A020878 -id:A020878 - cp's OEIS frontend

A098600 a(n) = Fibonacci(n-1) + Fibonacci(n+1) - (-1)^n.

1, 2, 2, 5, 6, 12, 17, 30, 46, 77, 122, 200, 321, 522, 842, 1365, 2206, 3572, 5777, 9350, 15126, 24477, 39602, 64080, 103681, 167762, 271442, 439205, 710646, 1149852, 1860497, 3010350, 4870846, 7881197, 12752042, 20633240, 33385281, 54018522

Offset: 0

Paul Barry, Sep 17 2004

Row sums of A098599.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and Milinko Mandić, The Golden Ratio, Factorials, and the Lambert W Function, Journal of Integer Sequences, Vol. 27 (2024), Article 24.5.7. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (0,2,1).

Cf. A000032, A000035, A000045, A001350, A008346.
Cf. A020878, A098599, A099925, A181716, A182143.
First differences of A014217 and A062724.

[Fibonacci(n-1) + Fibonacci(n+1) - (-1)^n: n in [0..50]]; // Vincenzo Librandi, Aug 31 2014

Table[-(-1)^n + LucasL[n], {n, 0, 39}] (* Alonso del Arte, Aug 30 2014 *)
Table[Fibonacci[n - 1] + Fibonacci[n + 1] - (-1)^n, {n, 0, 40}] (* Vincenzo Librandi, Aug 31 2014 *)
CoefficientList[ Series[-(1 + 2x)/(-1 + 2x^2 + x^3), {x, 0, 40}], x] (* or *)
LinearRecurrence[{0, 2, 1}, {1, 2, 2}, 40] (* Robert G. Wilson v, Mar 09 2018 *)

a(n)=fibonacci(n-1) + fibonacci(n+1) - (-1)^n; \\ Joerg Arndt, Oct 18 2014

Vec((1+2*x)/((1+x)*(1-x-x^2)) + O(x^30)) \\ Colin Barker, Jun 03 2016

[lucas_number2(n,1,-1) -(-1)^n for n in range(51)] # G. C. Greubel, Mar 26 2024

G.f.: (1+2*x) / ((1+x)*(1-x-x^2)).
a(n) = Sum_{k = 0..n} binomial(k, n-k) + binomial(k-1, n-k-1).
a(n) = A020878(n) - 1 = A001350(n) + 1.
a(n) = Lucas(n) - (-1)^n. - Paul Barry, Dec 01 2004
a(n) = A181716(n+1). - Richard R. Forberg, Aug 30 2014
a(n) = [x^n] ( (1 + x + sqrt(1 + 6*x + 5*x^2))/2 )^n. exp( Sum_{n >= 1} a(n)*x^n/n ) = Sum_{n >= 0} Fibonacci(n+2)*x^n. Cf. A182143. - Peter Bala, Jun 29 2015
From Colin Barker, Jun 03 2016: (Start)
a(n) = (-(-1)^n + ((1/2)*(1-sqrt(5)))^n + ((1/2)*(1+sqrt(5)))^n).
a(n) = 2*a(n-2) + a(n-3) for n > 2. (End)
E.g.f.: (2*exp(3*x/2)*cosh(sqrt(5)*x/2) - 1)*exp(-x). - Ilya Gutkovskiy, Jun 03 2016
a(n) = A014217(n) + A000035(n). - Paul Curtz, Jul 27 2023

A162483 a(n) is the number of perfect matchings of an edge-labeled 2 X (2n+1) Mobius grid graph.

Original entry on oeis.org

3, 6, 13, 31, 78, 201, 523, 1366, 3573, 9351, 24478, 64081, 167763, 439206, 1149853, 3010351, 7881198, 20633241, 54018523, 141422326, 370248453, 969323031, 2537720638, 6643838881, 17393796003, 45537549126, 119218851373, 312119004991, 817138163598

Offset: 0

Sarah-Marie Belcastro, Jul 04 2009

This is a specialization for m=2 of a general formula for the number of perfect matchings of an edge-labeled m X (2n+1) Mobius grid graph.

			G.f. = 3 + 6*x + 13*x^2 + 31*x^3 + 78*x^4 + 201*x^5 + 523*x^6 + 1366*x^7 + ...
a(0) = 3 because this is the number of perfect matchings of a 2 X 1 Mobius grid graph (one for each of the three multiple edges).

