A102002 - OEIS

%I #24 Sep 22 2025 16:00:46

%S 1,7,13,31,85,199,493,1231,3013,7447,18397,45343,111925,276199,681421,

%T 1681519,4149157,10237879,25262269,62334655,153810709,379529095,

%U 936489133,2310790159,5701884805,14069421655,34716351901,85662734431,211373124853,521564001319

%N Weighted tribonacci (1,2,4), companion to A102001.

%C a(n)/a(n-1) tends to 2.46750385...an eigenvalue of M and a root of the characteristic polynomial x^3 - x^2 - 2x - 4. A102001 is generated from [1 1 1 / 2 0 0 / 0 2 0] but has the same characteristic polynomial and recursive multipliers (1,2,4). A101000 uses the recursive multipliers (1,2,4,8).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,4).

%F a(n) = a(n-1) + 2*a(n-2) + 4*a(n-3), a>3. a(n) = center term in M^n * [1 1 1], where M = the 3X3 matrix [0 1 0 / 0 0 1 / 4 2 1]; M^n * [1 1 1] = [a(n-1) a(n) a(n+1)].

%F G.f.: -x*(4*x^2+6*x+1)/(4*x^3+2*x^2+x-1). [_Harvey P. Dale_, Apr 28 2012]

%e a(6) = 199 = 85 + 2*31 + 4*13 = a(5) + 2*a(4) + 4*a(3).

%e a(6) = 199 since M^6 * [1 1 1] = [85 199 493] = [a(5) a(6) a(7)].

%t LinearRecurrence[{1,2,4}, {1,7,13}, 50] (* _Harvey P. Dale_, Apr 28 2012 *)

%o (SageMath)

%o from sage.combinat.sloane_functions import recur_gen3

%o it = recur_gen3(1,1,1,1,2,4)

%o [next(it) for i in range(32)]

%o # _Zerinvary Lajos_, Jun 25 2008

%Y Cf. A102000, A102001.

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, Dec 23 2004

%E More terms from _Harvey P. Dale_, Apr 28 2012