Search: id:A000285
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%I A000285 M3246 N1309
%S A000285 1,4,5,9,14,23,37,60,97,157,254,411,665,1076,1741,2817,
%T A000285 4558,7375,11933,19308,31241,50549,81790,132339,214129,
%U A000285 346468,560597,907065,1467662,2374727,3842389,6217116,10059505
%N A000285 a(n) = a(n-1) + a(n-2).
%C A000285 a(n-1)=sum(P(4;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=3. These 
               are the sums over the SW-NE diagonals in P(4;n,k), the (4,1) Pascal 
               triangle A093561. Observation by Paul Barry (pbarry(AT)wit.ie, Apr 
               29 2004. Proof via recursion relations and comparison of inputs. 
               Also SW-NE diagonal sums in the Pascal (1,3) triangle A095660.
%D A000285 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000285 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci 
               Association, San Jose, CA, 1972, p. 53.
%D A000285 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224.
%D A000285 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000285 A. Brousseau, Seeking the lost gold mine or exploring Fibonacci factorizations, 
               Fib. Quart., 3 (1965), 129-130.
%H A000285 T. D. Noe, <a href="b000285.txt">Table of n, a(n) for n=0..500</a>
%H A000285 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A000285 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A000285 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A000285 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A000285 G.f.: (1+3*x)/(1-x-x^2).
%F A000285 Row sums of A131775 starting (1, 4, 5, 9, 14, 23,...). - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Jul 14 2007
%F A000285 a(n)=2*Fibonacci(n-2)+Fibonacci(n), n>=2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Oct 05 2007
%F A000285 a(n)=((1+sqrt5)^n-(1-sqrt5)^n)/(2^n*sqrt5)+ 1.5*((1+sqrt5)^(n-1)-(1-sqrt5)^(n-1))/
               (2^(n-2)*sqrt5). Offset 1. a(3)=5. [From Al Hakanson (hawkuu(AT)gmail.com), 
               Jan 14 2009]
%p A000285 BB := n->if n=1 then 3; > elif n=2 then 1; > else BB(n-2)+BB(n-1); > 
               fi: > L:=[]: for k from 2 to 34 do L:=[op(L),BB(k)]: od: L; - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 19 2007
%p A000285 with(combinat):a:=n->2*fibonacci(n-2)+fibonacci(n): seq(a(n), n=2..34); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007
%p A000285 A000285:=-(1+3*z)/(-1+z+z**2); [S. Plouffe in his 1992 dissertation.]
%t A000285 a=1;lst={a};s=6;Do[a=s-(a+1);AppendTo[lst,a];s+=a,{n,5!}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Oct 27 2009]
%Y A000285 Essentially the same as A104449.
%Y A000285 a(n) = A101220(3, 0, n+1).
%Y A000285 a(n) = A109754(3, n+1).
%Y A000285 a(k) = A090888(2, k-1), for k > 0.
%Y A000285 Cf. A131775.
%Y A000285 Sequence in context: A096818 A038099 A120740 this_sequence A042031 A041493 
               A042765
%Y A000285 Adjacent sequences: A000282 A000283 A000284 this_sequence A000286 A000287 
               A000288
%K A000285 easy,nonn,nice
%O A000285 0,2
%A A000285 N. J. A. Sloane (njas(AT)research.att.com).

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