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A027941
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Fibonacci(2n+1)-1.
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48
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0, 1, 4, 12, 33, 88, 232, 609, 1596, 4180, 10945, 28656, 75024, 196417, 514228, 1346268, 3524577, 9227464, 24157816, 63245985, 165580140, 433494436, 1134903169, 2971215072, 7778742048, 20365011073, 53316291172, 139583862444
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OFFSET
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0,3
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COMMENTS
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Also T(2n+1,n+1), T given by A027935. Also first row of Inverse Stolarsky array.
Number of Schroeder paths of length 2(n+1) having exactly one up step starting at an even height (a Schroeder path is a lattice path starting from (0,0), ending at a point on the x-axis, consisting only of steps U=(1,1) (up steps), D=(1,-1) (down steps) and H=(2,0) (level steps) and never going below the x-axis. Schroeder paths are counted by the large Schroeder numbers (A006318). Example. a(1)=4 because among the six Schroeder paths of length 4 only the paths (U)HD, (U)UDD, H(U)D, (U)DH have exactly one U step that starts at an even height (shown between parentheses). - Emeric Deutsch, Dec 19 2004
Also: smallest number not writeable as the sum of n or fewer Fibonacci numbers. E.g. a(4)=88 because it is the smallest number that needs at least 5 Fibonacci numbers: 88=55+21+8+3+1 - Johan Claes, Apr 19 2005
Except for first term, numbers a(n) that set a new record in the number of Fibonacci numbers needed to sum up to n. Position of records in sequence A007895. - R. Stephan, May 15 2005
Successive extremal petal bends beta(n) = a(n-2). See the Ring Lemma of Rodin and Sullivan in K. Stephenson, Introduction to Circle Packing (Cambridge U. P., 2005), pp. 73-74 and 318-321. - David W. Cantrell (DWCantrell(AT)sigmaxi.net)
a(n+1)= AAB^(n)(1), n>=1, with compositions of Wythoff's complementary A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link under A135817 for the Wythoff representation of numbers (with A as 1 and B as 0 and the argument 1 omitted). E.g. 4=`110`, 12=`1100`, 33=`11000`, 88=`110000,..., in Wythoff code. AA(1)=1=a(1) but for uniqueness reason 1=A(1) in Wythoff code.
Start with n. Each n generates a sublist {n-1,n-1,n-2,..,1}. Each element of each sublist also generates a sublist. Addnumbers in all terms. For example, 3->{2,2,1} and both 2->{1,1}, so a(3) = 3 + 2 + 2 + 1 + 1 + 1 + 1 + 1 = 12. - Jon Perry, Sep 01 2012
For n>0: smallest number such that the inner product of Zeckendorf binary representation and its reverse equals n: A216176(a(n)) = n, see also A189920. - Reinhard Zumkeller, Mar 10 2013
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REFERENCES
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A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 12.
C. Kimberling, "Interspersions and dispersions," Proceedings of the American Mathematical Society 117 (1993) 313-321.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..200
C. Kimberling, Interspersions
N. J. A. Sloane, Classic Sequences
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
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a(n) = sum(i=1, n, binomial(n+i, n-i)). - Benoit Cloitre, Oct 15 2002
G.f.: sum(k>=1, x^k/(1-x)^(2k+1)). - Benoit Cloitre, Apr 21 2003
Third diagonal of array defined by T(i, 1)=T(1, j)=1, T(i, j)=Max(T(i-1, j)+T(i-1, j-1); T(i-1, j-1)+T(i, j-1)). - Benoit Cloitre, Aug 05 2003
a(n) = sum(k=1, n, F(2k)), i.e. partial sums of A001906. - Benoit Cloitre, Oct 27 2003
a(n) = sum{k=0..n, U(k, 3/2)} = sum{k=0..n, S(k, 3)}, S(k, 3) = A001906(k+1) - Paul Barry, Nov 14 2003
G.f.: 1/((1-x)*(1-3*x+x^2)) = 1/(1-4*x+4*x^2-x^3).
a(n) = 4*a(n-1)-4*a(n-2)+a(n-3) with n>=2, a(-1)=0, a(0)=1, a(1)=4.
a(n) = 3*a(n-1)-a(n-2)+1 with n>=1, a(-1)=0, a(0)=1.
a(n) = Sum[ F(k)*L(k), {k,1,n} ], where L(k) = Lucas(k) = A000032(k) = F(k-1) + F(k+1). - Alexander Adamchuk, May 18 2007
a(n) = 2*a(n-1) + Sum[a(k), 0<k<n-1] + n. - Jon Perry, Sep 01 2012
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EXAMPLE
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a(5) = 88 = 2*33 + 12 + 4 + 1 + 5. a(6) = 232 = 2*88 + 33 + 12 + 4 + 1 + 6. - Jon Perry, Sep 01 2012
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MAPLE
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with(combinat): seq(fibonacci(2*n+1)-1, n=1..27); (Deutsch)
a:=n->sum(binomial(n+k+1, 2*k), k=0..n): seq(a(n), n=-1..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 02 2007
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MATHEMATICA
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Table[Fibonacci[2*n+1]-1, {n, 0, 17}] (* Vladimir Orlovsky, Jul 21 2008 *)
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PROG
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(MAGMA) [Fibonacci(2*n+1)-1: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011
(PARI) a(n)=fibonacci(2*n+1)-1 \\ Charles R Greathouse IV, Nov 20 2012
(Haskell)
a027941 = (subtract 1) . a000045 . (+ 1) . (* 2)
-- Reinhard Zumkeller, Mar 10 2013
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CROSSREFS
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Cf. A000045, A035507, A001906, A006318, A000032.
Related to partial sums of Fibonacci(k*n) over n: A000071, A099919, A058038, A138134, A05360; this sequence is the case k=2.
Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).
Cf. A000225 (sublist connection)
Sequence in context: A104747 A070050 A186025 * A219092 A135254 A000754
Adjacent sequences: A027938 A027939 A027940 * A027942 A027943 A027944
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Clark Kimberling
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EXTENSIONS
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More terms from James A. Sellers, Sep 08 2000
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STATUS
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approved
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