A065692 - OEIS

%I #15 Dec 29 2024 18:39:13

%S 0,3,24,117,417,1140,2583,7311,39840,251613,1329969,5738172,20838447,

%T 67344735,216818808,812592261,3660235905,17401156548,78077421639,

%U 318790895727,1197423287952,4316735957805,15912781695633,63028759626636,266607310489695,1146981374901183

%N Braided power sequence: this is b(n+1) = 3*b(n) + 2*d(n) - c(n), A065693 is c(n+1) = 3*c(n) + 2*b(n) - d(n) and A065694 is d(n+1) = 3*d(n) + 2*c(n) - b(n), starting with b(0) = 0, c(0) = 1 and d(0) = 2.

%C Tends to 4^n. "Braided" because the order of b(n), c(n) and d(n) changes between the six possibilities as n increases.

%H Andrew Howroyd, <a href="/A065692/b065692.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-33,52).

%F From _Colin Barker_, Sep 02 2016: (Start)

%F a(n) = 9*a(n-1) - 33*a(n-2) + 52*a(n-3) for n > 2.

%F G.f.: 3*x*(1 - x)/((1 - 4*x)*(1 - 5*x + 13*x^2)). (End)

%e a(1) = 3*0 + 2*2 - 1*1 = 3.

%o (PARI) a(n) = {[1,0,0]*[3,-1,2; 2,3,-1; -1,2,3]^n*[0,1,2]~} \\ _Andrew Howroyd_, Dec 29 2024

%Y Cf. A065693, A065694.

%K nonn,easy

%O 0,2

%A _Henry Bottomley_, Nov 14 2001

%E a(23) onwards from _Andrew Howroyd_, Dec 29 2024