A168175 Expansion of 1/(1 - 4*x + 7*x^2).

1, 4, 9, 8, -31, -180, -503, -752, 513, 7316, 25673, 51480, 26209, -255524, -1205559, -3033568, -3695359, 6453540, 51681673, 161551912, 284435937, 6880364, -1963530103, -7902282960, -17864421119, -16141703756, 60484132809

Offset: 0

Author

Roger L. Bagula, Nov 19 2009

Keywords

• sign
• easy

Comments

Also the coefficient of i of Q^(n+1), Q being the quaternion 2+i+j+k. The real part of the quaternion power is A213421, see also A087455, A088138, A128018. - Stanislav Sykora, Jun 11 2012

a(n)*(-1)^n gives the coefficient c(7^n) of (eta(z^6))^4, a modular cusp form of weight 2, when expanded in powers of q = exp(2*Pi*i*z), Im(z) > 0, assuming alpha-multiplicativity (but not for primes 2 and 3) with alpha(x) = x (weight 2) and input c(7) = -4. Eta is the Dedekind function. See the Apostol reference, p. 138, eq. (54) for alpha-multiplicativity and p. 130, eq. (39) with k=2. See also A000727(n) = b(n) where c(7^n) = b((7^n-1)/6) = b(A023000(n)), n >= 0. Proof: The alpha-multiplicity with alpha(1) = 1 and c(1) = 1 leads from p^n = p^(n-1)*p to the recurrence c_n = c*c_(n-1) - a*c(n-2), with c_n = c(p^n), c = c(p) and a = alpha(p). Inputs are c_{-1} = 0 and c_0 = c(1) = 1. This gives the polynomial c_n = sqrt(a)^n * S(n,c/sqrt(a)), with Chebyshev's S-polynomials (A049310). See the Apostol reference, Exercise 6., p. 139. Here p = 7, c = -4. - Wolfdieter Lang, Apr 27 2016

Examples

			G.f. = 1 + 4*x + 9*x^2 + 8*x^3 - 31*x^4 - 180*x^5 - 503*x^6 - 752*x^7 + ... - _Michael Somos_, Feb 23 2020
		

References

• Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 130, 138 - 139.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-7).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A023000, A049310.

Programs

Magma

I:=[1,4]; [n le 2 select I[n] else 4*Self(n-1)-7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 25 2012

Mathematica

CoefficientList[Series[1/(1-4x+7x^2),{x,0,30}],x] (* or *) LinearRecurrence[ {4,-7},{1,4},30] (* Harvey P. Dale, Nov 28 2014 *)

PARI

{a(n) = my(s=1, t=1); if( n<0, n=-2-n; s=-1; t=1/7); s * t^(n+1) * polcoeff(1 / (1 - 4*x + 7*x^2) + x * O(x^n), n)}; /* Michael Somos, Feb 23 2020 */

Formula

a(n) = (1/2 - i/sqrt(3))*(2 + i*sqrt(3))^n + (1/2 + i/sqrt(3))*(2 - i*sqrt(3))^n (Binet formula), where i is the imaginary unit.

a(n) = 4*a(n-1) - 7*a(n-2).

a(n) = sqrt(7)^n * S(n, 4/sqrt(7)), n >= 0, with Chebyshev's S polynomials (A049310). - Wolfdieter Lang, Apr 27 2016

E.g.f.: (2*sqrt(3)*sin(sqrt(3)*x) + 3*cos(sqrt(3)*x))*exp(2*x)/3. - Ilya Gutkovskiy, Apr 27 2016

a(n) = (-1) * 7^(n+1) * a(-2-n) for all n in Z. - Michael Somos, Feb 23 2020