A045925 -id:A045925 - cp's OEIS frontend

A116562 Duplicate of A045925.

0, 1, 2, 6, 12, 25, 48, 91, 168, 306, 550, 979, 1728, 3029, 5278, 9150, 15792, 27149, 46512, 79439, 135300, 229866, 389642, 659111, 1112832

Offset: 0

Roger L. Bagula, Mar 17 2006

dead

A114525 Triangle of coefficients of the Lucas (w-)polynomials.

Original entry on oeis.org

2, 0, 1, 2, 0, 1, 0, 3, 0, 1, 2, 0, 4, 0, 1, 0, 5, 0, 5, 0, 1, 2, 0, 9, 0, 6, 0, 1, 0, 7, 0, 14, 0, 7, 0, 1, 2, 0, 16, 0, 20, 0, 8, 0, 1, 0, 9, 0, 30, 0, 27, 0, 9, 0, 1, 2, 0, 25, 0, 50, 0, 35, 0, 10, 0, 1, 0, 11, 0, 55, 0, 77, 0, 44, 0, 11, 0, 1, 2, 0, 36, 0, 105, 0, 112, 0, 54, 0, 12, 0, 1

Offset: 0

Eric W. Weisstein, Dec 06 2005

Unsigned version of A108045.

The row reversed triangle is A162514. - Paolo Bonzini, Jun 23 2016

			2, x, 2 + x^2, 3*x + x^3, 2 + 4*x^2 + x^4, 5*x + 5*x^3 + x^5, ... give triangle
  n\k   0  1  2  3  4  5  6  7  8  9 10 ...
  0:    2
  1:    0  1
  2:    2  0  1
  3:    0  3  0  1
  4:    2  0  4  0  1
  5:    0  5  0  5  0  1
  6:    2  0  9  0  6  0  1
  7:    0  7  0 14  0  7  0  1
  8:    2  0 16  0 20  0  8  0  1
  9:    0  9  0 30  0 27  0  9  0  1
  10:   2  0 25  0 50  0 35  0 10  0  1
  n\k   0  1  2  3  4  5  6  7  8  9 10 ...
  .... reformatted by _Wolfdieter Lang_, Feb 10 2023

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group and Chebyshev polynomials, arXiv:2502.13673 [math.CO], 2025.
B. Johnson, Fibonacci numbers and matrices, 2009.
Jong Hyun Kim, Hadamard products and tilings, JIS 12 (2009) 09.7.4.
M. Pétréolle, Characterization of Cyclically Fully commutative elements in finite and affine Coxeter Groups, arXiv preprint arXiv:1403.1130 [math.GR], 2014.
Eric Weisstein's World of Mathematics, Fibonacci Polynomial
Eric Weisstein's World of Mathematics, Lucas Polynomial

Cf. A108045 (signed version).
Cf. A034807, A162514.
Cf. Sequences L(n,x): A000032(x = 1), A002203 (x = 2), A006497 (x = 3), A014448 (x = 4), A087130 (x = 5), A085447 (x = 6), A086902 (x = 7), A086594 (x = 8), A087798 (x = 9), A086927 (x = 10), A001946 (x = 11), A086928 (x = 12), A088316 (x = 13), A090300 (x = 14), A090301 (x = 15), A090305 (x = 16), A090306 (x = 17), A090307 (x = 18), A090308 (x = 19), A090309 (x = 20), A090310 (x = 21), A090313 (x = 22), A090314 (x = 23), A090316 (x = 24), A087281 (x = 29), A087287 (x = 76), A089772 (x = 199).

Lucas := proc(n,x)
    option remember;
    if  n=0 then
        2;
    elif n =1 then
        x ;
    else
        x*procname(n-1,x)+procname(n-2,x) ;
    end if;
end proc:
A114525 := proc(n,k)
    coeftayl(Lucas(n,x),x=0,k) ;
end proc:
seq(seq(A114525(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Aug 16 2019

row[n_] := CoefficientList[LucasL[n, x], x];
Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 11 2018 *)

