A142585 -id:A142585 - cp's OEIS frontend

A142710 a(n) = A142585(n) + A142586(n).

2, 2, 6, 14, 38, 112, 276, 814, 1998, 5702, 14226, 39404, 99908, 270922, 695106, 1859134, 4807518, 12748472, 33128916, 87394454, 227792678, 599050102, 1564242906, 4106054164, 10733283588, 28143585362, 73614464826, 192899714414, 504751433798, 1322156172352

Offset: 0

Paul Curtz, Sep 25 2008

Sum of the binomial and inverse binomial transforms of A014217.

Starting at a(1), the last digits form a period-4 sequence 2, 6, 4, 8.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,7,-12,-11,16,-4).

Cf. A000032, A014217, A142585, A142586, A147586.

[n eq 0 select 2 else (-1)^n*Lucas(n) +Lucas(2*n) -(1+(-1)^n)*2^(n-1): n in [0..50]]; // G. C. Greubel, Oct 26 2022

Join[{2},LinearRecurrence[{2,7,-12,-11,16,-4},{2,6,14,38,112,276},30]] (* Harvey P. Dale, Nov 25 2013 *)

def A142710(n): return (-1)^n*lucas_number2(n,1,-1) + lucas_number2(2*n,1,-1) - (1 + (-1)^n)*2^(n-1) -int(n==0)
[A142710(n) for n in range(51)] # G. C. Greubel, Oct 26 2022

a(n) = +2*a(n-1) +7*a(n-2) -12*a(n-3) -11*a(n-4) +16*a(n-5) -4*a(n-6), n>6. - R. J. Mathar, Jun 14 2010
G.f.: 2*(1-x-6*x^2+6*x^3+7*x^4-2*x^6)/((1-2*x)*(1+2*x)*(1+x-x^2)*(1-3*x+x^2)). - Colin Barker, Aug 13 2012
a(n) = (-1)^n*LucasL(n) + LucasL(2*n) - (1 + (-1)^n)*2^(n-1) - [n=0]. - G. C. Greubel, Oct 26 2022

Offset set to zero and extended - R. J. Mathar, Jun 14 2010

A175660 Eight bishops and one elephant on a 3 X 3 chessboard. a(n) = 2^(n+2) - 3*F(n+2).

Original entry on oeis.org

1, 2, 7, 17, 40, 89, 193, 410, 859, 1781, 3664, 7493, 15253, 30938, 62575, 126281, 254392, 511745, 1028281, 2064314, 4141171, 8302637, 16638112, 33329357, 66744685, 133628474, 267482023, 535328225, 1071245704, 2143444841

Offset: 0

Johannes W. Meijer, Aug 06 2010, Aug 10 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7, 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the central square the bishop turns into a raging elephant, see A175654.

The sequence above corresponds to four A[5] vectors with decimal values 171, 174, 234 and 426. These vectors lead for the side squares to A000079 and for the central square to A175661 (a(n) = 2^(n+2) - 3*F(n+1)).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, -1, -2).

Cf. A008466 (2^n-F(n+2)), A027934 (2^n-F(n+1)), A027974 (2^(n+3)-F(n+5)-F(n+3)), A074878 (3*2^n-2*F(n+2)), A142585 ((-1)^(n+1)*(2^(n-1)-F(n+1)-F(n-1))), A175661 (2^(n+2)-3*F(n+1)), A179610 (convolution of (-4)^n and F(n+1)).

nmax:=29; m:=1; A[5]:= [0,1,0,1,0,1,0,1,1]: A:=Matrix([[0,0,0,0,1,0,0,0,1], [0,0,0,1,0,1,0,0,0], [0,0,0,0,1,0,1,0,0], [0,1,0,0,0,0,0,1,0], A[5], [0,1,0,0,0,0,0,1,0], [0,0,1,0,1,0,0,0,0], [0,0,0,1,0,1,0,0,0], [1,0,0,0,1,0,0,0,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

Table[2^(n+2)-3Fibonacci[n+2],{n,0,30}] (* or *) LinearRecurrence[ {3,-1,-2},{1,2,7},30] (* Harvey P. Dale, Dec 28 2012 *)

G.f.: (1 - x + 2*x^2)/(1 - 3*x + x^2 + 2*x^3).
a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) with a(0)=1, a(1)=2 and a(2)=7.
a(n) = 2^(n+2) - 3*F(n+2) with F(n)=A000045(n).

A227200 a(n) = a(n-1) + a(n-2) - 2^(n-1) with a(0)=a(2)=0, a(1)=-a(3)=1, a(4)=-5.

Original entry on oeis.org

0, 1, 0, -1, -5, -14, -35, -81, -180, -389, -825, -1726, -3575, -7349, -15020, -30561, -61965, -125294, -252795, -509161, -1024100, -2057549, -4130225, -8284926, -16609455, -33282989, -66669660, -133507081, -267285605, -535010414, -1070731475

Offset: 0

Chandrakant N Phadte, Sep 18 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. N. Phadte and S. P. Pethe, On Second Order Non-homogeneous recurrence relation, Annales Mathematicae et informaticae, 41 (2013), pp. 205-210.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Cf. versions with different signs: A027974, A142585.

Cf. A000032, A000045.

LET N=0
LET L=0
LET M=1
PRINT L
PRINT M
FOR I=1 TO 30
LET N=M+L-(2)^(I-1)
PRINT N
LET L=M
LET M=N
NEXT I
END

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((1-3*x)/((1-2*x)*(1-x-x^2)))); // Bruno Berselli, Oct 03 2013

I:=[0,1,0,-1,-5]; [n le 5 select I[n] else Self(n-1)+Self(n-2)-2^(n-3): n in [1..35]]; // Vincenzo Librandi, Oct 05 2013

Table[LucasL[n + 1] - 2^n, {n, 0, 30}] (* Bruno Berselli, Oct 03 2013 *)
CoefficientList[Series[x (1 - 3 x)/((1 - 2 x) (1 - x - x^2)), {x, 0, 40}], x](* Vincenzo Librandi, Oct 05 2013 *)

a(n)=fibonacci(n)+fibonacci(n+2)-2^n \\ Charles R Greathouse IV, Oct 03 2013

G.f.: x*(1-3*x)/((1-2*x)*(1-x-x^2)).
a(n) = -(-1)^n*A142585(n+1) = A000032(n+1) - 2^n. [Bruno Berselli, Oct 03 2013]
a(n) = 3*a(n-1) -a(n-2) -2*a(n-3). [Bruno Berselli, Oct 03 2013]

More terms from Bruno Berselli, Oct 03 2013

cp's OEIS Frontend

A142710 a(n) = A142585(n) + A142586(n).

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A175660 Eight bishops and one elephant on a 3 X 3 chessboard. a(n) = 2^(n+2) - 3*F(n+2).

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A227200 a(n) = a(n-1) + a(n-2) - 2^(n-1) with a(0)=a(2)=0, a(1)=-a(3)=1, a(4)=-5.

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