A207361 Displacement under constant discrete unit surge.

0, 1, 11, 53, 173, 448, 994, 1974, 3606, 6171, 10021, 15587, 23387, 34034, 48244, 66844, 90780, 121125, 159087, 206017, 263417, 332948, 416438, 515890, 633490, 771615, 932841, 1119951, 1335943, 1584038, 1867688

Offset

0,3

Comments

Assume discrete times 0, 1, 2, 3, ...

Assume constant discrete unit surge (= jerk = rate of change of acceleration) starting surge(0) = 0.

Also assume acceleration(0) = velocity(0) = displacement(0) = 0.

So at t = 0, 1, 2, 3, 4, ... the acceleration = 0, 1, 2, 3, 4, ...

Then the velocity v(t) = v(t-1) + a(t)*t.

So the displacement = s(t) = s(t-1) + v(t)*t.

v(0,1,2,3,4,...) = 0, 1, 5, 14, 30, 55, 91, 140, ... = A000330(n).

The subsequence of primes is finite with three terms 11, 53, and 173.

Links

Table of n, a(n) for n=0..30.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

a(0) = 0; for n>0, a(n) = a(n-1) + n*A000330(n) = a(n-1) + n*(0^2 + 1^2 + 2^2 + ... + n^2) = a(n-1) + n^2*(n+1)*(2*n+1)/6 = n*(1+n)*(2+n)*(1 + 11*n + 8*n^2)/120 = (2*n + 25*n^2 + 50*n^3 + 35*n^4 + 8*n^5)/120.

G.f.: x*(2*x^2+5*x+1) / (x-1)^6. - Colin Barker, May 06 2013

a(n) = Sum_{i=0..n-1} A108678(i). - J. M. Bergot, May 02 2018

a(n) = Sum_{0<=i<=j<=n} i^2*j. - Robert FERREOL, May 24 2022

Example

s(4) = s(3) + v(4)*4 =  53 +  30*4 =  53 + 120 =  173;
s(5) = s(4) + v(5)*5 = 173 +  55*5 = 173 + 275 =  448;
s(6) = s(5) + v(6)*6 = 448 +  91*6 = 448 + 546 =  994;
s(7) = s(6) + v(7)*7 = 994 + 140*7 = 994 + 980 = 1974.

Maple

a:=n->sum(sum(i^2*j, j=i..n), i=0..n): seq(a(n), n=0..30); # Robert FERREOL, May 24 2022

Mathematica

a[0] = 0; a[n_] := a[n] = a[n-1] + n^2*(n+1)*(2*n+1)/6; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 22 2015 *)

Prog

(Maxima) A207361(n) := block(
        n*(1+n)*(2+n)*(1+11*n+8*n^2)/120
)$ /* R. J. Mathar, Mar 08 2012 */

Crossrefs

Cf. A000330, A101856, A108678.

Sequence in context: A164580 A141893 A093198 * A093864 A139967 A227965

Adjacent sequences: A207358 A207359 A207360 * A207362 A207363 A207364

Keyword

nonn,easy

Author

Jonathan Vos Post, Feb 18 2012

Status

approved