Search: id:A006645
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%I A006645
%S A006645 0,0,1,4,14,44,131,376,1052,2888,7813,20892,55338,145428,379655,985520,
%T A006645 2545720,6547792,16777993,42847988,109099078,277040572,701794187,
%U A006645 1773851304,4474555476,11266301976,28318897549,71070913036,178106093666
%N A006645 Self-convolution of Pell numbers (A000129).
%F A006645 a(n)= sum(b(k)*b(n-k), k=0..n) with b(k) := A000129(k).
%F A006645 a(n)= sum(2^(n-2)*(n-k-1)*binomial(n-2-k, k)*(1/4)^k, k=0..floor((n-2)/
               2)), n>=2.
%F A006645 a(n)= ((n-1)*P(n)+n*P(n-1))/4, P(n)=A000129(n). G.f.: (x/(1-2*x-x^2))^2 
               - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 
               11 2000
%F A006645 a(n)=F'(n, 2), the derivative of the n-th Fibonacci polynomial evaluated 
               at x=2. - T. D. Noe (noe(AT)sspectra.com), Jan 19 2006
%F A006645 G.f.: sage: taylor( mul(x/(1 - 2*x - x^2) for i in xrange(1,3)),x,0,28)# 
               [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 03 2009]
%e A006645 sage: taylor( mul(x/(1 - 2*x - x^2) for i in xrange(1,3)),x,0,28)# solution>
               > x^2 + 4*x^3 + 14*x^4 + 44*x^5 + 131*x^6 + 376*x^7 + 1052*x^8 + 
               2888*x^9 + 7813*x^10 + 20892*x^11 + 55338*x^12 +...+ 4474555476*x^24 
               + 11266301976*x^25 + 28318897549*x^26+ 71070913036*x^27 + 178106093666*x^28+etc... 
               [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 03 2009]
%p A006645 a:= n-> (Matrix(4, (i,j)-> if i=j-1 then 1 elif j=1 then [4, -2, -4, 
               -1][i] else 0 fi)^n) [1,3]: seq (a(n), n=0..40); [From Alois P. Heinz 
               (heinz(AT)hs-heilbronn.de), Oct 28 2008]
%Y A006645 a(n)= A054456(n-1, 1), n>=1 (second column of triangle), A054457.
%Y A006645 Sequence in context: A007466 A062109 A118042 this_sequence A094309 A000300 
               A005323
%Y A006645 Adjacent sequences: A006642 A006643 A006644 this_sequence A006646 A006647 
               A006648
%K A006645 nonn
%O A006645 0,4
%A A006645 N. J. A. Sloane (njas(AT)research.att.com).
%E A006645 Sum formulae and cross-references added.- Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Aug 07 2002
%E A006645 More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 28 2008

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