A101856 -id:A101856 - cp's OEIS frontend

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

Jonathan Vos Post, Feb 18 2012

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.

			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.

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

Cf. A000330, A101856, A108678.

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

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 *)

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

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

A292793 1/4 of the number of self-avoiding paths that are made of alternated vertical and horizontal n consecutive steps. (start point is different from end point.)

Original entry on oeis.org

1, 2, 4, 8, 16, 29, 54, 98, 176, 318, 572, 1026, 1826, 3255, 5794, 10233, 18172, 32012, 56488, 99469, 175034, 307479, 540068, 947235, 1659907, 2908958, 5095019, 8917677, 15598100, 27281252, 47718310, 83298748, 145405769, 253641303, 442352671, 770769569, 1343166519, 2339093953

Offset: 2

Seiichi Manyama, Sep 23 2017

nonn

			a(2) = 1;
   E
   |
   *
   |
S--*
a(3) = 2;
   *--*--*--E     E--*--*--*
   |                       |
   *                       *
   |                       |
S--*                    S--*
a(4) = 4;
            E                                      E
            |                                      |
            *                                      *
            |                                      |
            *                                      *
            |                                      |
            *                                      *
            |                                      |
   *--*--*--*        *--*--*--*     *--*--*--*     *--*--*--*
   |                 |        |     |        |              |
   *                 *        *     *        *              *
   |                 |        |     |        |              |
S--*              S--*        *     *     S--*           S--*
                              |     |
                              E     E

Cf. A101856.

def A292793(n)
  ary = [1]
  b_ary = [[[0, 0], [1, 0], [1, 1], [1, 2]]]
  s = 4
  (3..n).each{|i|
    s += i
    f_ary, b_ary = b_ary, []
    if i % 2 == 1
      f_ary.each{|a|
        b = a.clone
        x, y = *b[-1]
        b += (1..i).map{|j| [x + j, y]}
        b_ary << b if b.uniq.size == s
        c = a.clone
        x, y = *c[-1]
        c += (1..i).map{|j| [x - j, y]}
        b_ary << c if c.uniq.size == s
      }
    else
      f_ary.each{|a|
        b = a.clone
        x, y = *b[-1]
        b += (1..i).map{|j| [x, y + j]}
        b_ary << b if b.uniq.size == s
        c = a.clone
        x, y = *c[-1]
        c += (1..i).map{|j| [x, y - j]}
        b_ary << c if c.uniq.size == s
      }
    end
    ary << b_ary.size
  }
  ary
end
p A292793(16)

a(25)-a(39) from Bert Dobbelaere, Sep 14 2019

A101857 Number of possibly-self-intersecting walks that it is possible for an accelerating ant to produce with n steps (rotations & reflections not included). On step 1 the ant moves forward 1 unit, then turns left or right and proceeds 2 units, then turns left or right until at the end of its n-th step it arrives back at its starting place.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 16, 28, 0, 0, 0, 0, 0, 0, 1190, 2108, 0, 0, 0, 0, 0, 0

Offset: 1

Gordon Hamilton, Jan 27 2005

Accelerating ant walks can only arrive back at the starting place if the number of moves is -1 or 0 mod(8).

			a(7) = 1 because of the following solution:
655555XXX
6XXXX4XXX
6XXXX4XXX
6XXXX4XXX
6XXXX4333
6XXXXXXX2
777777712
where the ant starts at the "1" and moves right 1 space, up 2 spaces and so on...

Cf. A101856.

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A207361 Displacement under constant discrete unit surge.

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A292793 1/4 of the number of self-avoiding paths that are made of alternated vertical and horizontal n consecutive steps. (start point is different from end point.)

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