A163260 -id:A163260 - cp's OEIS frontend

A163259 Triangle T(n,k) read by rows: mod(A007318(n,k+1);A007318(n,k)).

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 4, 1, 0, 0, 0, 0, 5, 1, 0, 0, 3, 5, 15, 6, 1, 0, 0, 0, 14, 0, 21, 7, 1, 0, 0, 4, 0, 14, 56, 28, 8, 1, 0, 0, 0, 12, 42, 0, 84, 36, 9, 1, 0, 0, 5, 30, 90, 42, 210, 120, 45, 10, 1, 0, 0, 0, 0, 0, 132, 0, 330, 165, 55, 11, 1, 0, 0, 6, 22, 55, 297, 132, 792, 495

Offset: 1

Mats Granvik, Jul 23 2009

The zeros in this table form the pattern of ones in A051731.

			Table begins:
0
0...0
0...1...0
0...0...1...0
0...2...4...1...0
0...0...0...5...1...0
0...3...5..15...6...1...0
0...0..14...0..21...7...1...0
0...4...0..14..56..28...8...1...0
0...0..12..42...0..84..36...9...1...0
0...5..30..90..42.210.120..45..10...1...0

G. C. Greubel, Table of n, a(n) for the first 50 rows

Cf. A163260, A051731, A007318.

T[n_, k_] := Mod[Binomial[n, k + 1], Binomial[n, k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]// Flatten (* G. C. Greubel, Dec 12 2016 *)

T(n, k) = mod(binomial(n, k + 1), binomial(n, k)), for 0 <= k <= n, n>= 0. - G. C. Greubel, Dec 12 2016

Corrected the formula in the title Mats Granvik, Jul 24 2009

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

Roman Witula, Aug 04 2012

The Ramanujan-type sequence the number 9 for the argument 2*Pi/7. The sequence is connecting with the following decomposition: (s(4)/s(1))^(1/3)*s(1)^n + (s(1)/s(2))^(1/3)*s(2)^n + (s(2)/s(4))^(1/3)*s(4)^n = x(n)*(4-3*7^(1/3))^(1/3) + y(n)*(11-3*49^(1/3))^(1/3), where s(j) := sin(2*Pi*j/7), x(0)=1, x(1)=-7^(1/6)/2, x(2)=y(0)=y(1)=0, y(2)=7^(1/3)/4 and X(n)=sqrt(7)*(X(n-1)-X(n-3)) for every n=3,4,..., and X=x or X=y. It could be deduced the formula 4*y(n) = a(n)*7^(1/3 + (3+(-1)^n)/4), which implies a(0)=0, a(1)= 0, a(2)= 1/7, a(3)=1/7, a(4)=1, a(5)=6/7, i.e., A163260(n)=7*a(n) for every n=0,1,...,5. The sequence a(n) is discussed in third Witula paper.

			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.

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

LinearRecurrence[{0,7,0,-14,0,7}, {5,4,22,17,91,69}, {1,50}] (* G. C. Greubel, Apr 19 2018 *)

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

G.f.: -x*(1+x)*(6*x^4+x^3-12*x^2-x+5) / ( -1+7*x^2-14*x^4+7*x^6 ). - R. J. Mathar, Sep 14 2012

More terms from Michel Marcus, Apr 20 2016

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A163259 Triangle T(n,k) read by rows: mod(A007318(n,k+1);A007318(n,k)).

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