A215794 a(n) = -7^n*A(2*n+1), where A(n) = A(n-1) + A(n-2) + A(n-3)/7, with A(0)=3, A(1)=1, A(2)=3.

-1, -31, -609, -11711, -224833, -4315871, -82846113, -1590286719, -30526618241, -585978870687, -11248256653025, -215917815567167, -4144686996149441, -79560041170858591, -1527208244431770145, -29315784501060168447, -562736106255347592449

Offset: 0

Author

Roman Witula, Aug 23 2012

Keywords

• sign
• easy

Comments

The Berndt-type sequence number 12 for the argument 2Pi/7 defined by the relation sqrt(7)*a(n) = t(1)^(2*n+1) + t(2)^(2*n+1) + t(4)^(2*n+1) = (-sqrt(7) + 4*s(1))^(2*n+1) + (-sqrt(7) + 4*s(2))^(2*n+1) + (-sqrt(7) + 4*s(4))^(2*n+1), where t(j) := tan(2*Pi*j/7) and s(j) := sin(2*Pi*j/7) (the respective sum with even powers in A108716 are given, see also A215828). We note that sqrt(7)*a(n) = B(2*n+1), where B(n) is defined in the comments to A215575. From Witula-Slota's (Section 6) and Witula's (Remark 11) papers it follows that B(n) is equal to the product (-sqrt(7))^n by the value of big omega function with index n for the argument 2*i/sqrt(7). The last value is equal to A(n). The respective recurrence relation for A(n) from the following decomposition follow (see Witula-Slota's paper for details): (X-1-2*i*d*s(1))*(X-1-2*i*d*s(2))*(X-1- 2*i*d*s(4)) = X^3 - (3+i*sqrt(7))*X^2 + (3+i*2*sqrt(7)*d)*X - (1+i*sqrt(7)*d + i*sqrt(7)*d^3), since the big omega function with index n for the argument d is equal to the sum: (1 + 2*i*d*s(1))^n + (1 + 2*i*d*s(2))^n + (1 + 2*i*d*s(4))^n and it is equal to 3 for n=0, 3 + i*sqrt(7)*d for n=1, and lastly 3 + 2*i*sqrt(7)*d - 7*d^2 for n=2.

The sequence a(n+1)/a(n) is decreasing and convergent to (t(2))^2 = 19.195669... Moreover we have floor(a(n+1)/a(n)) = 19 for every n=1,2,...

Examples

			We have -31*sqrt(7) = t(1)^3 + t(2)^3 + t(4)^3.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• 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
• Index entries for linear recurrences with constant coefficients, signature (21,-35,7).

Crossrefs

Cf. A215575, A108716, A215828, A275195.

Programs

Magma

m:=17; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(-(1+10*x-7*x^2)/(1-21*x+35*x^2-7*x^3)));  // Bruno Berselli, Aug 30 2012

Magma

I:=[-1, -31, -609]; [n le 3 select I[n] else 21*Self(n-1)-35*Self(n-2)+7*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Mar 19 2013

Mathematica

  LinearRecurrence[{21, -35, 7}, {-1, -31, -609}, 17] (* Bruno Berselli, Aug 30 2012 *)

Formula

G.f.: -(1+10*x-7*x^2)/(1-21*x+35*x^2-7*x^3). [Bruno Berselli, Aug 30 2012]

a(n) = -A275195(2*n-1)/(7^n). - Kai Wang, Aug 02 2016