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A001924
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Apply partial sum operator twice to Fibonacci numbers.
(Formerly M2645 N1053)
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35
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Leading coefficients in certain rook polynomials (for n>=2; see p. 18 of the Riordan paper). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 08 2004
A107909(a(n)) = A000225(n) = 2^n - 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 28 2005
(1, 3, 7, 14,...) = row sums of triangle A141289. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 22 2008
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REFERENCES
| 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
| T. D. Noe, Table of n, a(n) for n=0..500
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.
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-2,-1,1).
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FORMULA
| 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 (wolfdieter.lang(AT)physik.uni-karlsruhe.de) ]
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 (se16(AT)btinternet.com), Jan 03 2003
a(n)=sum(k=0, n, sum(i=0, k, F(i))). - Benoit Cloitre (benoit7848c(AT)orange.fr), 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 (pbarry(AT)wit.ie), Mar 26 2003
a(n)=sum(k=0, n, F(k)*(n-k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 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 (qntmpkt(AT)yahoo.com), Sep 19 2007
a(n)=Sum_{k=1..n}{C(n-k+2,k+1)}, with n>=0. - Paolo P. Lava (paoloplava(AT)gmail.com), Apr 16 2008
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MAPLE
| A001924:=-1/(z**2+z-1)/(z-1)**2; [Conjectured by S. Plouffe in his 1992 dissertation.]
The conjecture by Simon Plouffe needs to have the numerator changed from -1 to z. (* RGWv *)
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MATHEMATICA
| Join[{b=0}, a=0; Table[c=b+a+n; a=b; b=c, {n, 1, 60}]] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 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] (* RGWv *)
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PROG
| (PARI) a(n)=fibonacci(n+4)-n-3 \\ Charles R Greathouse IV, Feb 24, 2011
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CROSSREFS
| Cf. A000045, A001891, A133640, A141289.
Right-hand column 4 of triangle A011794.
Sequence in context: A008646 A036830 A014153 * A079921 A014168 A132109
Adjacent sequences: A001921 A001922 A001923 * A001925 A001926 A001927
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Better description 1/97.
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