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%I A001595 M2453 N0974
%S A001595 1,1,3,5,9,15,25,41,67,109,177,287,465,753,1219,1973,3193,5167,8361,13529,
%T A001595 21891,35421,57313,92735,150049,242785,392835,635621,1028457,1664079,
%U A001595 2692537,4356617,7049155,11405773,18454929,29860703,48315633,78176337
%N A001595 a(n) = a(n-1) + a(n-2) + 1, with a(0) = a(1) = 1.
%C A001595 2-ranks of difference sets constructed from Segre hyperovals.
%C A001595 Sometimes called Leonardo numbers. - George Pollard (porges+oeis(AT)porg.es), 
               Jan 02 2008
%C A001595 A001595=Sum of first n Fibonacci numbers minus previous Fibonacci number. 
               [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 13 2009]
%D A001595 Dijkstra, E. W., 'Fibonacci numbers and Leonardo numbers', circulated 
               privately, July 1981.
%D A001595 Dijkstra, E. W., 'Smoothsort, an alternative for sorting in situ', Science 
               of Computer Programming, 1(3): 223-233, 1982.
%D A001595 R. Evans, H. D. L. Hollmann, C. Krattenthaler, Q. Xiang, Gauss Sums, 
               Jacobi Sums and p-Ranks of Cyclic Difference Sets, J. Combin. Theory 
               Ser. A 87 (1999), 74-119.
%D A001595 D. Singmaster, Some counterexamples and problems on linear recurrences, 
               Fib. Quart. 8 (1970), 264-267.
%D A001595 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001595 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001595 Q. Xiang, On Balanced Binary Sequences with Two-Level Autocorrelation 
               Functions, IEEE Trans. Inform. Theory 44 (1998), 3153-3156.
%H A001595 T. D. Noe, <a href="b001595.txt">Table of n, a(n) for n=0..500</a>
%H A001595 E. W. Dijkstra, <a href="http://www.cs.utexas.edu/users/EWD/ewd07xx/EWD796a.PDF">
               Smoothsort, an alternative for sorting in situ (EWD796a)</a>.
%H A001595 E. W. Dijkstra, <a href="http://www.cs.utexas.edu/users/EWD/ewd07xx/EWD797.PDF">
               Fibonacci numbers and Leonardo numbers (EWD797)</a>.
%H A001595 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=1019">
               Encyclopedia of Combinatorial Structures 1019</a>
%H A001595 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 A001595 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.
%H A001595 <a href="http://radon.mat.univie.ac.at/People/kratt/artikel/glynn.html">
               Supplement to "Gauss Sums, Jacobi Sums and p-Ranks ..."</a>
%F A001595 a(n)=2*Fibonacci(n+1)-1. - Richard L. Ollerton (r.ollerton(AT)uws.edu.au), 
               Mar 22 2002.
%F A001595 G.f. (1-x+x^2)/(1-2x+x^3) = 2/(1-x-x^2) - 1/(1-x). [Conjectured by S. 
               Plouffe in his 1992 dissertation; this is readily verified.]
%F A001595 G.f. for left-shifted sequence is (1+x-x^2)/(1-2*x+x^3).
%F A001595 a(n) = (2/sqrt(5))*((1+sqrt(5))/2)^(n+1) - 2/sqrt(5)*((1-sqrt(5))/2)^(n+1) 
               - 1.
%F A001595 a(n+1)/a(n) is asymptotic to Phi = (1+sqrt(5))/2. - Jonathan Vos Post 
               (jvospost3(AT)gmail.com), May 26 2005
%F A001595 For n >= 2, a(n+1) = ceiling(Phi*a(n)). - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), 
               Sep 30 2009
%F A001595 a(n) = Sum[A109754(n-k+1,k),{k,0,n+1}] - Sum[A109754(n-k,k),{k,0,n}] 
               = Sum[A101220(n-k+1,0,k),{k,0,n+1}] - Sum[A101220(n-k,0,k),{k,0,n}]. 
               - Ross La Haye (rlahaye(AT)new.rr.com), May 31 2006
%F A001595 a(n)=F(n)+F(n+3)-1 n>=-1 {where F(n) is the n-th Fibonacci number} - 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 31 2008
%p A001595 L := 1,3: for i from 3 to 100 do l := nops([ L ]): L := L,op(l,[ L ])+op(l-1,
               [ L ])+1: od: [ L ];
%p A001595 A001595:=(1-z+z**2)/(z-1)/(z**2+z-1); [S. Plouffe in his 1992 dissertation.]
%p A001595 with(combinat): seq(fibonacci(n)+fibonacci(n+3)-1, n=-1..32); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jan 31 2008
%t A001595 Join[ {1, 3}, Table[ a[ 1 ]=1; a[ 2 ]=3; a[ i ]=a[ i-1 ]+a[ i-2 ]+1, 
               {i, 3, 100} ] ]
%t A001595 a=0;lst={};Do[f=Fibonacci[n];a+=f;AppendTo[lst,a-Fibonacci[n-1]],{n,5!}];
               lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 13 2009]
%o A001595 sage: from sage.combinat.sloane_functions import recur_gen2b sage: it 
               = recur_gen2b(1,1,1,1, lambda n: 1) sage: [it.next() for i in xrange(1,
               38)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2008
%o A001595 (PARI) a(n) = 2*fibonacci(n+1)-1 - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net) 
               Sep 30 2009
%Y A001595 Cf. A049112, A049114, A000045, A128587.
%Y A001595 Sequence in context: A053522 A053521 A128587 this_sequence A092369 A061969 
               A034084
%Y A001595 Adjacent sequences: A001592 A001593 A001594 this_sequence A001596 A001597 
               A001598
%K A001595 nonn,easy,nice
%O A001595 0,3
%A A001595 N. J. A. Sloane (njas(AT)research.att.com).
%E A001595 Additional comments from Christian Krattenthaler (kratt(AT)ap.univie.ac.at).
%E A001595 Further edits from Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net) 
               Sep 30 2009 and N. J. A. Sloane, Oct 03 2009

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