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

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, 4575358578833, 13726075615106

Offset

0,3

Comments

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_{k<i} m(k,j) + Sum_{k<j} m(i,k), which is the sum of the terms above m(i,j) plus the sum of the terms to the left of m(i,j). The sum of the terms in antidiagonal(n) = a(n+1). - J. M. Bergot, May 27 2013

Links

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).

Formula

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

E.g.f.: (1/5)*(exp(3*x) + exp(x/2)*(sqrt(5)*sinh(sqrt(5)*(x/2))-cosh(sqrt(5)*(x/2)))). - Enrique Navarrete, Mar 05 2026

Example

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

Maple

a:= n-> (<<0|1|0>, <0|0|1>, <-3|-2|4>>^n. <<0, 1, 2>>)[1, 1]:
seq(a(n), n=0..29);  # Alois P. Heinz, Mar 05 2026

Mathematica

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 *)

Prog

(Magma) 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
(SageMath)
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

Crossrefs

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

Sequence in context: A173993 A270863 A027914 * A025272 A394411 A356783

Adjacent sequences: A098700 A098701 A098702 * A098704 A098705 A098706

Keyword

nonn,easy

Author

Ross La Haye, Oct 27 2004

Extensions

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

More terms from Ross La Haye, Dec 21 2004

Status

approved