A169630 -id:A169630 - cp's OEIS frontend

A282464 a(n) = Sum_{i=0..n} i*Fibonacci(i)^2.

0, 1, 3, 15, 51, 176, 560, 1743, 5271, 15675, 45925, 133056, 381888, 1087645, 3077451, 8658951, 24245655, 67602608, 187789616, 519924075, 1435228575, 3951341811, 10852291273, 29740435200, 81340229376, 222058995001, 605201766675, 1646862596223, 4474969884411

Offset: 0

Bruno Berselli, Feb 16 2017

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4,-10,10,4,-5,1).

Cf. A000045.
Partial sums of A169630.
Cf. A014286: partial sums of i*Fibonacci(i).
Cf. A064831: partial sums of (n+1-i)*Fibonacci(i)^2.

[&+[i*Fibonacci(i)^2: i in [0..n]]: n in [0..30]];

with(combinat): P:=proc(q) local a,n; a:=0; print(a); for n from 1 to q do
a:=a+n*fibonacci(n)^2; print(a); od; end: P(100); # Paolo P. Lava, Feb 17 2017

a[n_] := Sum[i*Fibonacci[i]^2, {i, 0, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 16 2017 *)
LinearRecurrence[{5,-4,-10,10,4,-5,1},{0,1,3,15,51,176,560},30] (* Harvey P. Dale, May 15 2021 *)

makelist(sum(i*fib(i)^2, i, 0, n), n, 0, 30);

a(n) = sum(i=0, n, i*fibonacci(i)^2) \\ Colin Barker, Feb 16 2017

[sum(i*fibonacci(i)^2 for i in [0..n]) for n in range(30)]

O.g.f.: x*(1 - 2*x + 4*x^2 - 2*x^3 + x^4)/((1 - x)*(1 + x)^2*(1 - 3*x + x^2)^2).
a(n) = 5*a(n-1) - 4*a(n-2) - 10*a(n-3) + 10*a(n-4) + 4*a(n-5) - 5*a(n-6) + a(n-7).
a(n) = ((n-1)*Fibonacci(n) + n*Fibonacci(n-1))*Fibonacci(n) + (1 - (-1)^n)/2.

A259451 a(n) = n^2*Fibonacci(n).

Original entry on oeis.org

0, 1, 4, 18, 48, 125, 288, 637, 1344, 2754, 5500, 10769, 20736, 39377, 73892, 137250, 252672, 461533, 837216, 1509341, 2706000, 4827186, 8572124, 15159553, 26707968, 46890625, 82061668, 143188722, 249163824, 432466589, 748836000, 1293764509, 2230588416, 3838265442, 6592537372, 11303644625, 19349736192

Offset: 0

N. J. A. Sloane, Jun 27 2015

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

Cf. A000045, A000290, A169630.

a:= n-> n^2*(<<1|1>, <1|0>>^n)[1, 2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 30 2015

a[n_] := n^2 MatrixPower[{{1, 1}, {1, 0}}, n][[1, 2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 14 2016, after Alois P. Heinz *)

concat(0, Vec(-x*(x^4-x^3+6*x^2+x+1)/(x^2+x-1)^3 + O(x^100))) \\ Colin Barker, Jun 29 2015

From Colin Barker, Jun 29 2015: (Start)
a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6).
G.f.: -x*(x^4 - x^3 + 6*x^2 + x + 1)/(x^2 + x - 1)^3. (End)
E.g.f.: exp(x/2)*x*(sqrt(5)*(1 + x)*cosh(sqrt(5)*x/2) + (1 + 3*x)*sinh(sqrt(5)*x/2))/sqrt(5). - Stefano Spezia, Mar 04 2023

A005822 G.f.: x(1-x^2)(x^4+x^3-x^2+x+1) / (x^8-4x^6-x^4-4x^2+1).