Clark Kimberling, Table of n, a(n) for n = 0..1000
G. Tesler, Matchings in graphs on non-orientable surfaces, Journal of Combinatorial Theory B, 78(2000), 198-231.
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A020878, A256233.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((3-6*x+x^2)/((1-x)*(x^2-3*x+1)))); // G. C. Greubel, Sep 22 2018

Table[Re[(1 - I) (2*I + Fibonacci[2 + 2*n] + 1/2 (-Fibonacci[1 + 2*n] + LucasL[1 + 2*n]))], {n, 0, 30}]
Table[LucasL[2*n + 1] + 2, {n, 0, 30}] (* Clark Kimberling, Oct 26 2012 *)
LinearRecurrence[{4, -4, 1}, {3, 6, 13}, 30] (* or *) CoefficientList[Series[(-3 + 6 x - x^2)/(-1 + 4 x - 4 x^2 + x^3), {x, 0, 30}], x] (* Stefano Spezia, Sep 23 2018 *)

{a(n) = 2 + fibonacci(2*n) + fibonacci(2*n+2)}; /* Michael Somos, Nov 03 2016 */

a(n) = Real((1-I) * ((L(2*n+1) - F(2*n+1))/2 + F(2*n+2) + 2*I)).
From R. J. Mathar, Aug 08 2009: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).
G.f.: (3-6*x+x^2)/((1-x)*(x^2-3*x+1)). (End)
a(n+1)-a(n) = A005248(n+1). - R. J. Mathar, Dec 18 2010
a(n) = A000032(2n+1)+2. - Clark Kimberling, Oct 26 2012
a(n) = 2^(-1-n)*(2^(2+n)-(3-sqrt(5))^n*(-1+sqrt(5))+(1+sqrt(5))*(3+sqrt(5))^n). - Colin Barker, Nov 03 2016
a(n) = 2 + L(2*n+1), A256233(n) = -a(-n-1) for all n in Z. - Michael Somos, Nov 03 2016

A263200 Number of perfect matchings on a Möbius strip of width 3 and length 2n.

Original entry on oeis.org

28, 104, 388, 1448, 5404, 20168, 75268, 280904, 1048348, 3912488, 14601604, 54493928, 203374108, 759002504, 2832635908, 10571541128, 39453528604, 147242573288, 549516764548, 2050824484904, 7653781175068, 28564300215368, 106603419686404, 397849378530248

Offset: 2

Sergey Perepechko, Oct 12 2015

This sequence obeys the same recurrence relation as A001835.

Colin Barker, Table of n, a(n) for n = 2..1000
W. T. Lu and F. Y. Wu, Close-packed dimers on nonorientable surfaces, Physics Letters A, 293(2002), 235-246.
S. N. Perepechko, Recurrence relations for the number of perfect matchings on the Mobius strips (in Russian), Proc. of XIX international conference on computational mechanics and modern applied software systems (CMMASS'2015), Alushta, Crimea, 2015, 98-100.
Sergey Perepechko, Graph view
G. Tesler, Matchings in graphs on non-orientable surfaces, Journal of Combinatorial Theory B, 78(2000), 198-231.
Index entries for linear recurrences with constant coefficients, signature (4,-1).

Cf. A001075, A001835, A020878.

I:=[28,104]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 12 2015

CoefficientList[Series[4 (7 - 2 x)/(1 - 4 x + x^2), {x, 0, 33}], x] (* Vincenzo Librandi, Oct 12 2015 *)

Vec(4*x^2*(7-2*x)/(1-4*x+x^2) + O(x^30)) \\ Altug Alkan, Oct 12 2015

a(n) = Product_{k=1..n} (10 + 2*cos(Pi*(4*k-1)/n) - 12*cos(1/2*Pi*(4*k-1)/n)).
G.f.: 4*x^2*(7-2*x)/(1-4*x+x^2).
From Colin Barker, Oct 12 2015: (Start)
a(n) = 2*((2-sqrt(3))^n + (2+sqrt(3))^n).
a(n) = 4*a(n-1) - a(n-2). (End)
a(n) = 4*A001075(n) for n >= 2. - Philippe Deléham, Mar 03 2023

A263201 Number of perfect matchings on a Möbius strip of width 4 and length n.

Original entry on oeis.org

11, 37, 71, 252, 539, 1813, 4271, 13519, 34276, 103803, 276119, 813417, 2226851, 6455052, 17965151, 51604017, 144948419, 414258603, 1169523076, 3333192319, 9436433171, 26853404413, 76139155439, 216490730652, 614339685971, 1745997031837, 4956888901511

Offset: 2

Sergey Perepechko, Oct 12 2015

nonn

This sequence obeys the same recurrence relation as A252054.

Colin Barker, Table of n, a(n) for n = 2..1000
W. T. Lu and F. Y. Wu, Close-packed dimers on nonorientable surfaces, Physics Letters A, 293(2002), 235-246.
S. N. Perepechko, Recurrence relations for the number of perfect matchings on the Mobius strips (in Russian), Proc. of XIX international conference on computational mechanics and modern applied software systems (CMMASS'2015), Alushta, Crimea, 2015, 98-100.
Sergey Perepechko, Graph view
G. Tesler, Matchings in graphs on non-orientable surfaces, Journal of Combinatorial Theory B, 78(2000), 198-231.
Index entries for linear recurrences with constant coefficients, signature (1,13,-7,-61,12,128,0,-128,-12,61,7,-13,-1,1).

