A014917 -id:A014917 - cp's OEIS frontend

A378728 The total number of fires in a rooted undirected infinite 5-ary tree with a self-loop at the root, when the chip-firing process starts with (5^n-1)/4 chips at the root.

0, 1, 12, 98, 684, 4395, 26856, 158692, 915528, 5187989, 28991700, 160217286, 877380372, 4768371583, 25749206544, 138282775880, 739097595216, 3933906555177, 20861625671388, 110268592834474, 581145286560060, 3054738044738771, 16018748283386232, 83819031715393068

Offset: 1

Tanya Khovanova and the MIT PRIMES STEP senior group, Dec 05 2024

Each vertex of this tree has degree 6. If a vertex has at least 6 chips, the vertex fires, and one chip is sent to each neighbor. The root sends 1 chip to each of its five children and one chip to itself.
The order of the firings doesn't affect the number of firings.
This number of chips is interesting because the stable configuration has 1 chip for every vertex in the top n layers.
a(n) is partial sums of A014917.
For binary trees, the corresponding sequence is A045618.
For ternary trees, the corresponding sequence is A212337.
For 4-ary trees, the corresponding sequence is A378727.
a(2k-1) is divisible by 12.

Yifan Xie, Table of n, a(n) for n = 1..1000
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See p. 16.
Wikipedia, Chip-firing game.
Index entries for linear recurrences with constant coefficients, signature (12,-46,60,-25).

Cf. A014917, A045618, A212337, A378727.

Table[((2 n - 3) 5^n + 2 n + 3)/32, {n, 30}]

a(n) = ((2*n - 3)*5^n + 2*n + 3)/32.

G.f.: x^2/(1-6*x+5*x^2)^2. - Jinyuan Wang, Jan 24 2025

A059045 Square array T(n,k) read by antidiagonals where T(0,k) = 0 and T(n,k) = 1 + 2k + 3k^2 + ... + n*k^(n-1).

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 5, 1, 0, 1, 10, 17, 7, 1, 0, 1, 15, 49, 34, 9, 1, 0, 1, 21, 129, 142, 57, 11, 1, 0, 1, 28, 321, 547, 313, 86, 13, 1, 0, 1, 36, 769, 2005, 1593, 586, 121, 15, 1, 0, 1, 45, 1793, 7108, 7737, 3711, 985, 162, 17, 1, 0, 1, 55, 4097, 24604, 36409

Offset: 0

Henry Bottomley, Dec 18 2000

			   0,   0,   0,    0,     0,      0,      0,      0,       0, ...
   1,   1,   1,    1,     1,      1,      1,      1,       1, ...
   1,   3,   5,    7,     9,     11,     13,     15,      17, ...
   1,   6,  17,   34,    57,     86,    121,    162,     209, ...
   1,  10,  49,  142,   313,    586,    985,   1534,    2257, ...
   1,  15, 129,  547,  1593,   3711,   7465,  13539,   22737, ...
   1,  21, 321, 2005,  7737,  22461,  54121, 114381,  219345, ...
   1,  28, 769, 7108, 36409, 131836, 380713, 937924, 2054353, ...

Cf. A000004, A000012, A005408, A056109, A056578, A056579 (rows 1-6).

Cf. A057427, A000217, A000337, A014915, A014916, A014917, A014918, A014920 (columns 1-7).

A059045 := proc(n,k)
    if k = 1 then
        n*(n+1) /2 ;
    else
        (1+n*k^(n+1)-k^n*(n+1))/(k-1)^2 ;
    end if;
end proc: # R. J. Mathar, Mar 29 2013

T(n,k) = n*k^(n-1)+T(n-1, k) = (n*k^(n+1)-(n+1)*k^n+1)/(k-1)^2.

A108283 Triangle read by rows, generated from (..., 3, 2, 1).

Original entry on oeis.org

1, 1, 3, 1, 5, 6, 1, 7, 17, 10, 1, 9, 34, 49, 15, 1, 11, 57, 142, 129, 21, 1, 13, 86, 313, 547, 321, 28, 1, 15, 121, 586, 1593, 2005, 769, 36, 1, 17, 162, 985, 3711, 7737, 7108, 1793, 45, 1, 19, 209, 1534, 7465, 22461, 36409, 24604, 4097, 55, 1, 21, 262, 2257, 13539, 54121, 131836, 167481, 83653, 9217, 66

Offset: 1

Gary W. Adamson, May 30 2005

Inverse binomial transforms of each column form the rows of A108284. Rightmost diagonal = triangular numbers, (A000217); while diagonals going to the left from (1, 3, 6, ...) are A000337 starting with 1: (1, 5, 17, 49, ...); A014915: (1, 7, 34, 142, ...); A014916: (1, 9, 57, ...); A014917: (1, 11, 86, ...).

			4th column = 10, 49, 142, 313, ... = f(x), x = 1, 2, 3; 4x^3 + 3x^2 + 2x + 1. f(3) = 142.
First few rows of the triangle:
  1;
  1,  3;
  1,  5,  6;
  1,  7, 17,  10;
  1,  9, 34,  49,  15;
  1, 11, 57, 142, 129, 21;
  ...

Cf. A059045, A108284, A000217, A000337, A014915, A014916, A014917.

A108283 := proc(n,k)
    local x ;
    x := n-k+1 ;
    add( i*x^(i-1),i=1..k) ;
end proc:
seq(seq( A108283(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Sep 14 2016

T[, 1] := 1; T[n, n_] := n (n + 1)/2; T[n_, k_] := (1 - (n - k + 1)^k*(k^2 - k*n + 1))/(n - k)^2; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 13 2016 *)

n-th column = f(x), x = 1, 2, 3; n*x^(n-1) + (n-1)*x^(n-2) + (n-3)*x^(n-3) + ... + 1.

T(n,k) = (1+ (n-k+1)^k*(n*k-k^2-1))/ (n-k)^2, n>k. - Jean-François Alcover, Sep 13 2016

More terms from Jean-François Alcover, Sep 13 2016

cp's OEIS Frontend

A378728 The total number of fires in a rooted undirected infinite 5-ary tree with a self-loop at the root, when the chip-firing process starts with (5^n-1)/4 chips at the root.

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A059045 Square array T(n,k) read by antidiagonals where T(0,k) = 0 and T(n,k) = 1 + 2k + 3k^2 + ... + n*k^(n-1).

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A108283 Triangle read by rows, generated from (..., 3, 2, 1).

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