Original entry on oeis.org

0, 1, 1, 2, 4, 11, 16, 49, 72, 214, 319, 947, 1408, 4187, 6223, 18502, 27504, 81769, 121552, 361379, 537196, 1597106, 2374129, 7058377, 10492416, 31194361, 46371025, 137862866, 204935836, 609282227, 905709904, 2692710841, 4002767136, 11900382694, 17690150767

Offset: 0

N. J. A. Sloane

This is a rescaled version of the number of spanning trees in the cube of an n-cycle. See A331905 for details. - N. J. A. Sloane, Feb 06 2020

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Colin Barker, Table of n, a(n) for n = 0..1000 [Jul 09 2015; a(0) inserted by _Georg Fischer_, Jan 27 2020]
G. Baron et al., The number of spanning trees in the square of a cycle, Fib. Quart., 23 (1985), 258-264.
Tsuyoshi Miezaki, A note on spanning trees.
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 sequences related to trees
Index entries for linear recurrences with constant coefficients, signature (0,4,0,1,0,4,0,-1).

Cf. A169630, A331905.

m:=40; R:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!( x*(1-x^2)*(x^4+x^3-x^2+x+1) / (x^8-4*x^6-x^4-4*x^2+1))); // Vincenzo Librandi, Jan 28 2020

A005822:=-z*(z-1)*(1+z)*(z**4+z**3-z**2+z+1)/(-4*z**6-z**4-4*z**2+1+z**8); # [Conjectured (correctly) by Simon Plouffe in his 1992 dissertation; adapted to offset 0 by Georg Fischer, Jan 27 2020]

CoefficientList[Series[x (1 - x^2) (x^4 + x^3 - x^2 + x + 1) / (x^8 - 4 x^6 - x^4 - 4 x^2 + 1), {x, 0, 35}], x] (* Vincenzo Librandi, Jan 28 2020 *)

Vec(-x*(x-1)*(x+1)*(x^4+x^3-x^2+x+1)/(x^8-4*x^6-x^4-4*x^2+1) + O(x^50)) \\ Colin Barker, Jul 09 2015

G.f. adapted to the offset from Colin Barker, Jul 09 2015

Entry revised by N. J. A. Sloane, Jan 25 2020 and Feb 06 2020.

A078692 Triangle reads by rows: T(n,k) = coefficient of x^k in (x^3-2x^2-2x+1)^n.

Original entry on oeis.org

1, -2, -2, 1, 1, -4, 0, 10, 0, -4, 1, 1, -6, 6, 19, -24, -24, 19, 6, -6, 1, 1, -8, 16, 20, -80, -8, 134, -8, -80, 20, 16, -8, 1, 1, -10, 30, 5, -160, 128, 330, -340, -340, 330, 128, -160, 5, 30, -10, 1, 1, -12, 48, -34, -240, 468, 399, -1416, -192, 2020, -192, -1416, 399, 468, -240, -34, 48, -12, 1

Offset: 1

Mohammad K. Azarian, Feb 01 2003

Original name: Coefficients of polynomials in the denominator of the generating function f(x)=(x-x^2)/(x^3-2x^2-2x+1) for F(n)^2 (where F(n) is the Fibonacci sequence) and its successive derivatives starting with the highest power of x.

			Triangle begins:
  1, -2, -2, 1; # see A007598
  1, -4, 0, 10, 0, -4, 1;  # see A169630
  1, -6, 6, 19, -24, -24, 19, 6, -6, 1;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11050 (rows 1..85 of triangle, flattened).

Cf. A007598, A169630.

A078692row[n_] := Module[{x}, CoefficientList[(x^3 - 2*x^2 - 2*x + 1)^n, x]];
Array[A078692row, 6] (* Paolo Xausa, Jul 14 2025 *)

row(n) = Vec((x^3-2*x^2-2*x+1)^n); \\ Michel Marcus, Jul 11 2025

(d^(n)/d(x^n)) f(x), where f(x) = (x-x^2) / (x^3-2*x^2-2*x+1), for n=0, 1, 2, 3, ...

