal, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial, J. Stat. Phys. 135 (2009) 279-373; arXiv:0711.1738 [cond-mat.stat-mech], 2007-2009. Mentions this sequence. - _N. J. A. Sloane_, Mar 14 2014
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Cf. A001519, A111262.
Cf. A136376.
Cf. A004146 (number of spanning trees of wheel graph).

Table[n Fibonacci[2n-1],{n,0,26}] (* or *) Table[Sum[Fibonacci[2k]*BernoulliB[2n-2k]*Binomial[2n,2k],{k,1,n}],{n,0,26}] (* or *) CoefficientList[Series[x(1-2x+2x^2)/(1-3x+x^2)^2 ,{x,0,26}],x] (* Indranil Ghosh, Feb 26 2017 *)

a(n) = n*fibonacci(2*n-1); \\ Indranil Ghosh, Feb 26 2017

concat(0, Vec(x*(1-2*x+2*x^2) / (1-3*x+x^2)^2 + O(x^30))) \\ Colin Barker, Feb 26 2017

G.f.: x*(1-2x+2x^2)/(1-3x+x^2)^2.
a(n) = 6*a(n-1)-11*a(n-2)+6*a(n-3)-a(n-4).
a(n) = Sum_{k=0..n} C(n,k)*k*F(k).
From Benoit Cloitre, Nov 29 2006: (Start)
a(n) = Sum_{k=1..n} F(2k)*B(2n-2k)*binomial(2n,2k) where F=Fibonacci numbers and B=Bernoulli numbers;
a(n) = n*F(2n-1). (End)
a(n) = (2^(-1-n)*(-(-5+sqrt(5))*(3+sqrt(5))^n + (3-sqrt(5))^n*(5+sqrt(5)))*n) / 5. - Colin Barker, Feb 26 2017
a(n) = (1/sqrt(5)) * n * (((1 + sqrt(5)) / 2)^(2*n-1) - ((1 - sqrt(5)) / 2)^(2*n-1)). - Harry Richman, Jul 31 2023
a(n) = round((1/sqrt(5)) * n * phi^(2n-1)), where phi = (1+sqrt(5))/2 is the golden ratio A001622. - Harry Richman, Jul 31 2023

A136391 a(n) = nF(n) - (n-1)F(n-1), where the F(j)'s are the Fibonacci numbers (F(0)=0, F(1)=1).

Original entry on oeis.org

1, 1, 4, 6, 13, 23, 43, 77, 138, 244, 429, 749, 1301, 2249, 3872, 6642, 11357, 19363, 32927, 55861, 94566, 159776, 269469, 453721, 762793, 1280593, 2147068, 3595422, 6013933, 10048559, 16773139, 27971549, 46605186, 77587084, 129063117, 214531397, 356346557

Offset: 1

Gary W. Adamson, Dec 28 2007

By definition, the arithmetic mean of a(1) ... a(n) is equal to A000045(n).

Proof of the three-term recurrence formula: a(n+1) - a(n) - a(n-1) = ((n+1)*F(n+1) - n*F(n)) - (n*F(n) - (n-1)*F(n-1)) - ((n-1)*F(n-1) - (n-2)*F(n-2)) = (n+1)*F(n+1) - 2*n*F(n) + (n-2)*F(n-2) = (n+1)*(2*F(n) - F(n-2)) - 2*n*F(n) + (n-2)*F(n-2) = 2*F(n) - 3*F(n-2) = F(n-1) + F(n-3) = L(n-2). - Giuseppe Coppoletta, Sep 01 2014

			a(6) = 23 = 6*F(6) - 5*F(5) = 6*8 - 5*5 = 48 - 25.

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

Cf. A000045, A246715, A354044.

# The function 'fibrec' is defined in A354044.
function A136391(n)
    a, b = fibrec(n - 1)
    n*b - (n - 1)*a
end
println([A136391(n) for n in 1:35]) # Peter Luschny, May 18 2022

with(combinat): seq(n*fibonacci(n)-(n-1)*fibonacci(n-1),n=1..30); # Emeric Deutsch, Jan 01 2008

Table[n Fibonacci[n] - (n-1) Fibonacci[n-1], {n, 1, 20}] (* Vladimir Reshetnikov, Oct 28 2015 *)

Vec(x*(1-x)*(1+x^2)/(1-x-x^2)^2 + O(x^100)) \\ Altug Alkan, Oct 28 2015

Equals A128064 * A000045.
From R. J. Mathar, Nov 25 2008: (Start)
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) = A045925(n) - A045925(n-1).
G.f.: x*(1 - x)*(1 + x^2)/(1 - x - x^2)^2.
a(n) = A014286(n-1) - A014286(n-2), n>3. (End)
Recurrence: a(n+1) = a(n) + a(n-1) + L(n-2) for n>1, where L = A000032 (see proof in Comments section). - Giuseppe Coppoletta, Sep 01 2014
E.g.f.: (exp(x*phi)/phi+exp(-x/phi)*phi)*(x+1)/sqrt(5)-1, where phi=(1+sqrt(5))/2. - Vladimir Reshetnikov, Oct 28 2015
a(n) = F(n-1) + n*F(n-2). - Bruno Berselli, Jul 26 2017

More terms from Emeric Deutsch, Jan 01 2008

cp's OEIS Frontend

A317408 a(n) = n * Fibonacci(2n).

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A094588 a(n) = n*F(n-1) + F(n), where F = A000045.

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Julia

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A136376 a(n) = n*F(n) + (n-1)*F(n-1).

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A169630 a(n) = n times the square of Fibonacci(n).

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A190062 a(n) = n*Fibonacci(n) - Sum_{i=0..n-1} Fibonacci(i).

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A131410 A127647 * A000012.

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A146005 a(n) = n*Lucas(n).

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A136376 a(n) = nF(n) + (n-1)F(n-1).

A136391 a(n) = nF(n) - (n-1)F(n-1), where the F(j)'s are the Fibonacci numbers (F(0)=0, F(1)=1).