A084178 -id:A084178 - cp's OEIS frontend

A084179 Expansion of the g.f. x/((1+2x)(1-x-x^2)).

0, 1, -1, 4, -5, 15, -22, 57, -93, 220, -385, 859, -1574, 3381, -6385, 13380, -25773, 53143, -103702, 211585, -416405, 843756, -1669801, 3368259, -6690150, 13455325, -26789257, 53774932, -107232053, 214978335, -429124630, 859595529, -1717012749, 3437550076

Offset: 0

Paul Barry, May 18 2003

Sums of consecutive pairs yield A084178.

Number of walks of length n+1 between two vertices at distance 2 in the cycle graph C_5. In general a(n,m) = 2^n/m*Sum_{k=0..m-1} cos(4*Pi*k/m)*cos(2*Pi*k/m)^n is the number of walks of length n between two vertices at distance 2 in the cycle graph C_m. - Herbert Kociemba, May 31 2004

Robert Israel, Table of n, a(n) for n = 0..3260
Index entries for linear recurrences with constant coefficients, signature (-1,3,2).

Cf. A000032, A007598, A084178.

I:=[0,1,-1]; [n le 3 select I[n] else -Self(n-1)+3*Self(n-2)+2*Self(n-3): n in [1..45]]; // Vincenzo Librandi, Nov 10 2014

f:= gfun:-rectoproc({a(n) = -a(n-1)+3*a(n-2)+2*a(n-3),
   a(0)=0,a(1)=1,a(2)=-1},a(n),remember):
seq(f(n), n=0..100); # Robert Israel, Dec 11 2015

CoefficientList[Series[x / ((1 + 2 x) (1 - x - x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Nov 10 2014 *)

concat(0, Vec(x/((1+2*x)*(1-x-x^2)) + O(x^100))) \\ Altug Alkan, Dec 11 2015

Binomial transform is A007598. The unsigned sequence has G.f. x/((1-2x)(1+x-x^2)) with a(n) = 2*2^n/5-(-1)^n*A000032(n)/5. - Paul Barry, Apr 17 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n, k)*Fib(k)^2; a(n) = ((1/2-sqrt(5)/2)^n+(1/2+sqrt(5)/2)^n-2(-2)^n)/5; a(n) = A000032(n)/5-2(-2)^n/5. - Paul Barry, Apr 17 2004
a(n) = 2^n/5*Sum_{k=0..4} cos(4*Pi*k/5)*cos(2*Pi*k/5)^n. - Herbert Kociemba, May 31 2004
a(n) = -a(n-1) + 3*a(n-2) + 2*a(n-3) for n>2. - Paul Curtz, Mar 09 2008

A098703 a(n) = (3^n + phi^(n-1) + (-phi)^(1-n)) / 5, where phi denotes the golden ratio A001622.

Original entry on oeis.org

0, 1, 2, 6, 17, 50, 148, 441, 1318, 3946, 11825, 35454, 106328, 318929, 956698, 2869950, 8609617, 25828474, 77484812, 232453449, 697358750, 2092073666, 6276216817, 18828643686, 56485920112, 169457742625, 508373199218, 1525119551286

Offset: 0

Ross La Haye, Oct 27 2004

Sums of antidiagonals of A090888.
Partial sums of A099159.
Form an array with m(0,n) = A000045(n), the Fibonacci numbers, and m(i,j) = Sum_{kJ. M. Bergot, May 27 2013

			a(2) = 2 because 3^2 = 9, Luc(1) = 1 and (9 + 1) / 5 = 2.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein, Golden Ratio
Eric Weisstein, Lucas Number
Eric Weisstein, Fibonacci Number
Index entries for linear recurrences with constant coefficients, signature (4,-2,-3).

Cf. A000032, A000045, A000244, A000285, A001622, A001654, A022383, A052964, A084178, A090888, A094688, A099159.

I:=[0,1,2]; [n le 3 select I[n] else 4*Self(n-1)-2*Self(n-2)-3*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 18 2018

f[n_] := (3^n + Fibonacci[n] + Fibonacci[n - 2])/5; Table[ f[n], {n, 0, 27}] (* Robert G. Wilson v, Nov 04 2004 *)
LinearRecurrence[{4, -2, -3}, {0, 1, 2}, 30] (* Jean-François Alcover, Feb 17 2018 *)

def A098703(n): return (3**n + lucas_number2(n-1,1,-1))//5
print([A098703(n) for n in range(21)]) # G. C. Greubel, Jun 02 2025

a(n) = (((1 + sqrt(5))^n - (1 - sqrt(5))^n) / (2^n*sqrt(5))) + ((3^n - (((1 + sqrt(5)) / 2)^(n+1) + ((1 - sqrt(5)) / 2)^(n+1))) / 5).
a(n) = (3^n + (((1 + sqrt(5)) / 2)^(n-1) + ((1 - sqrt(5)) / 2)^(n-1))) / 5.
a(n) = (3^n + A000032(n-1))/5 = A000045(n) + (3^n - A000032(n+1))/5.
a(n) = (3^n + A000045(n) + A000045(n-2))/5.
a(n) = (3^n + 4*A000045(n) - A000045(n+2))/5.
a(n) = Sum_{k=0...n-1} (A000045(k)*3^(n-k-1) - A000045(k-2)*2^(n-k-1)).
a(n) = 4*a(n-1) - 2*a(n-2) - 3*a(n-3).
a(n) = A000045(n) + A094688(n-1).
a(n) = 3^1 * a(n-1) - A000045(n-3), for n > 2.
a(n) = 3^2 * a(n-2) - A000285(n-4), for n > 3.
a(n) = 3^3 * a(n-3) - A022383(n-5), for n > 4.
Limit_{n -> oo} a(n) / a(n-1) = 3.
From Ross La Haye, Dec 21 2004: (Start)
a(n) = A101220(1,3,n).
Binomial transform of unsigned A084178.
Binomial transform of signed A084178 gives the Fibonacci oblongs (A001654). (End)
G.f.: x*(1-2*x)/((1-3*x)*(1-x-x^2)). - Ross La Haye, Aug 09 2005
a(0) = 0, a(1) = 1, a(n) = a(n-1) + a(n-2) + 3^(n-2) for n > 1. - Ross La Haye, Aug 20 2005
Binomial transform of A052964 beginning {0,1,0,3,1,10,...}. - Ross La Haye, May 31 2006

More terms from Robert G. Wilson v, Nov 04 2004

More terms from Ross La Haye, Dec 21 2004

cp's OEIS Frontend

A084179 Expansion of the g.f. x/((1+2x)(1-x-x^2)).

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