Missing a(9) inserted and entry revised by Sean A. Irvine, Jul 11 2025

A331905 Number of spanning trees in the multigraph cube of an n-cycle.

Original entry on oeis.org

1, 4, 12, 128, 605, 3072, 16807, 82944, 412164, 2035220, 9864899, 47579136, 227902597, 1084320412, 5134860060, 24207040512, 113664879137, 531895993344, 2481300851179, 11543181696640, 53565699079956, 248005494380204, 1145875775104967, 5284358088818688

Offset: 1

David J. Seal, Jan 31 2020

nonn

The multigraph cube of an n-cycle has n nodes V1, V2, ... Vn, with one edge Vi to Vj for each pair (i,j) such that j = i+1, i+2 or i+3 modulo n. It is a multigraph when n <= 6 because this produces instances of multiple edges between the same two vertices, and it also produces loops if n <= 3.
Baron et al. (1985) describes the corresponding sequence A169630 for the multigraph square of a cycle.
I conjecture that a(n) = gcd(n,2) * n * (A005822(n))^2. [This is correct - see the Formula section. - N. J. A. Sloane, Feb 06 2020]
Terms a(7) to a(18) calculated by Brendan McKay, and terms a(1) to a(6) by David J. Seal, in both cases using Kirchhoff's matrix tree theorem.

			The multigraph cube of a 4-cycle has four vertices, with two edges between each pair of distinct vertices - i.e., it is a doubled-edge cover of the complete graph on 4 vertices. The complete graph on 4 vertices has 4^2 = 16 spanning trees, and each of those spanning trees corresponds to 8 spanning trees of the multigraph tree because there are independent choices of 2 multigraph edges to be made for each of the three edges in the graph's spanning tree. So a(4) = 16 * 8 = 128.

Alois P. Heinz, Table of n, a(n) for n = 1..1546
G. Baron et al., The number of spanning trees in the square of a cycle, Fib. Quart., 23 (1985), 258-264.
Yoshiaki Doi et al., A note on spanning trees
Min Li, Zhibing Chen, Xiaoqing Ruan, and Xuerong Yong, The formulas for the number of spanning trees in circulant graphs, Disc. Math. 338 (11) (2015) 1883-1906, Lemma 1.

Cf. A005822, A169630 (corresponding sequence for the multigraph square of an n-cycle).

a:= n-> ((<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|4|1|4>>^iquo(n, 2, 'd').
       <[<0, 1, 4, 16>, <1, 2, 11, 49>][d+1]>)[1, 1])^2*n*(2-irem(n, 2)):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 06 2020

The following formulas were provided by Tsuyoshi Miezaki on Feb 05 2020 (see Doi et al. link). Let z1=(-3+sqrt(-7))/4, z2=(-3-sqrt(-7))/4; T(n,z) = cos(n*arccos(z)). Then a(n) = (2*n/7)*(T(n,z1)-1)*(T(n,z2)-1). Furthermore a(n) = 2*n*A005822(n)^2 if n is even, or n*A005822(n)^2 if n is odd. - N. J. A. Sloane, Feb 06 2020

More terms from Alois P. Heinz, Feb 06 2020

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A282464 a(n) = Sum_{i=0..n} i*Fibonacci(i)^2.

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A259451 a(n) = n^2*Fibonacci(n).

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A005822 G.f.: x*(1-x^2)*(x^4+x^3-x^2+x+1) / (x^8-4*x^6-x^4-4*x^2+1).

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A078692 Triangle reads by rows: T(n,k) = coefficient of x^k in (x^3-2*x^2-2*x+1)^n.

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A331905 Number of spanning trees in the multigraph cube of an n-cycle.

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Maple

Formula

Extensions

A005822 G.f.: x(1-x^2)(x^4+x^3-x^2+x+1) / (x^8-4x^6-x^4-4x^2+1).

A078692 Triangle reads by rows: T(n,k) = coefficient of x^k in (x^3-2x^2-2x+1)^n.