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a(n) = Sum_{i=0..n} i*Fibonacci(i)^2.
+10
4
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
FORMULA
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.
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) a(n) = sum(i=0, n, i*fibonacci(i)^2) \\ Colin Barker, Feb 16 2017 (Sage) [sum(i*fibonacci(i)^2 for i in [0..n]) for n in range(30)]
(Maxima) makelist(sum(i*fib(i)^2, i, 0, n), n, 0, 30)
(Magma) [&+[i*Fibonacci(i)^2: i in [0..n]]: n in [0..30]];
CROSSREFS
Cf. A014286: partial sums of i*Fibonacci(i). Cf. A064831: partial sums of (n+1-i)*Fibonacci(i)^2.
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
FORMULA
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
MAPLE
a:= n-> n^2*(<<1|1>, <1|0>>^n)[1, 2]:
MATHEMATICA
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 *)
PROG
(PARI) 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
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).
(Formerly M1243)
+10
2
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
COMMENTS
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
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
MAPLE
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]
MATHEMATICA
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 *)
PROG
(PARI) 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 (Magma) m:=40; R<x>:=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
Number of spanning trees in the multigraph cube of an n-cycle.
+10
2
1, 4, 12, 128, 605, 3072, 16807, 82944, 412164, 2035220, 9864899, 47579136, 227902597, 1084320412, 5134860060, 24207040512, 113664879137, 531895993344, 2481300851179, 11543181696640, 53565699079956, 248005494380204, 1145875775104967, 5284358088818688
COMMENTS
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.
FORMULA
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
EXAMPLE
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.
MAPLE
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)):
CROSSREFS
Cf. A005822, A169630 (corresponding sequence for the multigraph square of an n-cycle).
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.
+10
1
1, -2, -2, 1, 1, -4, 0, 10, -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
FORMULA
(d^(n)/d(x^n))f(x), where f(x)=(x-x^2)/(x^3-2x^2-2x+1), for n=0, 1, 2, 3, ...
EXAMPLE
The coefficients of the first 2 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:
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