A005362 Hoggatt sequence with parameter d=4.

1, 2, 7, 32, 177, 1122, 7898, 60398, 494078, 4274228, 38763298, 366039104, 3579512809, 36091415154, 373853631974, 3966563630394, 42997859838010, 475191259977060, 5344193918791710, 61066078557804360, 707984385321707910, 8318207051955884772, 98936727936728464152

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn

Comments

Let V be the vector representation of SL(4) (of dimension 4) and let E be the exterior algebra of V (of dimension 16). Then a(n) is the dimension of the subspace of invariant tensors in the n-th tensor power of E. - Bruce Westbury, Feb 18 2021

This is the number of 4-vicious walkers (aka vicious 4-watermelons) - see Essam and Guttmann (1995). This is the 4-walker analog of A001181. - N. J. A. Sloane, Mar 22 2021

References

• D. C. Fielder and C. O. Alford, "An investigation of sequences derived from Hoggatt sums and Hoggatt triangles", in G. E. Bergum et al., editors, Applications of Fibonacci Numbers: Proc. Third Internat. Conf. on Fibonacci Numbers and Their Applications, Pisa, Jul 25-29, 1988. Kluwer, Dordrecht, Vol. 3, 1990, pp. 77-88.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..845
• J. W. Essam and A. J. Guttmann, Vicious walkers and directed polymer networks in general dimensions, Physical Review E, 52(6), (1995) pp. 5849-5862. See (60) and (63).
• D. C. Fielder, Letter to N. J. A. Sloane, Jun 1988
• D. C. Fielder and C. O. Alford, On a conjecture by Hoggatt with extensions to Hoggatt sums and Hoggatt triangles, Fib. Quart., 27 (1989), 160-168.
• D. C. Fielder and C. O. Alford, An investigation of sequences derived from Hoggatt Sums and Hoggatt Triangles, Application of Fibonacci Numbers, 3 (1990) 77-88. Proceedings of 'The Third Annual Conference on Fibonacci Numbers and Their Applications,' Pisa, Italy, July 25-29, 1988. (Annotated scanned copy)
• Nick Hobson, Python program for this sequence
• Vaclav Kotesovec, Calculation of the asymptotic formula for the sequence A005366

Crossrefs

Cf. A005364, A005365, A005366, A056940, A116925.

Programs

Magma

A056940:= func< n,k | (&*[Binomial(n+j,k)/Binomial(k+j,k): j in [0..3]]) >;
A005362:= func< n | (&+[A056940(n,k): k in [0..n]]) >;
[A005362(n): n in [0..30]]; // G. C. Greubel, Nov 14 2022

Maple

a := n -> hypergeom([-3-n, -2-n, -1-n, -n], [2, 3, 4], 1):
seq(simplify(a(n)), n=0..25); # Peter Luschny, Feb 18 2021

Mathematica

A005362[n_]:=HypergeometricPFQ[{-3-n,-2-n,-1-n,-n},{2,3,4},1] (* Richard L. Ollerton, Sep 12 2006 *)

SageMath

def A005362(n): return simplify(hypergeometric([-3-n, -2-n, -1-n, -n],[2,3,4], 1))
[A005362(n) for n in range(41)] # G. C. Greubel, Nov 14 2022

Formula

From Richard L. Ollerton, Sep 12 2006: (Start)

a(n) = Hypergeometric4F3([-3-n, -2-n, -1-n, -n], [2, 3, 4], 1).

(n+3)*(n+4)*(n+5)*(n+6)*a(n) = 6*(n+1)*(n+3)*(n+4)*(2*n+5)*a(n-1) + 4*(n-1)*n*(4*n+7)*(4*n+9)*a(n-2); a(0)=1, a(1)=2. (End)

a(n) = S(4,n) where S(d,n) is defined in A005364. - Sean A. Irvine, May 29 2016

a(n) ~ 3 * 2^(4*n + 29/2) / (Pi^(3/2) * n^(15/2)). - Vaclav Kotesovec, Apr 01 2021