From Peter Bala, Mar 18 2015: (Start)
The Lucas polynomials L(n,x) satisfy the recurrence L(n+1,x) = x*L(n,x) + L(n-1,x) with L(0,x) = 2 and L(1,x) = x.
O.g.f.: Sum_{n >= 0} L(n,x)*t^n = (2 - x*t)/(1 - t^2 - x*t) = 2 + x*t + (x^2 + 2)*t^2 + (3*x + x^3)*t^3 + ....
L(n,x) = trace( [ x, 1; 1, 0 ]^n ).
exp( Sum_{n >= 1} L(n,x)*t^n/n ) = Sum_{n >= 0} F(n+1,x)*t^n, where F(n,x) denotes the n-th Fibonacci polynomial. (see Appendix A3 in Johnson).
exp( Sum_{n >= 1} L(n,x)*L(2*n,x)*t^n/n ) = 1/( F(1,x)*F(2*x)*F(3,x) ) * Sum_{n >= 0} F(n+1,x)*F(n+2,x)*F(n+3,x)*t^n.
exp( Sum_{n >= 1} L(3*n,x)/L(n,x)*t^n/n ) = Sum_{n >= 0} L(2*n + 1,x)*t^n.
L(n,1) = Lucas(n) = A000032(n); L(n,4) = Lucas(3*n) = A014448(n); L(n,11) = Lucas(5*n) = A001946(n); L(n,29) = Lucas(7*n) = A087281(n); L(n,76) = Lucas(9*n) = A087287(n); L(n,199) = Lucas(11*n) = A089772(n). The general result is L(n,Lucas(2*k + 1)) = Lucas((2*k + 1)*n). (End)
From Jeremy Dover, Jun 10 2016: (Start)
Read as a triangle T(n,k), n >= 0, n >= k >= 0, T(n,k) = (Binomial((n+k)/2,k) + Binomial((n+k-2)/2,k))*(1+(-1)^(n-k))/2.
T(n,k) = A046854(n-1,k-1) + A046854(n-1,k) + A046854(n-2,k) for even n+k with n+k > 0, assuming A046854(n,k) = 0 for n < 0, k < 0, k > n.
T(n,k) is the number of binary strings of length n with exactly k pairs of consecutive 0's and no pair of consecutive 1's, where the first and last bits are considered consecutive. (End)
From Peter Bala, Sep 03 2022: (Start)
L(n,x) = 2*(i)^n*T(n,-i*x/2), where i = sqrt(-1) and T(n,x) is the n-th Chebyshev polynomial of the first kind.
d/dx(L(n,x)) = n*F(n,x), where F(n,x) denotes the n-th Fibonacci polynomial.
Let P_n(x,y) = (L(n,x) - L(n,y))/(x - y). Then {P_n(x,y): n >= 1} is a fourth-order linear divisibility sequence of polynomials in the ring Z[x,y]: if m divides n in Z then P_m(x,y) divides P_n(x,y) in Z[x,y].
P_n(1,1) = A045925(n); P_n(1,4) = A273622; P_n(2,2) = A093967(n).
L(2*n,x)^2 - L(2*n-1,x)*L(2*n+1,x) = x^2 + 4 for n >= 1.
Sum_{n >= 1} L(2*n,x)/( L(2*n-1,x) * L(2*n+1,x) ) = 1/x^2 and
Sum_{n >= 1} (-1)^(n+1)/( L(2*n,x) + x^2/L(2*n,x) ) = 1/(x^2 + 4), both valid for all nonzero real x. (End)
From Peter Bala, Nov 18 2022: (Start)
L(n,x) = Sum_{k = 0..floor(n/2)} (n/(n-k))*binomial(n-k,k)*x^(n-2*k) for n >= 1.
For odd m, L(n, L(m,x)) = L(n*m, x).
For integral x, the sequence {u(n)} := {L(n,x)} satisfies the Gauss congruences: u(m*p^r) == u(m*p^(r-1)) (mod p^r) for all positive integers m and r and all primes p.
Let p be an odd prime and let 0 <= k <= p - 1. Let alpha_k = the p-adic limit_{n -> oo} L(p^n,k). Then alpha_k is a well-defined p-adic-integer and the polynomial L(p,x) - x of degree p factorizes as L(p,x) - x = Product_{k = 0..p-1} (x - alpha_k). For example, L(5,x) - x = x^5 + 5*x^3 + 4*x = x*(x - A269591)*(x - A210850)*(x - A210851)*(x - A269592) in the ring of 5-adic integers. (End)
The formula for L(n,x) given in the first line of the preceding section, with L(0, x) = 2, is rewritten L(n, x) = Sum_{k = 0..floor(n/2)} A034807(n, k)*x^(n - 2*k). See the formula by Alexander Elkins in A034807. - Wolfdieter Lang, Feb 10 2023

A127670 Discriminants of Chebyshev S-polynomials A049310.

