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A001924
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Apply partial sum operator twice to Fibonacci numbers.
(Formerly M2645 N1053)
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45
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0, 1, 3, 7, 14, 26, 46, 79, 133, 221, 364, 596, 972, 1581, 2567, 4163, 6746, 10926, 17690, 28635, 46345, 75001, 121368, 196392, 317784, 514201, 832011, 1346239, 2178278, 3524546, 5702854, 9227431, 14930317, 24157781, 39088132, 63245948, 102334116, 165580101
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OFFSET
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0,3
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COMMENTS
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Leading coefficients in certain rook polynomials (for n>=2; see p. 18 of the Riordan paper). - Emeric Deutsch, Mar 08 2004
A107909(a(n)) = A000225(n) = 2^n - 1. - Reinhard Zumkeller, May 28 2005
(1, 3, 7, 14,...) = row sums of triangle A141289. - Gary W. Adamson, Jun 22 2008
a(n) is the number of nonempty subsets of {1,2,...,n} such that the difference of successive elements is at most 2. See example below. Generally, the O.g.f. for the number of nonempty subsets of {1,2,...,n} such that the difference of successive elements is <= k is: x/((1-x)*(1-2x+x^(k+1)). Cf.A000217 the case for k=1, A001477 the case for k=0 (counts singleton subsets). - Geoffrey Critzer, Feb 17 2012
-fibonacci(n-2) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, Dec 31 2012
a(n) is the number of bit strings of length n+1 with the pattern 00 and without the pattern 011, see example. [John M. Campbell, Feb 10 2013].
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REFERENCES
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W. Lang, Problem B-858, Fibonacci Quarterly, 36 (1998), 373-374, ibid. 37 (1999) 183-184.
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
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).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..500
_Simon 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.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-2,-1,1).
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FORMULA
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G.f.: x/((1-x-x^2)*(1-x)^2). Convolution of natural numbers n >= 1 with Fibonacci numbers F(k). a(n)=F(n+4)-(3+n) [ Wolfdieter Lang ]
a(n) = a(n-1)+a(n-2)+n = Fib(n+4)-n-3 = a(n-1)+A000071(n+2) = A001891(n)-a(n-1) = n+A001891(n-1) = A065220(n+4)+1 = A000126(n+1)-1. - Henry Bottomley, Jan 03 2003
a(n) = sum(k=0, n, sum(i=0, k, F(i))). - Benoit Cloitre, Jan 26 2003
a(n) = (sqrt(5)/2+1/2)^n*(7*sqrt(5)/10+3/2)+(3/2-7*sqrt(5)/10)*(sqrt(5)/2-1/2)^n*(-1)^n-n-3 - Paul Barry, Mar 26 2003
a(n) = sum(k=0, n, F(k)*(n-k)) - Benoit Cloitre, Jun 07 2004
a(n) - a(n-1) = A101220(1,1,n). - Ross La Haye (rlahaye(AT)new.rr.com), May 31 2006
F(n) + a(n-3) = A133640(n). - Gary W. Adamson, Sep 19 2007
a(n) = Sum_{k=1..n} C(n-k+2,k+1), with n>=0. - Paolo P. Lava, Apr 16 2008
a(n) = A077880(-3-n) = 2*a(n-1) - a(n-3) + 1. - Michael Somos, Dec 31 2012
INVERT transform is A122595. PSUM transform is A014162. PSUMSIGN transform is A129696. BINOMIAL transform of A039834 with 0,1 prepended is this sequence. - Michael Somos, Dec 31 2012
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EXAMPLE
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a(5) = 26 because there are 31 nonempty subsets of {1,2,3,4,5} but 5 of these have successive elements that differ by 3 or more: {1,4}, {1,5}, {2,5}, {1,2,5}, {1,4,5}. - Geoffrey Critzer, Feb 17 2012
From John M. Campbell, Feb 10 2013: (Start)
There are a(5) = 26 bit strings with the pattern 00 and without the pattern 011 of length 5+1:
000000, 000001, 000010, 000100, 000101,
001000, 001001, 001010, 010000, 010001,
010010, 010100, 100000, 100001, 100010,
100100, 100101, 101000, 101001, 110000,
110001, 110010, 110100, 111000, 111001,
111100.
(End)
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MAPLE
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A001924:=-1/(z**2+z-1)/(z-1)**2; [Conjectured by Simon Plouffe in his 1992 dissertation.]
The conjecture by Simon Plouffe needs to have the numerator changed from -1 to z. (* Robert G. Wilson v *)
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a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <1|-1|-2|3>>^n.
<<0, 1, 3, 7>>)[1, 1]:
seq (a(n), n=0..40); # Alois P. Heinz, Oct 05 2012
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MATHEMATICA
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Join[{b=0}, a=0; Table[c=b+a+n; a=b; b=c, {n, 1, 60}]] [From Vladimir Joseph Stephan Orlovsky, Dec 10 2008]
f[n_] := Fibonacci[n + 4] - 3 - n; Array[f, 32, 0] (* Or *)
Accumulate@ Accumulate@ Fibonacci@ Range[0, 31] (* Or *)
a[n_] := a[n] = a[n - 1] + a[n - 2] + n; a[0] = 0; a[1] = 1; Array[a, 32, 0] (* Or )
gf = x/(1 - 3 x + 2 x^2 + x^3 - x^4); CoefficientList[ Series[ gf, {x, 0, 31}], x] (* Robert G. Wilson v *)
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PROG
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(PARI) a(n)=fibonacci(n+4)-n-3 \\ Charles R Greathouse IV, Feb 24, 2011
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CROSSREFS
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Cf. A000045, A001891, A133640, A141289.
Right-hand column 4 of triangle A011794.
Cf. A014162, A039834, A077880, A122595, A129696.
Sequence in context: A036830 A014153 * A079921 A014168 A132109 A099854
Adjacent sequences: A001921 A001922 A001923 * A001925 A001926 A001927
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Better description 1/97.
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STATUS
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approved
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