Cf. A020878, A263200.

CoefficientList[Series[(11 + 26 x - 109 x^2 - 223 x^3 + 294 x^4 + 620 x^5 - 306 x^6 - 764 x^7 + 100 x^8 + 414 x^9 + 5 x^10 - 92 x^11 - 3 x^12 + 7 x^13)/((1 - x) (1 + x) (1 + x - 3 x^2 - x^3 + x^4) (1 - x - 3 x^2 + x^3 + x^4) (1 - x - 5 x^2 - x^3 + x^4)), {x, 0, 33}], x] (* Vincenzo Librandi, Oct 12 2015 *)

Vec(z^2*(11 + 26*z - 109*z^2 - 223*z^3 + 294*z^4 + 620*z^5 - 306*z^6 -764*z^7 + 100*z^8 + 414*z^9 + 5*z^10 - 92*z^11 - 3*z^12 + 7*z^13)/((1 - z)*(1 + z)*(1 + z - 3*z^2 - z^3 + z^4)*(1 - z - 3*z^2 + z^3 + z^4)*(1 - z - 5*z^2 - z^3 + z^4)) + O(z^50)) \\ Altug Alkan, Oct 12 2015

G.f.: z^2*(11 + 26*z - 109*z^2 - 223*z^3 + 294*z^4 + 620*z^5 - 306*z^6 -764*z^7 + 100*z^8 + 414*z^9 + 5*z^10 - 92*z^11 - 3*z^12 + 7*z^13)/((1 - z)*(1 + z)*(1 + z - 3*z^2 - z^3 + z^4)*(1 - z - 3*z^2 + z^3 + z^4)*(1 - z - 5*z^2 - z^3 + z^4)).

A290470 Number of minimal edge covers in the n-Moebius ladder.

Original entry on oeis.org

3, 7, 15, 59, 143, 367, 1039, 2755, 7395, 20007, 53727, 144635, 389535, 1048159, 2821535, 7595267, 20443523, 55029319, 148125295, 398712379, 1073232175, 2888862159, 7776059055, 20931132355, 56341155043, 151655701607, 408217663167, 1098815603707, 2957725352255

Offset: 1

Eric W. Weisstein, Aug 03 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Edge Cover
Eric Weisstein's World of Mathematics, Minimal Edge Cover
Eric Weisstein's World of Mathematics, Moebius Ladder
Index entries for linear recurrences with constant coefficients, signature (1, 2, 6, 2, 2, -2, -2, -1, 1).

Cf. A020866, A020877, A020878, A284710.

Table[2 Cos[n Pi/2] - RootSum[-1 + # + #^2 + #^3 &, #^n &] +
  RootSum[1 - 2 #^2 - 2 #^3 + #^4 &, #^n &], {n, 20}]
LinearRecurrence[{1, 2, 6, 2, 2, -2, -2, -1, 1}, {3, 7, 15, 59, 143, 367, 1039, 2755, 7395}, 20]
CoefficientList[Series[-(((1 + x) (-3 - x - x^2 + x^3) (-1 - 4 x^3 + 3 x^4))/((1 + x^2) (-1 - x - x^2 + x^3) (1 - 2 x - 2 x^2 + x^4))), {x, 0, 20}], x]

Vec((1 + x)*(1 + 4*x^3 - 3*x^4)*(3 + x + x^2 - x^3)/((1 + x^2)*(1 + x + x^2 - x^3)*(1 - 2*x - 2*x^2 + x^4)) + O(x^30)) \\ Andrew Howroyd, Aug 04 2017

From Andrew Howroyd, Aug 04 2017: (Start)
a(n) = a(n-1) + 2*a(n-2) + 6*a(n-3) + 2*a(n-4) + 2*a(n-5) - 2*a(n-6) - 2*a(n-7) - a(n-8) + a(n-9) for n > 9.
G.f.: x*(1 + x)*(1 + 4*x^3 - 3*x^4)*(3 + x + x^2 - x^3)/((1 + x^2)*(1 + x + x^2 - x^3)*(1 - 2*x - 2*x^2 + x^4)).
(End)

a(1)-a(2) and terms a(9) and beyond from Andrew Howroyd, Aug 04 2017

cp's OEIS Frontend

A098600 a(n) = Fibonacci(n-1) + Fibonacci(n+1) - (-1)^n.

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A162483 a(n) is the number of perfect matchings of an edge-labeled 2 X (2n+1) Mobius grid graph.

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A263200 Number of perfect matchings on a Möbius strip of width 3 and length 2n.

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A263201 Number of perfect matchings on a Möbius strip of width 4 and length n.

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Mathematica

PARI

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A290470 Number of minimal edge covers in the n-Moebius ladder.

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