A216486 a(n) is equal to the rational part (considering of the field Q(sqrt(13))) of the numbers A(n)/sqrt(13), where we have A(n) = ((sqrt(13) - 1)/2)*A(n-1) + A(n-2) + ((3-sqrt(13))/2)*A(n-3), with A(0) = 6, A(1) = sqrt(13) - 1, and A(2) = 11 - sqrt(13).

0, 1, -1, 4, -3, 14, -10, 48, -37, 166, -144, 582, -570, 2067, -2260, 7421, -8923, 26878, -35020, 98039, -136612, 359649, -529990, 1325491, -2046310, 4903786, -7868991, 18199354, -30157768, 67720279, -115255425, 252540383, -439456837, 943488036

Offset: 0

Author

Roman Witula, Sep 11 2012

Keywords

• sign
• easy

Comments

The Berndt-type sequence number 2 for the argument 2*Pi/13 defined by the following relation: A216605(n) + a(n)*sqrt(13) = A(n) = 2*(c(1)^n + c(3)^n + c(4)^n), where c(j) := 2*cos(2*Pi*j/13), j=1..6. The numbers a(n), n=0,1,..., are all positive integers. We note that we also have A216605(n) - a(n)*sqrt(13) = B(n) = 2*(c(2)^n + c(5)^n + c(6)^n) and the following recurrence relation holds: B(n) = -((sqrt(13)+1)/2)*B(n-1) + B(n-2) + ((3+sqrt(13))/2)*B(n-3), with B(0) = 6, B(1) = -sqrt(13) - 1, and B(2) = 11 + sqrt(13).

We note that the sums a(2*n+1) + a(2*n+2) are nonnegative only for n = 0..5.

Examples

			We have a(5) + a(6) + a(4) + a(2) = a(7) + a(8) + a(6) + a(2) = a(9) + a(5) + a(1) + a(10) + a(8) = 0 and
  a(6) + a(9) + a(10) = a(11) + a(12) = 12.
Moreover, the following relations hold: A(3) = 4*A(1), B(3) = 4*B(1), A(5) = 4*A(3) + 2*sqrt(13), B(5) = 4*B(3)-2*sqrt(13), A(7) = 4*A(5) + 8*sqrt(13), B(7) = 4*B(5)-8*sqrt(13), A(4) = 3*A(2) - 2, B(4) = 3*B(2) + 2, 6 + A(6) = 3*A(4) + A(2), and A(8) - 3*A(6) = 25 - A(5)/2.
		

References

• R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, Thirteenth International Conference on Fibonacci Numbers and Their Applications, Congressus Numerantium, 201 (2010), 89-107.
• R. Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500
• R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, (abstract) see p. 15.
• Index entries for linear recurrences with constant coefficients, signature (-1,5,4,-6,-3,1).

Crossrefs

Cf. A216605.

Programs

Mathematica

LinearRecurrence[{-1, 5, 4, -6, -3, 1}, {0, 1, -1, 4, -3, 14}, 30]

PARI

concat([0],Vec((1-2*x^2+2*x^3+x^4)/(1+x-5*x^2-4*x^3+6*x^4+3*x^5-x^6) + O(x^30))) \\ Andrew Howroyd, Feb 25 2018

Formula

G.f.: x*(1 - 2*x^2 + 2*x^3 + x^4)/(1 + x - 5*x^2 - 4*x^3 + 6*x^4 + 3*x^5 - x^6).

a(n) = - a(n-1) + 5*a(n-2) + 4*a(n-3) - 6*a(n-4) - 3*a(n-5) + a(n-6), which from the respective polynomial-type formula follows given by Witula in section "Formula" in A216605.