Original entry on oeis.org

1, 4, 32, 400, 6912, 153664, 4194304, 136048896, 5120000000, 219503494144, 10567230160896, 564668382613504, 33174037869887488, 2125764000000000000, 147573952589676412928, 11034809241396899282944, 884295678882933431599104, 75613185918270483380568064

Offset: 1

Wolfdieter Lang, Jan 23 2007

a(n-1) is the number of fixed n-cell polycubes that are proper in n-1 dimensions (Barequet et al., 2010).
From Rigoberto Florez, Sep 02 2018: (Start)
a(n-1) is the discriminant of the Morgan-Voyce Fibonacci-type polynomial B(n).
Morgan-Voyce Fibonacci-type polynomials are defined as B(0) = 0, B(1) = 1 and B(n) = (x+2)*B(n-1) - B(n-2) for n > 1.
The absolute value of the discriminant of Fibonacci polynomial F(n) is a(n-1).
Fibonacci polynomials are defined as F(0) = 0, F(1) = 1 and F(n) = x*F(n-1) + F(n-2) for n > 1. (End)
The first 6 values are the dimensions of the polynomial ring in 3n variables xi, yi, zi for 1 <= i <= n modulo the ideal generated by x1^a y1^b z1^c + ... + xn^a yn^b zn^c for 0 < a+b+c <= n (see Fact 2.8.1 in Haiman's paper). - Mike Zabrocki, Dec 31 2019

			n=3: The zeros are [sqrt(2),0,-sqrt(2)]. The Vn(xn[1],...,xn[n]) matrix is [[1,1,1],[sqrt(2),0,-sqrt(2)],[2,0,2]]. The squared determinant is 32 = a(3). - _Wolfdieter Lang_, Aug 07 2011

Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016, http://www.csun.edu/~ctoth/Handbook/chap14.pdf.
G. Barequet and M. Shalah, Automatic Proofs for Formulae Enumerating Proper Polycubes, 31st International Symposium on Computational Geometry (SoCG'15). Editors: Lars Arge and János Pach; pp. 19-22, 2015.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990; p. 219 for T and U polynomials.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Andrei Asinowski, Gill Barequet, Ronnie Barequet, and Gunter Rote, Proper n-Cell Polycubes in n - 3 Dimensions, Journal of Integer Sequences, Vol. 15 (2012), #12.8.4.
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012. See Th. 1. [From _N. J. A. Sloane_, Oct 16 2010]
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica 30 (2010), pp. 257-275.
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, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
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, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
M. Haiman, Conjectures on the quotient ring by diagonal invariants, preprint, 1993.
M. Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), no. 1, 17--76.
O. Khorunzhiy, Enumeration of tree-type diagrams assembled from oriented chains of edges, arXiv:2207.00766 [math.CO], 2022.
Andrew Snowden, Measures for the colored circle, arXiv:2302.08699 [math.CO], 2023.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials
Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Cf. A007701 (T-polynomials), A086804 (U-polynomials), A171860 and A191092 (fixed n-cell polycubes proper in n-2 and n-3 dimensions, resp.).
Cf. A243953, A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.
A317403 is essentially the same sequence.
Diagonal 1 of A195739.

[((n+1)^n/(n+1)^2)*2^n: n in [1..20]]; // Vincenzo Librandi, Jun 23 2014

Table[((n + 1)^n)/(n + 1)^2 2^n, {n, 1, 30}] (* Vincenzo Librandi, Jun 23 2014 *)

a(n) = ((n+1)^(n-2))*2^n, n >= 1.
a(n) = (Det(Vn(xn[1],...,xn[n])))^2 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]:=2*cos(Pi*i/(n+1)), i=1..n, are the zeros of S(n,x):=U(n,x/2).
a(n) = ((-1)^(n*(n-1)/2))*Product_{j=1..n} ((d/dx)S(n,x)|_{x=xn[j]}), n >= 1, with the zeros xn[j], j=1..n, given above.
a(n) = A007830(n-2)*A000079(n), n >= 2. - Omar E. Pol, Aug 27 2011
E.g.f.: -LambertW(-2*x)*(2+LambertW(-2*x))/(4*x). - Vaclav Kotesovec, Jun 22 2014

Slightly edited by Gill Barequet, May 24 2011

A099920 a(n) = (n+1)*F(n), F(n) = Fibonacci numbers A000045.

