cp's OEIS Frontend

A200541 Product of Fibonacci and tribonacci numbers: a(n) = A000045(n+1)*A000073(n+2).

%I A200541 #19 Mar 25 2024 09:37:04
%S A200541 1,1,4,12,35,104,312,924,2754,8195,24386,72576,215991,642785,1912960,
%T A200541 5693016,16942573,50421592,150056090,446571180,1329008590,3955167387,
%U A200541 11770690808,35029911168,104250013425,310251009501,923315841860,2747814245904,8177573467339,24336691577000
%N A200541 Product of Fibonacci and tribonacci numbers: a(n) = A000045(n+1)*A000073(n+2).
%C A200541 Limit a(n+1)/a(n) = (sqrt(5)+1)/2 * (1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3))/3 = 2.9760284849940...
%H A200541 Paul D. Hanna, <a href="/A200541/b200541.txt">Table of n, a(n) for n = 0..200</a>
%H A200541 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,5,2,-1,1).
%F A200541 G.f.: (1 - x^2 - x^3) / (1 - x - 4*x^2 - 5*x^3 - 2*x^4 + x^5 - x^6).
%e A200541 G.f.: A(x) = 1 + x + 4*x^2 + 12*x^3 + 35*x^4 + 104*x^5 + 312*x^6 + 924*x^7 + 2754*x^8 +...+ A000045(n+1)*A000073(n+2)*x^n +...
%e A200541 where tribonacci numbers (A000073) begin:
%e A200541 [1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136,5768,10609,...],
%e A200541 and Fibonacci numbers (A000045) begin:
%e A200541 [1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,...].
%t A200541 Module[{nn=30,fs,ts},fs=Fibonacci[Range[nn]];ts=LinearRecurrence[{1,1,1},{1,1,2},nn];Times@@@Thread[{fs,ts}]] (* or *) LinearRecurrence[ {1,4,5,2,-1,1},{1,1,4,12,35,104},30] (* _Harvey P. Dale_, Dec 14 2016 *)
%o A200541 (PARI) {a(n)=polcoeff((1-x^2-x^3)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6 +x*O(x^n)),n)}
%o A200541 (PARI) {A000073(n)=polcoeff(x^2/(1-x-x^2-x^3+x^3*O(x^n)),n)}
%o A200541 {a(n)=fibonacci(n+1)*A000073(n+2)}
%Y A200541 Cf. A001582, A000045, A000073.
%K A200541 nonn
%O A200541 0,3
%A A200541 _Paul D. Hanna_, Nov 18 2011