login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A094430 a(n) is the rightmost term of M^n * [1 0 0], where M is the 3 X 3 matrix [0 1 0 / 0 0 1 / 7 -14 7]. 9

%I

%S 7,49,245,1078,4459,17836,69972,271313,1044435,4002467,15294370,

%T 58337097,222255768,846131608,3219700183,12247849145,46582062709,

%U 177142452214,673583231587,2561162729076,9737971026812,37024601601729

%N a(n) is the rightmost term of M^n * [1 0 0], where M is the 3 X 3 matrix [0 1 0 / 0 0 1 / 7 -14 7].

%C In A094429 the multiplier is [1 1 1] instead of [1 0 0]. The matrix M is derived from the 3rd-order Lucas polynomial x^3 - 7x^2 + 14x - 7, with a convergent of the series = 3.801937735... = (2 sin 3*Pi/7)^2; (an eigenvalue of the matrix and a root of the polynomial).

%C From _Roman Witula_, Sep 29 2012: (Start)

%C This sequence is the Berndt-type sequence number 17 for the argument 2*Pi/7 (see Formula section and Crossrefs for other Berndt-type sequences for the argument 2*Pi/7 - for numbers from 1 to 18 without 17).

%C Note that all numbers of the form a(n)*7^(-floor((n+4)/3)) are integers. (End)

%H G. C. Greubel, <a href="/A094430/b094430.txt">Table of n, a(n) for n = 1..1000</a>

%H Roman Witula and Damian Slota, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Slota/witula13.html">New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7</a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,7).

%F From _Colin Barker_, Jun 19 2012: (Start)

%F a(n) = 7*a(n-1)-14*a(n-2)+7*a(n-3).

%F G.f.: 7*x/(1-7*x+14*x^2-7*x^3). (End)

%F -a(n) = s(2)*s(1)^(2*n+3) + s(4)*s(2)^(2*n+3) + s(1)*s(4)^(2*n+3), where s(j) := 2*sin(2*Pi*j/7); for the proof see A215494 and the Witula-Slota paper. This formula and the respective recurrence also give a(0)=a(-1)=0. - _Roman Witula_, Aug 13 2012

%e a(4) = 1078 since M^4 * [1 0 0] = [49 245 1078] = [a(2), a(3), a(4)].

%e We have a(2)=7*a(1), a(3)=5*a(2), 22*a(3)=5*a(4), and a(6)=4*a(5), which implies s(2)*s(1)^15 + s(4)*s(2)^15 + s(1)*s(4)^15 = 4*(s(2)*s(1)^13 + s(4)*s(2)^13 + s(1)*s(4)^13). - _Roman Witula_, Sep 29 2012

%t Table[(MatrixPower[{{0, 1, 0}, {0, 0, 1}, {7, -14, 7}}, n].{1, 0, 0})[[3]], {n, 22}] (* _Robert G. Wilson v_, May 08 2004 *)

%t Join[{7}, LinearRecurrence[{7,-14,7}, {49,245,1078}, 50]] (* _Roman Witula_, Aug 13 2012 *)(* corrected by _G. C. Greubel_, May 09 2018 *)

%o (PARI) x='x+O('x^30); Vec(7*x/(1-7*x+14*x^2-7*x^3)) \\ _G. C. Greubel_, May 09 2018

%o (PARI) a(n) = (([0, 1, 0; 0, 0, 1; 7, -14, 7]^n)*[1,0,0]~)[3]; \\ _Michel Marcus_, May 10 2018

%o (MAGMA) I:=[49,245,1078]; [7] cat [n le 3 select I[n] else 7*Self(n-1) -14*Self(n-2) + 7*Self(n-3): n in [1..30]]; // _G. C. Greubel_, May 09 2018

%Y Cf. A215007, A215008, A215143, A215493, A215494, A215510, A215512, A215575, A215694, A215695, A108716, A215794, A215828, A215817, A215877, A094429, A217274.

%K nonn,easy

%O 1,1

%A _Gary W. Adamson_, May 02 2004

%E More terms from _Robert G. Wilson v_, May 08 2004

%E Name edited by _Michel Marcus_, May 10 2018

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 10 20:36 EST 2019. Contains 329909 sequences. (Running on oeis4.)