Original entry on oeis.org

0, 2, 3, 8, 15, 30, 56, 104, 189, 340, 605, 1068, 1872, 3262, 5655, 9760, 16779, 28746, 49096, 83620, 142065, 240812, 407353, 687768, 1159200, 1950650, 3277611, 5499704, 9216519, 15426870, 25793240, 43080608, 71884197, 119835652

Offset: 0

Paul Barry and Ralf Stephan, Oct 15 2004

A Fibonacci-Lucas convolution.
The number of edges in the Lucas cube L_(n+1) [Klavzar]. - R. J. Mathar, Nov 05 2008
Sums of rows of the triangle in A108037. - Reinhard Zumkeller, Oct 07 2012
a(n-1) is the total binary weight of all bimultus bitstrings of length n. A bitstring is bimultus if each of its 1's possess at least one neighboring 1 and each of its 0's possess at least one neighboring 0. - Steven Finch, May 26 2020

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 35.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
S. Klavzar, On median nature and enumerative properties of Fibonacci-like cubes, Discr. Math. 299 (2005), 145-153.
Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.
Steven Finch, Variance of longest run duration in a random bitstring, arXiv:2005.12185 [math.CO], 2020.
Tamás Szakács, Convolution of second order linear recursive sequences. II. Commun. Math. 25, No. 2, 137-148 (2017), remark 2.
Tamás Szakács, Linear recursive sequences and factorials, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See p. 35.
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, Lucas Cube Graph.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1)

Equals A010049(n) + A001629(n+1).

Cf. A000045, A000032, A045925, A023607.

a099920 n = a099920_list !! n
a099920_list = zipWith (*) [1..] a000045_list
-- Reinhard Zumkeller, Oct 07 2012

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

Table[(n + 1) Fibonacci[n], {n, 0, 40}] (* Harvey P. Dale, Jan 18 2012 *)
LinearRecurrence[{2, 1, -2, -1}, {0, 2, 3, 8}, 40] (* Harvey P. Dale, Jan 18 2012 *)
CoefficientList[Series[(2 - x) x/(-1 + x + x^2)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Jul 28 2023 *)

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

G.f.: x*(2-x)/(1-x-x^2)^2;
a(n) = Sum_{k=0..n} F(n-k)*(L(k-1) + 0^k).
a(n) = Sum_{k=0..n+1} F(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4); a(0)=0, a(1)=2, a(2)=3, a(3)=8. - Harvey P. Dale, Jan 18 2012
a(n) = a(n-1) + a(n-2) + A000032(n-1) (Lucas numbers). - Bob Selcoe, Aug 19 2015
a(n) = 2*A001629(n+1) - A001629(n). - R. J. Mathar, Feb 04 2022

Entry revised by N. J. A. Sloane, Jan 23 2006. The offset changed, so some of the formulas may now be slightly off.

A004798 Convolution of Fibonacci numbers 1,2,3,5,... with themselves.

Original entry on oeis.org

1, 4, 10, 22, 45, 88, 167, 310, 566, 1020, 1819, 3216, 5645, 9848, 17090, 29522, 50793, 87080, 148819, 253610, 431086, 731064, 1237175, 2089632, 3523225, 5930668, 9968122, 16730830, 28045221, 46954360, 78524159, 131181406, 218933030, 365044788, 608135635, 1012268592

Offset: 1

Clark Kimberling

From Emeric Deutsch, Feb 15 2010: (Start)
a(n) is the number of subwords of the form 0000 in all binary words of length n+3 that have no pair of adjacent 1's. Example: a(2)=4 because in the 13 (=A000045(7)) binary words of length 5 that have no pair of adjacent 1's, namely 00000, 00001, 00010, 00100, 00101, 01000, 01001, 01010, 10000, 10001, 10010, 10100, 10101, we have 2 + 1 + 0 + 0 + 0 + 0 + 0 + 0 + 1 + 1 + 0 + 0 + 0 = 4 subwords of the form 0000.
a(n) = Sum_{k>=0} k*A171855(n + 3,k). (End)
a(n) is the total number of 0's in all binary words of length n that have no pair of adjacent 1's. Example: a(5) = 45 because in the binary words listed in the above example there are respectively 5 + 4 + 4 + 4 + 3 + 4 + 3 + 3 + 4 + 3 + 3 + 3 + 2 = 45. - Geoffrey Critzer, Jul 22 2013

			a(6) = 45 + 22 + A000045(6+2) = 45 + 22 + 21 = 88. - _Philippe Deléham_, Jan 22 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
É. Czabarka, R. Flórez, and L. Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.
Bridget Eileen Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A171855, A000045.

Cf. A000032, A001629, A030528, A045925.

