A137742 - OEIS

%I #57 Mar 31 2024 16:26:37

%S 1,8,21,40,66,100,143,196,260,336,425,528,646,780,931,1100,1288,1496,

%T 1725,1976,2250,2548,2871,3220,3596,4000,4433,4896,5390,5916,6475,

%U 7068,7696,8360,9061,9800,10578,11396,12255,13156,14100,15088,16121,17200,18326,19500

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

%C 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.

%C 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

%H Vincenzo Librandi, <a href="/A137742/b137742.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Do#repeat">Index entries for doubling substrings</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F From _Bruno Berselli_, Aug 23 2011: (Start)

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

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

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

%F From _Ilya Gutkovskiy_, Jul 01 2016: (Start)

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

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

%F a(n) = binomial(n+4,3) - 2*(n+4) for n > 1. - _Michael Chu_, Dec 09 2021

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

%t 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 *)

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

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

%o (Magma) [1] cat [(n^3+9*n^2+14*n-24)/6: n in [2..46]];  // _Bruno Berselli_, Aug 23 2011

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

%Y See A275874 for another version.

%K nonn,easy

%O 1,2

%A _M. F. Hasler_, Feb 10 2008