A215484 a(n) = 21*a(n-2) + 7*a(n-3), with a(0)=a(1)=0, a(2)=9.

0, 0, 9, 0, 189, 63, 3969, 2646, 83790, 83349, 1778112, 2336859, 37923795, 61520823, 812757708, 1557403848, 17498557629, 38394784764, 378371537145, 928780383447, 8214565773393, 22152988812402, 179007343925382, 522714725474193, 3914225144119836, 12230060642435727, 85857731104835907, 284230849499989119

Offset: 0

Author

Roman Witula, Aug 13 2012

Keywords

• nonn
• easy

Comments

We have a(n)=C(n;3), where C(n;d), n=1,2,..., d in C, denote one of the quasi-Fibonacci numbers defined in the comments to A121449 and in the Witula-Slota-Warzynski paper. Its conjugate sequences A(n;3) and B(n;3) are discussed in A121458 and A215492, respectively. Similarly as in A121458, we deduce that each of the elements a(3*n), a(3*n+1), a(3*n+2) are divisible by 9*7^n for every n=0,1,... . Some additional facts connecting all three sequences a(n), A121458, and A215492 are given in the comments to A121458.

Examples

			We have a(5)=7*a(2), a(4)=21*a(2), a(4)=3*a(5), a(6)=21*a(4), a(7)=14*a(4), 3*a(7)=2*a(6), a(8)-a(9)=7*a(5), a(9)=21*a(6), 2*a(9)=63*a(7), a(12)=455*a(9) - especially the values and the relations connecting with a(8) and a(9) are very attractive.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• R. Witula, D. Slota and A. Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
• Index entries for linear recurrences with constant coefficients, signature (0, 21, 7).

Programs

Magma

I:=[0,0,9]; [n le 3 select I[n] else 21*Self(n-2) +7*Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 19 2018

Mathematica

LinearRecurrence[{0,21,7}, {0,0,9}, 50]
CoefficientList[Series[9x^2/(1-21x^2-7x^3),{x,0,30}],x] (* Harvey P. Dale, Jul 06 2021 *)

PARI

x='x+O('x^30); concat([0,0], Vec(9*x^2/(1-21*x^2-7*x^3))) \\ G. C. Greubel, Apr 19 2018

Formula

a(n) = (1/7)*((c(2)-c(4))*(1+3*c(1))^n + (c(4)-c(1))*(1+3*c(2))^n + (c(1)-c(2))*(1+3*c(4))^n), where c(j):=2*cos(2*Pi*j/7) (for the proof see formula (3.17) for d=3 in the Witula-Slota-Warzynski paper).

G.f.: 9*x^2/(1-21*x^2-7*x^3).