List([1..40], n-> (n*Lucas(1,-1,n+3)[2] - 2*Fibonacci(n))/5); # G. C. Greubel, Jul 07 2019

I:=[1,4,10,22]; [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, Apr 08 2018

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|-2|1|2>>^n. <<0, 1, 4, 10>>)[1, 1]:
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 04 2013
# Alternative:
a := n -> n*(hypergeom([-(n+1)/2,-n/2],[-n-1],-4) - hypergeom([(1-n)/2,1-n/2],[-n], -4)): seq(simplify(a(n)), n=1..40); # Peter Luschny, Apr 10 2018

nn=40; Drop[CoefficientList[Series[D[(1+x)/(1-y x -y x^2),y]/.y->1,{x,0,nn}],x],1] (* Geoffrey Critzer, Jul 22 2013 *)
Table[n Fibonacci[n] + 2/5 (n LucasL[n] - Fibonacci[n]), {n, 40}] (* Vladimir Reshetnikov, Sep 27 2016 *)
a[n_] := ListConvolve[f = Fibonacci[Range[2, n+1]], f][[1]]; Array[a, 40] (* Jean-François Alcover, Feb 15 2018 *)
LinearRecurrence[{2, 1, -2, -1}, {1, 4, 10, 22}, 40] (* Vincenzo Librandi, Apr 08 2014 *)

Vec(((1+x)/(1-x-x^2))^2+O(x^66)) \\ Joerg Arndt, Jul 04 2013

[(n*lucas_number2(n+3,1,-1) - 2*fibonacci(n))/5 for n in (1..40)] # G. C. Greubel, Jul 07 2019

O.g.f.: (x+1)^2*x/(1-x-x^2)^2. - Len Smiley, Dec 11 2001
a(n) = a(n-1) + a(n-2) + Fibonacci(n+2). - Philippe Deléham, Jan 22 2012
O.g.f. is the derivative of A(x,y) with respect to y and then evaluated at y = 1, where A(x,y) is the o.g.f. for A030528. - Geoffrey Critzer, Jul 22 2013
a(n) = A001629(n+1) + A001629(n-1) + 2*A001629(n). - R. J. Mathar, Oct 30 2015
a(n) = n*Fibonacci(n) + (2/5)*(n*Lucas(n) - Fibonacci(n)) = A045925(n) + 2*A001629(n), where Lucas = A000032, Fibonacci = A000045. - Vladimir Reshetnikov, Sep 27 2016
a(n) = Sum_{i=0..floor((n+1)/2)} binomial(n+1-i,i)*(n-i). - John M. Campbell, Apr 07 2018
From Peter Luschny, Apr 10 2018: (Start)
a(n) = n*(hypergeom([-(n+1)/2, -n/2], [-n - 1], -4) - hypergeom([(1-n)/2, 1 - n/2], [-n], -4)).
a(n) = n*A000045(n+2) - A001629(n+1). (End)
E.g.f.: exp(x/2)*(35*x*cosh(sqrt(5)*x/2) + sqrt(5)*(15*x - 4)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Dec 04 2023

A065220 a(n) = Fibonacci(n) - n.

Original entry on oeis.org

0, 0, -1, -1, -1, 0, 2, 6, 13, 25, 45, 78, 132, 220, 363, 595, 971, 1580, 2566, 4162, 6745, 10925, 17689, 28634, 46344, 75000, 121367, 196391, 317783, 514200, 832010, 1346238, 2178277, 3524545, 5702853, 9227430, 14930316, 24157780, 39088131, 63245947, 102334115, 165580100, 267914254

Offset: 0

Henry Bottomley, Oct 22 2001

E(n) = Fib(n+4)-(n+4): cost of maximum height Huffman tree of size n for Fibonacci sequence (Fibonacci sequence is minimizing absolutely ordered sequence of Huffman tree). - Alex Vinokur (alexvn(AT)barak-online.net), Oct 26 2004

Vinokur A.B, Huffman trees and Fibonacci numbers, Kibernetika Issue 6 (1986) 9-12 (in Russian); English translation in Cybernetics 21, Issue 6 (1986), 692-696.

Harry J. Smith, Table of n, a(n) for n = 0..300
Gregory Dresden, On the Brousseau sums Sum_{i=1..n} i^p*Fibonacci(i), arxiv.org:2206.00115 [math.NT], 2022.
Albert Frank, International Contest Of Logical Sequences, 2002 - 2003. Item 7
Albert Frank, Solutions of International Contest Of Logical Sequences, 2002 - 2003.
A. B. Vinokur, Huffman trees and Fibonacci numbers, Cybernetics 21, Issue 6 (1986), 692-696; also at Research Gate.
Alex Vinokur, Fibonacci connection between Huffman codes and Wythoff array, arXiv:cs/0410013 [cs.DM], 2004-2005.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A002062, A045925.

