A172012 - OEIS

%I #25 Jun 30 2026 23:46:35

%S 2,3,15,54,207,783,2970,11259,42687,161838,613575,2326239,8819442,

%T 33437043,126769455,480619494,1822166847,6908359023,26191577610,

%U 99299809899,376474162527,1427321917278,5411388239415,20516130470079,77782556128482,294896059795683

%N Expansion of (2-3*x)/(1-3*x-3*x^2).

%C The case k=3 in a family of sequences a(n) = L(k,n), L(k,n)=k*(L(k,n-1)+L(k,n-2)), L(k,0)=2 and L(k,1)=k.

%C The case k=1 is A000032 (classic Lucas sequence), k=2 is A080040, this here is essentially A085480.

%H Vladimir V. Kruchinin and Maria Y. Perminova, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Perminova/perm3.html">Identities and Hadamard Product of the Generalized Fibonacci, Lucas, Catalan, and Harmonic Numbers</a>, Journal of Integer Sequences, Vol. 28 (2025), Article 25.8.8. See p. 5.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence: Specific names</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,3).

%F a(n) = 3*( a(n-1)+a(n-2) ) = 2*A030195(n+1)-3*A030195(n).

%F L(k,n) = c^n+b^n where c=(k+d)/2 ; b=(k-d)/2; d=sqrt(k*(k+4)) (Binet formula).

%F a(0)=2, a(1)=3, a(n) = 3*a(n-1)+3*a(n-2). - _Harvey P. Dale_, Aug 24 2011

%F a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 21*x^2))/2 )^n for n >= 1. - _Peter Bala_, Jun 23 2015

%F E.g.f.: 2*exp(3*x/2)*cosh(sqrt(21)*x/2). - _Stefano Spezia_, Dec 21 2025

%t CoefficientList[Series[(2-3x)/(1-3x-3x^2),{x,0,30}],x] (* _Harvey P. Dale_, Aug 24 2011 *)

%t (* Alternative: *)

%t LinearRecurrence[{3,3},{2,3},31] (* _Harvey P. Dale_, Aug 24 2011 *)

%Y Cf. A030195.

%K nonn,easy,changed

%O 0,1

%A Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Jan 22 2010

%E Edited and extended by _R. J. Mathar_, Jan 23 2010