A066804 - OEIS

%I #21 Sep 08 2022 08:45:05

%S 1,8,34,100,235,476,868,1464,2325,3520,5126,7228,9919,13300,17480,

%T 22576,28713,36024,44650,54740,66451,79948,95404,113000,132925,155376,

%U 180558,208684,239975,274660,312976,355168,401489,452200,507570,567876

%N Sum of diagonal elements and those below it for a square matrix of integers, starting with 1.

%D T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

%H Vincenzo Librandi, <a href="/A066804/b066804.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).

%F a(n) = n*(n+1)*(2*n^2-n+2)/6.

%F G.f.: x*(1+3*x+4*x^2)/(1-x)^5. [_Bruno Berselli_, Jun 22 2013]

%F a(n) = n*A000217(n) - sum((n-4*i)*A000217(i), i=0..n-1). [_Bruno Berselli_, Jun 22 2013]

%F a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4)+a(n-5), with n>4, a(0)=1, a(1)=8, a(2)=34, a(3)=100, a(4)=235. [_Yosu Yurramendi_, Sep 03 2013]

%e a(7) = 7*28 - (7*0+3*1-1*3-5*6-9*10-13*15-17*21) = 868. [_Bruno Berselli_, Jun 22 2013]

%t Table[n(n+1)(2n^2-n+2)/6,{n,50}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,8,34,100,235},50] (* _Harvey P. Dale_, Dec 02 2016 *)

%o (Magma) [n*(n+1)*(2*n^2-n+2)/6: n in [1..30]]; // _Vincenzo Librandi_, May 22 2011

%o (R)

%o a <- c(1, 8, 34, 100, 235)

%o for(n in (length(a)+1):30) a[n] <- 5*a[n-1] -10*a[n-2] +10*a[n-3] -5*a[n-4]+a[n-5]

%o a

%o [_Yosu Yurramendi_, Sep 03 2013]

%K nonn,easy

%O 1,2

%A Aaron Gulliver (agullive(AT)ece.uvic.ca), Jan 19 2002

%E More terms from _Sascha Kurz_, Mar 23 2002