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 A001595 a(n) = a(n-1) + a(n-2) + 1, with a(0) = a(1) = 1. (Formerly M2453 N0974) 35
 1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, 13529, 21891, 35421, 57313, 92735, 150049, 242785, 392835, 635621, 1028457, 1664079, 2692537, 4356617, 7049155, 11405773, 18454929, 29860703, 48315633, 78176337 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 2-ranks of difference sets constructed from Segre hyperovals. Sometimes called Leonardo numbers. - George Pollard, Jan 02 2008 a(n) is the number of nodes in the Fibonacci tree of order n. A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node (see the Knuth reference, p. 417). - Emeric Deutsch, Jun 14 2010 Also odd numbers whose index is a Fibonacci number: odd(Fib(k)). - Carmine Suriano, Oct 21 2010 This is the sequence A(1,1;1,1;1) of the family of sequences [a,b:c,d:k] considered by Gary Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 17 2010 REFERENCES 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. D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417. 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). J. Ziegenbalg, Algorithmen, Spektrum Akademischer Verlag, 1996, p. 172. LINKS T. D. Noe, Table of n, a(n) for n = 0..500 Paula M. M. C. Catarino and Anabela Borges, On Leonardo numbers, Acta Mathematica Universitatis Comenianae (2019), 1-12. P. Catarino and A. Borges, A Note on Incomplete Leonardo Numbers= 2, a(n+1) = ceiling(Phi*a(n)). - Franklin T. Adams-Watters, Sep 30 2009 a(n) = Sum_{k=0..n+1} A109754(n-k+1,k) - Sum_{k=0..n} A109754(n-k,k) = Sum_{k=0..n+1} A101220(n-k+1,0,k) - Sum_{k=0..n} A101220(n-k,0,k). - Ross La Haye, May 31 2006 a(n) = A000071(n+3) - A000045(n). - Vladimir Joseph Stephan Orlovsky, Oct 13 2009 a(n) = Fibonacci(n-1) + Fibonacci(n+2) - 1. - Zerinvary Lajos, Jan 31 2008, corrected by R. J. Mathar, Dec 17 2010 a(n) = 2*a(n-1) - a(n-3); a(0)=1, a(1)=1, a(2)=3. - Harvey P. Dale, Aug 07 2012 EXAMPLE a(7) = odd(F(7)) = odd(8) = 15. - Carmine Suriano, Oct 21 2010 MAPLE L := 1, 3: for i from 3 to 40 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); # Simon Plouffe in his 1992 dissertation with(combinat): seq(fibonacci(n-1)+fibonacci(n+2)-1, n=0..40); # Zerinvary Lajos, Jan 31 2008 MATHEMATICA Join[{1, 3}, Table[a=1; a=3; a[i]=a[i-1]+a[i-2]+1, {i, 3, 40} ] ] Table[2*Fibonacci[n+1]-1, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Oct 13 2009; modified by G. C. Greubel, Jul 10 2019 *) RecurrenceTable[{a==a==1, a[n]==a[n-1]+a[n-2]+1}, a, {n, 40}] (* or *) LinearRecurrence[{2, 0, -1}, {1, 1, 3}, 40] (* Harvey P. Dale, Aug 07 2012 *) PROG (PARI) a(n) = 2*fibonacci(n+1)-1 \\ Franklin T. Adams-Watters, Sep 30 2009 (Haskell) a001595 n = a001595_list !! n a001595_list =    1 : 1 : (map (+ 1) \$ zipWith (+) a001595_list \$ tail a001595_list) -- Reinhard Zumkeller, Aug 14 2011 (Magma) [2*Fibonacci(n+1)-1: n in [0..40]]; // G. C. Greubel, Jul 10 2019 (Sage) [2*fibonacci(n+1)-1 for n in (0..40)] # G. C. Greubel, Jul 10 2019 (GAP) List([0..40], n-> 2*Fibonacci(n+1) -1); # G. C. Greubel, Jul 10 2019 CROSSREFS Cf. A049112, A049114, A000045, A128587, A033538. Sequence in context: A053522 A053521 A128587 * A092369 A298340 A061969 Adjacent sequences:  A001592 A001593 A001594 * A001596 A001597 A001598 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Additional comments from Christian Krattenthaler (kratt(AT)ap.univie.ac.at) Further edits from Franklin T. Adams-Watters, Sep 30 2009, and N. J. A. Sloane, Oct 03 2009 STATUS approved

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Last modified September 24 19:48 EDT 2022. Contains 356949 sequences. (Running on oeis4.)