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

a065220 n = a065220_list !! n
a065220_list = zipWith (-) a000045_list [0..]
-- Reinhard Zumkeller, Nov 06 2012

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

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+a[n-2] od: seq(a[n]-n, n=0..42); # Zerinvary Lajos, Mar 20 2008

lst={};Do[f=Fibonacci[n]-n;AppendTo[lst,f],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 21 2009 *)
Table[Fibonacci[n]-n,{n,0,50}] (* or *) LinearRecurrence[{3,-2,-1,1},{0,0,-1,-1},50] (* Harvey P. Dale, May 29 2017 *)

a(n) = { fibonacci(n) - n } \\ Harry J. Smith, Oct 14 2009

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

a(n) = A000045(n) - A001477(n) = A000126(n-3) - 2 = A001924(n-4) - 1.
a(n) = a(n-1) + a(n-2) + n - 3 = a(n-1) + A000071(n-2).
G.f.: x^2*(2x-1)/((1-x-x^2)*(1-x)^2).
a(n) = Sum_{i=0..n} (i - 2)*F(n-i) for F(n) the Fibonacci sequence A000045. - Greg Dresden, Jun 01 2022

A014286 a(n) = Sum_{j=0..n} j*Fibonacci(j).

Original entry on oeis.org

0, 1, 3, 9, 21, 46, 94, 185, 353, 659, 1209, 2188, 3916, 6945, 12223, 21373, 37165, 64314, 110826, 190265, 325565, 555431, 945073, 1604184, 2717016, 4592641, 7748859, 13052145, 21950853, 36863494, 61824694, 103559033, 173264921, 289575995, 483474153

Offset: 0

N. J. A. Sloane

Equals row sums of triangle A143061. - Gary W. Adamson, Jul 20 2008

Colin Barker, Table of n, a(n) for n = 0..1000
Carlos Alirio Rico Acevedo, and 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. A000045.
Cf. A143061.
Partial sums of A045925.
Cf. A282464: partial sums of j*Fibonacci(j)^2.

List([0..50], n-> n*Fibonacci(n+2)-Fibonacci(n+3)+2); # G. C. Greubel, Jun 13 2019

[n*Fibonacci(n+2)-Fibonacci(n+3)+2: n in [0..50]]; // Vincenzo Librandi, Mar 31 2011

A014286 := proc(n)
    add(i*combinat[fibonacci](i),i=0..n) ;
end proc: # R. J. Mathar, Apr 11 2016

Accumulate[Table[Fibonacci[n]*n, {n, 0, 50}]] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
a[0] = 0; a[1] = 1; a[2] = 3; a[3] = 9; a[n_] := a[n] = 2 a[n-1] + a[n-2] - 2 a[n-3] - a[n-4] + 2; Table[a[n], {n, 0, 50}] (* Vladimir Reshetnikov, Oct 28 2015 *)

concat(0, Vec(x*(1+x^2)/((1-x)*(1-x-x^2)^2) + O(x^50))) \\ Altug Alkan, Oct 28 2015

[n*fibonacci(n+2)-fibonacci(n+3)+2 for n in (0..50)] # G. C. Greubel, Jun 13 2019

G.f.: x*(1+x^2)/((1-x)*(1-x-x^2)^2).
a(n) = n*F(n+2) - F(n+3) + 2.
Recurrences, from Vladimir Reshetnikov, Oct 28 2015: (Start)
6-term, homogeneous, constant coefficients: a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 9, a(4) = 21, a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + a(n-4) + a(n-5).
5-term, non-homogeneous, constant coefficients: a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 9, a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) + 2. (End)

A007502 Les Marvin sequence: a(n) = F(n) + (n-1)*F(n-1), F() = Fibonacci numbers.

Original entry on oeis.org

1, 2, 4, 9, 17, 33, 61, 112, 202, 361, 639, 1123, 1961, 3406, 5888, 10137, 17389, 29733, 50693, 86204, 146246, 247577, 418299, 705479, 1187857, 1997018, 3352636, 5621097, 9412937, 15744681, 26307469, 43912648

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Denominators of convergents of the continued fraction with the n partial quotients: [1;1,1,...(n-1 1's)...,1,n], starting with [1], [1;2], [1;1,3], [1;1,1,4], ... Numerators are A088209(n-1). - Paul D. Hanna, Sep 23 2003

			a(7) = F(7) + 6*F(6) = 13 + 6*8 = 61.

