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%I #44 Sep 08 2022 08:45:13
%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
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