A001891 Hit polynomials; convolution of natural numbers with Fibonacci numbers F(2), F(3), F(4), ....

0, 1, 4, 10, 21, 40, 72, 125, 212, 354, 585, 960, 1568, 2553, 4148, 6730, 10909, 17672, 28616, 46325, 74980, 121346, 196369, 317760, 514176, 831985, 1346212, 2178250, 3524517, 5702824, 9227400, 14930285, 24157748, 39088098, 63245913, 102334080, 165580064

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

Comments

a(n) is the sum of the n-th row of the triangle in A119457 for n > 0. - Reinhard Zumkeller, May 20 2006

Convolution of odds (A005408) with Fibonacci numbers (A000045). - Graeme McRae, Jun 06 2006

Equals row sums of triangle A152203. - Gary W. Adamson, Nov 29 2008

Define a triangle by T(n,0) = n*(n+1)+1, T(n,n) = 1, and T(r,c) = T(r-1,c) + T(r-2,c-1). This triangle starts: 1; 3,1; 7,2,1; 13,5,2,1; 21,12,4,2,1; the sum of terms in row n is a(n+1). - J. M. Bergot, Apr 23 2013

a(n) = number of k-tuples (u(1), u(2), ..., u(k)) with 1 <= u(1) < u(2) < ... < u(k) <= n such that u(i) - u(i-1) <= 2 for i = 2,...,k. Changing the bound from 2 to 3, then 4, then 5, yields A356619, A356620, A356621. The patterns suggest that the limiting sequence as the bound increases is A000295. - Clark Kimberling, Aug 24 2022

References

• J. Riordan, The enumeration of permutations with three-ply staircase restrictions, unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963. (See A001883)
• 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).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• N. J. A. Sloane, Annotated copy of Riordan's Three-Ply Staircase paper (unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963)
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Crossrefs

Partial sums of A001911.

A diagonal of triangle in A080061.

Right-hand column 5 of triangle A011794.

Cf. A001883-A001890. A152203.

Cf. A000295, A356619, A356620, A356621.

Programs

GAP

List([0..40], n-> Fibonacci(n+5) -2*n-5); # G. C. Greubel, Jul 06 2019

Magma

[Fibonacci(n+5)-(5+2*n): n in [0..40]]; // Vincenzo Librandi, Jun 07 2013

Mathematica

LinearRecurrence[{3,-2,-1,1}, {0,1,4,10}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
Table[Fibonacci[n+5] -(2*n+5), {n,0,40}] (* G. C. Greubel, Jul 06 2019 *)
maxDiff = 2;
Map[Length[Select[Map[{#, Max[Differences[#]]} &,
  Drop[Subsets[Range[#]], # + 1]], #[[2]] <= maxDiff &]] &,
  Range[16]] (* Peter J. C. Moses, Aug 14 2022 *)

PARI

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 1,-1,-2,3]^n*[0;1;4;10])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

Sage

[fibonacci(n+5) -2*n-5 for n in (0..40)] # G. C. Greubel, Jul 06 2019

Formula

G.f.: x*(1+x)/((1-x-x^2)*(1-x)^2). - Simon Plouffe in his 1992 dissertation

a(n) = Fibonacci(n+5) - (5+2*n). - Wolfdieter Lang

a(n) = a(n-1) + a(n-2) + (2n+1); a(-x)=0. - Barry E. Williams, Mar 27 2000

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4). - Sam Lachterman (slachterman(AT)fuse.net), Sep 22 2003

a(n) - a(n-1) = A101220(2,1,n). - Ross La Haye, May 31 2006

a(n) = (-3 + (2^(-1-n)*((1-sqrt(5))^n*(-11+5*sqrt(5)) + (1+sqrt(5))^n*(11+5*sqrt(5)))) / sqrt(5) - 2*(1+n)). - Colin Barker, Mar 11 2017