Les Marvin, Problem, J. Rec. Math., Vol. 10 (No. 3, 1976-1977), p. 213.
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 = 1..500
Ignas Gasparavičius, Andrius Grigutis, and Juozas Petkelis, Picturesque convolution-like recurrences and partial sums' generation, arXiv:2507.23619 [math.NT], 2025. See p. 27.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045, A045925, A088209 (numerators), A101220, A109754.

a007502 n = a007502_list !! (n-1)
a007502_list = zipWith (+) a045925_list $ tail a000045_list
-- Reinhard Zumkeller, Oct 01 2012, Mar 04 2012

# The function 'fibrec' is defined in A354044.
function A007502(n)
    n == 0 && return BigInt(1)
    a, b = fibrec(n-1)
    (n-1)*a + b
end
println([A007502(n) for n in 1:32]) # Peter Luschny, May 18 2022

A007502:= func< n | Fibonacci(n) +(n-1)*Fibonacci(n-1) >;
[A007502(n): n in [1..40]]; // G. C. Greubel, Aug 26 2025

Table[Fibonacci[n]+(n-1)*Fibonacci[n-1], {n,40}] (* or *) LinearRecurrence[ {2,1,-2,-1}, {1,2,4,9}, 40](* Harvey P. Dale, Jul 13 2011 *)
f[n_] := Denominator@  FromContinuedFraction@ Join[ Table[1, {n}], {n + 1}]; Array[f, 30, 0] (* Robert G. Wilson v, Mar 04 2012 *)

Vec((1-x^2+x^3)/(1-x-x^2)^2+O(x^99)) \\ Charles R Greathouse IV, Mar 04 2012

def A007502(n): return fibonacci(n) +(n-1)*fibonacci(n-1)
print([A007502(n) for n in range(1,41)]) # G. C. Greubel, Aug 26 2025

G.f.: (1-x^2+x^3)/(1-x-x^2)^2. - Paul D. Hanna, Sep 23 2003
a(n+1) = A109754(n, n+1) = A101220(n, 0, n+1). - N. J. A. Sloane, May 19 2006
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) for n>3, a(0)=1, a(1)=2, a(2)=4, a(3)=9. - Harvey P. Dale, Jul 13 2011
E.g.f.: exp(x/2)*( ((3 + 2*x)/sqrt(5))*sinh(sqrt(5)*x/2) - cosh(sqrt(5)*x/2) ) + 1. - G. C. Greubel, Aug 26 2025

A088209 Numerators of convergents of the continued fraction with the n+1 partial quotients: [1;1,1,...(n 1's)...,1,n+1], starting with [1], [1;2], [1;1,3], [1;1,1,4], ...

Original entry on oeis.org

1, 3, 7, 14, 28, 53, 99, 181, 327, 584, 1034, 1817, 3173, 5511, 9527, 16402, 28136, 48109, 82023, 139481, 236631, 400588, 676822, 1141489, 1921993, 3231243, 5424679, 9095126, 15230452, 25475429, 42566379, 71052157, 118489383

Offset: 0

Paul D. Hanna, Sep 23 2003

Denominators form the Les Marvin sequence: A007502(n+1).

			a(3)/A007502(4) = [1;1,1,4] = 14/9.

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

a(n) = A109754(n, n+2) = A101220(n, 0, n+2).

Cf. A007502 (the denominators), A000045, A045925.

a088209 n = a088209_list !! n
a088209_list = zipWith (+) a000045_list $ tail a045925_list
-- Reinhard Zumkeller, Oct 01 2012, Mar 04 2012

# The function 'fibrec' is defined in A354044.
function A088209(n)
    n == 0 && return BigInt(1)
    a, b = fibrec(n)
    a + (n + 1)*b
end
println([A088209(n) for n in 0:32]) # Peter Luschny, May 18 2022

f[n_] := Numerator@  FromContinuedFraction@ Join[ Table[1, {n}], {n + 1}]; Array[f, 30, 0] (* Robert G. Wilson v, Mar 04 2012 *)
CoefficientList[Series[(1+x-x^3)/(-1+x+x^2)^2,{x,0,40}],x] (* or *) LinearRecurrence[{2,1,-2,-1},{1,3,7,14},40] (* Harvey P. Dale, Jul 13 2021 *)

G.f.: (1+x-x^3)/(1-x-x^2)^2. [Corrected by Georg Fischer, Aug 16 2021]
a(n) = Fibonacci(n) + (n+1)*Fibonacci(n+1). - Paul Barry, Apr 20 2004
a(n) = a(n-1) + a(n-2) + Lucas(n). - Yuchun Ji, Apr 23 2023

A104509 Matrix inverse of triangle A104505, which is the right-hand side of triangle A084610 of coefficients in (1 + x - x^2)^n.

