A137742 a(n) = (n-1)*(n+4)*(n+6)/6 for n > 1, a(1)=1.

1, 8, 21, 40, 66, 100, 143, 196, 260, 336, 425, 528, 646, 780, 931, 1100, 1288, 1496, 1725, 1976, 2250, 2548, 2871, 3220, 3596, 4000, 4433, 4896, 5390, 5916, 6475, 7068, 7696, 8360, 9061, 9800, 10578, 11396, 12255, 13156, 14100, 15088, 16121, 17200, 18326, 19500

Offset: 1

Author

M. F. Hasler, Feb 10 2008

Keywords

• nonn
• easy

Comments

Also the number of different strings of length n+3 obtained from "123...n" by iteratively duplicating any substring (see A137743 for comments and examples). This is the principal (although not simplest) definition of this sequence and explains why a(1)=1 and not 0.

For n >= 3, sequence appears (not yet proved by induction) to give the number of multiplications between two nonzero matrix elements in calculating the product of two n X n Hessenberg matrices (square matrices which have 0's below the subdiagonal, other elements being in general nonzero). - John M. Coffey, Jun 21 2016

Examples

			a(5) = (5-1)*(5+4)*(5+6)/6 = 4*9*11/6 = 66. - _Michael B. Porter_, Jul 02 2016
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for doubling substrings
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A137740-A137743, A135473, A137744-A137748.

See A275874 for another version.

Programs

Magma

[1] cat [(n^3+9*n^2+14*n-24)/6: n in [2..46]];  // Bruno Berselli, Aug 23 2011

Mathematica

Join[{1},Table[Binomial[n,3]-2*n,{n,6,60}]] (*or*) Join[{1},Table[(n-1)(n+4)(n+6)/6,{n,2,56}]] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

PARI

A137742(n)=if(n<2,1,n=A135473(n+3,n);n[ #n]) /* function A135473 defined in A137743 */

PARI

A137742(n)=if(n<2,1,(n - 1)*(n + 4)*(n + 6)/6)

Formula

From Bruno Berselli, Aug 23 2011: (Start)

G.f.: x*(1+4*x-5*x^2+x^4)/(1-x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

a(-n-7) = -A000297(n). (End)

From Ilya Gutkovskiy, Jul 01 2016: (Start)

E.g.f.: 4 + x + (-24 + 24*x + 12*x^2 + x^3)*exp(x)/6.

Sum_{n>=1} 1/a(n) = 1542/1225. (End)

a(n) = binomial(n+4,3) - 2*(n+4) for n > 1. - Michael Chu, Dec 09 2021