0,3

2-ranks of difference sets constructed from Segre hyperovals.

Sometimes called Leonardo numbers. - George Pollard (porges+oeis(AT)porg.es), Jan 02 2008

A001595=Sum of first n Fibonacci numbers minus previous Fibonacci number. [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 13 2009]

Dijkstra, E. W., 'Fibonacci numbers and Leonardo numbers', circulated privately, July 1981.

Dijkstra, E. W., 'Smoothsort, an alternative for sorting in situ', Science of Computer Programming, 1(3): 223-233, 1982.

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. Singmaster, Some counterexamples and problems on linear recurrences, Fib. Quart. 8 (1970), 264-267.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Q. Xiang, On Balanced Binary Sequences with Two-Level Autocorrelation Functions, IEEE Trans. Inform. Theory 44 (1998), 3153-3156.

T. D. Noe, Table of n, a(n) for n=0..500

E. W. Dijkstra, Smoothsort, an alternative for sorting in situ (EWD796a).

E. W. Dijkstra, Fibonacci numbers and Leonardo numbers (EWD797).

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1019

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Supplement to "Gauss Sums, Jacobi Sums and p-Ranks ..."

a(n)=2*Fibonacci(n+1)-1. - Richard L. Ollerton (r.ollerton(AT)uws.edu.au), Mar 22 2002.

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.]

G.f. for left-shifted sequence is (1+x-x^2)/(1-2*x+x^3).

a(n) = (2/sqrt(5))*((1+sqrt(5))/2)^(n+1) - 2/sqrt(5)*((1-sqrt(5))/2)^(n+1) - 1.

a(n+1)/a(n) is asymptotic to Phi = (1+sqrt(5))/2. - Jonathan Vos Post (jvospost3(AT)gmail.com), May 26 2005

For n >= 2, a(n+1) = ceiling(Phi*a(n)). - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Sep 30 2009

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

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

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 ];

A001595:=(1-z+z**2)/(z-1)/(z**2+z-1); [S. Plouffe in his 1992 dissertation.]

with(combinat): seq(fibonacci(n)+fibonacci(n+3)-1, n=-1..32); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 31 2008

Join[ {1, 3}, Table[ a[ 1 ]=1; a[ 2 ]=3; a[ i ]=a[ i-1 ]+a[ i-2 ]+1, {i, 3, 100} ] ]

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]

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

(PARI) a(n) = 2*fibonacci(n+1)-1 - Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net) Sep 30 2009

Cf. A049112, A049114, A000045, A128587.

Sequence in context: A053522 A053521 A128587 this_sequence A092369 A061969 A034084

Adjacent sequences: A001592 A001593 A001594 this_sequence A001596 A001597 A001598

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Additional comments from Christian Krattenthaler (kratt(AT)ap.univie.ac.at).

Further edits from Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net) Sep 30 2009 and N. J. A. Sloane, Oct 03 2009