A120758
The (1,3)-entry in the matrix M^n, where M is the 3 X 3 matrix [0,2,1; 2,1,2; 1,2,2] (n>=1).
Original entry on oeis.org
1, 6, 25, 116, 517, 2338, 10517, 47400, 213481, 961726, 4332145, 19515036, 87908397, 395998298, 1783838637, 8035595600, 36197658961, 163058307446, 734522939465, 3308779311556, 14904940203477, 67141752851858, 302451060668357, 1362440511764600, 6137337207120441
Offset: 1
a(7)=10517 because M^7= [6682,9842,10517;9842,14401,15438;10517,15438,16524].
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with(linalg): M[1]:=matrix(3,3,[0,2,1,2,1,2,1,2,2]): for n from 2 to 25 do M[n]:=multiply(M[1],M[n-1]) od: seq(M[n][3,1],n=1..25);
A215139
a(n) = (a(n-1) - a(n-3))*7^((1+(-1)^n)/2) with a(6)=5, a(7)=4, a(8)=22.
Original entry on oeis.org
5, 4, 22, 17, 91, 69, 364, 273, 1428, 1064, 5537, 4109, 21315, 15778, 81683, 60368, 312130, 230447, 1190553, 878423, 4535832, 3345279, 17267992, 12732160, 65708167, 48440175, 249956105, 184247938, 950654341, 700698236, 3615152086, 2664497745, 13746596563, 10131444477
Offset: 6
From values of x(2),y(2) and the identity 2*sin(t)^2=1-cos(2*t) we obtain (s(4)/s(1))^(1/3)*c(1) + (s(1)/s(2))^(1/3)*c(4) + (s(2)/s(4))^(1/3)*c(1) = (4-3*7^(1/3))^(1/3) - (1/2)*(7*(11-3*49^(1/3)))^(1/3), where c(j):=cos(2*Pi*j/7). Further, from values of x(1),x(3),y(1),y(3) and the identity 4*sin(t)^3=3*sin(t)-sin(3*t) we obtain (s(4)/s(1))^(1/3)*s(4) + (s(1)/s(2))^(1/3)*s(1) + (s(2)/s(4))^(1/3)*s(2) = (-3*7^(1/6)/2 +4*7^(1/2))*(4-3*7^(1/3))^(1/3) - 7^(5/6)*(11-3*49^(1/3))^(1/3).
- R. Witula, E. Hetmaniok and D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the Fifteenth International Conference on Fibonacci Numbers and Their Applications, Eger, Hungary, 2012.
- G. C. Greubel, Table of n, a(n) for n = 6..1005
- 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.
- Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2*Pi/7, Journal of Integer Sequences, Vol. 12 (2009), Article 09.8.5.
- Roman Witula, Full Description of Ramanujan Cubic Polynomials, Journal of Integer Sequences, Vol. 13 (2010), Article 10.5.7.
- Roman Witula, Ramanujan Cubic Polynomials of the Second Kind, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.5.
- Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012) 779-796.
- Index entries for linear recurrences with constant coefficients, signature (0,7,0,-14,0,7).
Cf.
A214683,
A215112,
A006053,
A006054,
A215076,
A215100,
A120757,
A215560,
A215569,
A215572,
A214699.
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I:=[5,4,22,17,91,69]; [n le 6 select I[n] else 7*Self(n-2) - 14*Self(n-4) + 7*Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 19 2018
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LinearRecurrence[{0,7,0,-14,0,7}, {5,4,22,17,91,69}, {1,50}] (* G. C. Greubel, Apr 19 2018 *)
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Vec(-x*(1+x)*(6*x^4+x^3-12*x^2-x+5)/(-1+7*x^2-14*x^4+7*x^6) + O(x^50)) \\ Michel Marcus, Apr 20 2016
A120775
a(n) = 3*a(n-1) + 5*a(n-2) + a(n-3).
Original entry on oeis.org
1, 6, 23, 100, 421, 1786, 7563, 32040, 135721, 574926, 2435423, 10316620, 43701901, 185124226, 784198803, 3321919440, 14071876561, 59609425686, 252509579303, 1069647742900, 4531100550901, 19194049946506, 81307300336923, 344423251294200, 1459000305513721
Offset: 1
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