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%I M2645 N1053 #178 Feb 08 2024 01:34:00
%S 0,1,3,7,14,26,46,79,133,221,364,596,972,1581,2567,4163,6746,10926,
%T 17690,28635,46345,75001,121368,196392,317784,514201,832011,1346239,
%U 2178278,3524546,5702854,9227431,14930317,24157781,39088132,63245948,102334116,165580101
%N Apply partial sum operator twice to Fibonacci numbers.
%C Leading coefficients in certain rook polynomials (for n>=2; see p. 18 of the Riordan paper). - _Emeric Deutsch_, Mar 08 2004
%C A107909(a(n)) = A000225(n) = 2^n - 1. - _Reinhard Zumkeller_, May 28 2005
%C (1, 3, 7, 14, ...) = row sums of triangle A141289. - _Gary W. Adamson_, Jun 22 2008
%C 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)*(1-2*x+x^(k+1)). Cf.A000217 the case for k=1, A001477 the case for k=0 (counts singleton subsets). - _Geoffrey Critzer_, Feb 17 2012
%C -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
%C 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
%C a(n) = A228074(n+1,3) for n > 1. - _Reinhard Zumkeller_, Aug 15 2013
%D J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001924/b001924.txt">Table of n, a(n) for n = 0..500</a>
%H Bader AlBdaiwi, <a href="https://arxiv.org/abs/1603.01807">On the Number of Cycles in a Graph</a>, arXiv preprint arXiv:1603.01807 [cs.DM], 2016.
%H J.-L. Baril and J.-M. Pallo, <a href="http://jl.baril.u-bourgogne.fr/Motzkin.pdf">Motzkin subposet and Motzkin geodesics in Tamari lattices</a>, 2013.
%H N-N. Cao and F-Z. Zhao, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Cao2/cao5r.html">Some Properties of Hyperfibonacci and Hyperlucas Numbers</a>, J. Int. Seq. 13 (2010) # 10.8.8.
%H Hung Viet Chu, <a href="https://arxiv.org/abs/2005.10081">Various Sequences from Counting Subsets</a>, arXiv:2005.10081 [math.CO], 2020.
%H Hung Viet Chu, <a href="https://arxiv.org/abs/2106.03659">Partial Sums of the Fibonacci Sequence</a>, arXiv:2106.03659 [math.CO], 2021.
%H Ligia Loretta Cristea, Ivica Martinjak, and Igor Urbiha, <a href="http://arxiv.org/abs/1606.06228">Hyperfibonacci Sequences and Polytopic Numbers</a>, arXiv:1606.06228 [math.CO], 2016.
%H E. Kilic and P. Stanica, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Kilic/kilic6.html">Generating matrices for weighted sums of second order linear recurrences</a>, JIS 12 (2009) 09.2.7.
%H Wolfdieter Lang, <a href="http://www.fq.math.ca/Scanned/36-4/elementary36-4.pdf">Problem B-858</a>, Fibonacci Quarterly, 36 (1998), 373-374, <a href="http://www.fq.math.ca/Scanned/36-4/elementary36-4.pdf">Solution</a>, ibid. 37 (1999) 183-184.
%H Candice A. Marshall, <a href="http://hdl.handle.net/11603/10353">Construction of Pseudo-Involutions in the Riordan Group</a>, Dissertation, Morgan State University, 2017.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H J. Riordan, <a href="/A000211/a000211.pdf">Discordant permutations</a>, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]
%H Stacey Wagner, <a href="http://via.library.depaul.edu/depaul-disc/vol2/iss1/2">Enumerating Alternating Permutations with One Alternating Descent</a>, DePaul Discoveries: Vol. 2: Iss. 1, Article 2.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F From _Wolfdieter Lang_: (Start)
%F G.f.: x/((1-x-x^2)*(1-x)^2).
%F Convolution of natural numbers n >= 1 with Fibonacci numbers F(k).
%F a(n) = Fibonacci(n+4) - (3+n). (End)
%F From _Henry Bottomley_, Jan 03 2003: (Start)
%F a(n) = a(n-1) + a(n-2) + n = a(n-1) + A000071(n+2).
%F a(n) = A001891(n) - a(n-1) = n + A001891(n-1).
%F a(n) = A065220(n+4) + 1 = A000126(n+1) - 1. (End)
%F a(n) = Sum_{k=0..n} Sum_{i=0..k} Fibonacci(i). - _Benoit Cloitre_, Jan 26 2003
%F 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
%F a(n) = Sum_{k=0..n} Fibonacci(k)*(n-k). - _Benoit Cloitre_, Jun 07 2004
%F a(n) - a(n-1) = A101220(1,1,n). - _Ross La Haye_, May 31 2006
%F F(n) + a(n-3) = A133640(n). - _Gary W. Adamson_, Sep 19 2007
%F a(n) = A077880(-3-n) = 2*a(n-1) - a(n-3) + 1. - _Michael Somos_, Dec 31 2012
%F 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
%F a(n) = Sum_{k=0..n} Sum_{i=0..n} i * C(n-k,k-i). - _Wesley Ivan Hurt_, Sep 21 2017
%F E.g.f.: exp(x/2)*(15*cosh(sqrt(5)*x/2) + 7*sqrt(5)*sinh(sqrt(5)*x/2))/5 - exp(x)*(3 + x). - _Stefano Spezia_, Jun 25 2022
%e 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
%e From _John M. Campbell_, Feb 10 2013: (Start)
%e There are a(5) = 26 bit strings with the pattern 00 and without the pattern 011 of length 5+1:
%e 000000, 000001, 000010, 000100, 000101, 001000,
%e 001001, 001010, 010000, 010001, 010010, 010100,
%e 100000, 100001, 100010, 100100, 100101, 101000, 101001,
%e 110000, 110001, 110010, 110100, 111000, 111001, 111100.
%e (End)
%p A001924:=-1/(z**2+z-1)/(z-1)**2; # Conjectured by _Simon Plouffe_ in his 1992 dissertation.
%p ##
%p a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <1|-1|-2|3>>^n.
%p <<0, 1, 3, 7>>)[1, 1]:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Oct 05 2012
%t a[n_]:= Fibonacci[n+4] -3-n; Array[a, 40, 0] (* _Robert G. Wilson v_ *)
%t LinearRecurrence[{3,-2,-1,1},{0,1,3,7},40] (* _Harvey P. Dale_, Jan 24 2015 *)
%t Nest[Accumulate,Fibonacci[Range[0,40]],2] (* _Harvey P. Dale_, Jun 15 2016 *)
%o (PARI) a(n)=fibonacci(n+4)-n-3 \\ _Charles R Greathouse IV_, Feb 24 2011
%o (Haskell)
%o a001924 n = a001924_list !! n
%o a001924_list = drop 3 $ zipWith (-) (tail a000045_list) [0..]
%o -- _Reinhard Zumkeller_, Nov 17 2013
%o (Magma) [Fibonacci(n+4)-(n+3): n in [0..40]]; // _Vincenzo Librandi_, Jun 23 2016
%o (Sage) [fibonacci(n+4) -n-3 for n in (0..40)] # _G. C. Greubel_, Jul 08 2019
%o (GAP) List([0..40], n-> Fibonacci(n+4) -n-3) # _G. C. Greubel_, Jul 08 2019
%Y Cf. A000045, A001891, A133640, A141289.
%Y Right-hand column 4 of triangle A011794.
%Y Cf. A014162, A039834, A077880, A122595, A129696.
%Y Cf. A065220.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E Description improved by _N. J. A. Sloane_, Jan 01 1997
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