This site is supported by donations to The OEIS Foundation. 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!)
 A094429 Given the 3 X 3 matrix M = [0 1 0 / 0 0 1 / 7 -14 7], a(n) = (-) rightmost term of M^n * [1 1 1]. 10
 0, 7, 42, 196, 833, 3381, 13377, 52136, 201341, 773122, 2958032, 11291903, 43042727, 163918671, 623875840, 2373568575, 9028148962, 34334213564, 130560389505, 496440779373, 1887579497489, 7176808297736, 27286630574917 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS M is derived from the Lucas polynomial: x^3 - 7*x^2 + 14*x - 7 with a root (and eigenvalue of the matrix): 3.801377358... = (2*sin(3*Pi/7))^2, the convergent of the sequence. From Roman Witula, Sep 29 2012: (Start) The Berndt-type sequence number 16 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 16). Note that all numbers of the form a(n)*7^(-1 - floor((n-1)/3)) are integers and even a(10) and a(11) are divisible by 7^5. (End) LINKS G. C. Greubel, Table of n, a(n) for n = 1..1700 Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6. Index entries for linear recurrences with constant coefficients, signature (7,-14,7). FORMULA From Colin Barker, Jun 19 2012: (Start) a(n) = 7*a(n-1) - 14*a(n-2) + 7*a(n-3). G.f.: 7*x^2*(1-x)/(1 - 7*x + 14*x^2 - 7*x^3). (End) a(n) = c(4)*(s(1))^(2*n) + c(2)*(s(4))^(2*n) + c(1)*(s(2))^(2*n) = (-1/sqrt(7))*(c(1)*(s(1))^(2*n+3) + c(2)*(s(2))^(2*n+3) + c(3)*(s(3))^(2*n+3)) = (-1/sqrt(7))*(s(2)*(s(1))^(2*n+2) + s(4)*(s(2))^(2*n+2) + s(1)*(s(4))^(2*n+2)), where c(j) := 2*cos(2*Pi*j/7) and s(j) := 2*sin(2*Pi*j/7) (for the sums of the respective odd powers see A217274, see also A215493 and comments to A215494). For the proof of these formulas see Witula-Slota's paper. - Roman Witula, Jul 24 2012 EXAMPLE a(5) = 833. M^5 * [1 1 1] = [ -42 -196 -833]. We have 4*a(4) - a(5) = 4*a(5) - a(6) = 7*a(2) = 49, 88*a(10) = 23*a(11), and a(3) = 6*a(2), which implies the equalities c(4)*(s(1))^6 + c(2)*(s(4))^6 + c(1)*(s(2))^6 = 6*(c(4)*(s(1))^4 + c(2)*(s(4))^4 + c(1)*(s(2))^4) and s(2)*(s(1))^8 + s(4)*(s(2))^8 + s(1)*(s(4))^8 = 6*( s(2)*(s(1))^6 + s(4)*(s(2))^6 + s(1)*(s(4))^6). - Roman Witula, Sep 29 2012 MATHEMATICA Table[(MatrixPower[{{0, 1, 0}, {0, 0, 1}, {7, -14, 7}}, n].{-1, -1, -1})[], {n, 23}] (* Robert G. Wilson v, May 08 2004 *) LinearRecurrence[{7, -14, 7}, {0, 7, 42}, 50] (* Roman Witula, Aug 13 2012 *) PROG (PARI) x='x+O('x^30); concat(, Vec(7*x^2*(1-x)/(1-7*x+14*x^2-7*x^3))) \\ G. C. Greubel, May 09 2018 (PARI) a(n) = -(([0, 1, 0; 0, 0, 1; 7, -14, 7]^n)*[1, 1, 1]~); \\ Michel Marcus, May 10 2018 (MAGMA) I:=[0, 7, 42]; [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 CROSSREFS Cf. A215007, A215008, A215143, A215493, A215494, A215510, A215512, A215575, A215694, A215695, A108716, A215794, A215828, A215817, A215877, A094430, A217274. Sequence in context: A057425 A248329 A073376 * A246434 A255614 A022731 Adjacent sequences:  A094426 A094427 A094428 * A094430 A094431 A094432 KEYWORD nonn,easy AUTHOR Gary W. Adamson, May 02 2004 EXTENSIONS More terms from Robert G. Wilson v, May 08 2004 STATUS approved

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.

Last modified December 11 02:24 EST 2019. Contains 329910 sequences. (Running on oeis4.)