Original entry on oeis.org

1, 1, -1, 3, -2, 1, 4, -6, 3, -1, 7, -12, 10, -4, 1, 11, -25, 25, -15, 5, -1, 18, -48, 60, -44, 21, -6, 1, 29, -91, 133, -119, 70, -28, 7, -1, 47, -168, 284, -296, 210, -104, 36, -8, 1, 76, -306, 585, -699, 576, -342, 147, -45, 9, -1, 123, -550, 1175, -1580, 1485, -1022, 525, -200, 55, -10, 1, 199, -979, 2310, -3454, 3641

Offset: 0

Paul D. Hanna, Mar 11 2005

Riordan array ( (1 + x^2)/(1 - x - x^2), -x/(1 - x - x^2) ) belonging to the hitting time subgroup of the Riordan group (see Peart and Woan). - Peter Bala, Jun 29 2015

The sums of absolute values along steep diagonals in this triangle are: 1, 1, 3, 4 + |-1|, 7 + |-2|, 11 + |-6|, 18 + |-12| + 1, ... and these are the tribonacci numbers A000213 that begin with 1, 1, 1, 3. To see this, replace the y in the g.f. A(x,y) = (1 + x^2)/(1-x-x^2 + x*y) with y=-x^2, multiply by x, and add 1, to obtain the g.f. (1 - x^2)/(1-x-x^2-x^3) for A000213. - Noah Carey and Greg Dresden, Nov 02 2021

			Rows begin:
   1;
   1,   -1;
   3,   -2,   1;
   4,   -6,   3,   -1;
   7,  -12,  10,   -4,   1;
  11,  -25,  25,  -15,   5,   -1;
  18,  -48,  60,  -44,  21,   -6,   1;
  29,  -91, 133, -119,  70,  -28,   7,  -1;
  47, -168, 284, -296, 210, -104,  36,  -8, 1;
  76, -306, 585, -699, 576, -342, 147, -45, 9, -1; ...

Robert Israel, Table of n, a(n) for n = 0..10152 (rows 0 to 141, flattened).
P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
Wikipedia, Lucas polynomials.

Leftmost column is A000204 (Lucas numbers). Other columns include: A045925, A067988. Row sums are: {1,0,2,0,2,0,2,...}. Absolute row sums form: A099425. Antidiagonal sums are: {1,1,2,2,2,2,2,...}. Absolute antidiagonal sums are: A084214.

Cf. A104505, A084610, A000213.

S:= series((1 + x^2)/(1-x-x^2 + x*y),x, 20):
for n from 0 to 19 do R[n]:= coeff(S,x,n) od:
seq(seq(coeff(R[n],y,j),j=0..n), n=0..19); # Robert Israel, Jun 30 2015

nmax = 11;
T[n_, k_] := Coefficient[(1 + x - x^2)^n, x, n + k];
M = Table[T[n, k], {n, 0, nmax}, {k, 0, nmax}] // Inverse;
Table[M[[n+1, k+1]], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 27 2019 *)

{ T(n,k) = my(X=x+x*O(x^n),Y=y+y*O(y^k)); polcoeff(polcoeff((1 + X^2)/(1-X-X^2 + X*Y),n,x),k,y); }

{ tabl(nn) = my(m = matrix(nn, nn, n, k, n--; k--; if((nMichel Marcus, Jun 30 2015

{ A104509(n,k) = if(n==0, k==0, (-1)^k * sum(i=0, (n-k)\2, n/(n-i) * binomial(n-k-i,i) * binomial(n-i,k) )); } \\ Max Alekseyev, Oct 11 2021

For n>=1, a(n,k) = (-1)^k * Sum_{i=0..[(n-k)/2]} n/(n-i) * binomial(n-i,i) * binomial(n-2*i,k) = (-1)^k * Sum_{i=0..[(n-k)/2]} n/(n-i) * binomial(n-k-i,i) * binomial(n-i,k). - Max Alekseyev, Oct 11 2021
G.f.: A(x, y) = (1 + x^2)/(1-x-x^2 + x*y).
G.f. for column k: g_k(x) = -(x^2+1)*x^k/(x^2+x-1)^(k+1). - Robert Israel, Jun 30 2015
G.f. for row n>=1 is the Lucas polynomial L_n(1-x). - Max Alekseyev, Oct 11 2021

cp's OEIS Frontend

A116562 Duplicate of A045925.

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A114525 Triangle of coefficients of the Lucas (w-)polynomials.

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A127670 Discriminants of Chebyshev S-polynomials A049310.

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A099920 a(n) = (n+1)*F(n), F(n) = Fibonacci numbers A000045.

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A004798 Convolution of Fibonacci numbers 1,2,3,5,... with themselves.

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A065220 a(n) = Fibonacci(n) - n.

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A014286 a(n) = Sum_{j=0..n} j*Fibonacci(j).