A212337 -id:A212337 - cp's OEIS frontend

A261064 a(n) = (3^n-1)*(n+1)/4.

1, 6, 26, 100, 363, 1274, 4372, 14760, 49205, 162382, 531438, 1727180, 5580127, 17936130, 57395624, 182948560, 581130729, 1840247318, 5811307330, 18305618100, 57531942611, 180441092746, 564859072956, 1765184603000, 5507375961373, 17157594341214, 53379182394902

Offset: 1

R. J. Mathar, Aug 08 2015

Second column of A201730.

Number of non-selfintersecting broken lines in a convex (n+1)-gon. (National Math Contest "Atanas Radev" 2020, Bulgaria) - Ivaylo Kortezov, Jan 18 2020

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 21.
Ivaylo Kortezov, problem 8.4 ("Задача 8.4" in Bulgarian) in National Math Contest "Atanas Radev" 2020.
Mark Shattuck, Enumeration of consecutive patterns in flattened Catalan words, arXiv:2502.10661 [math.CO], 2025. See pp. 3, 6.
Index entries for linear recurrences with constant coefficients, signature (8,-22,24,-9).

Cf. A001792, A027907, A201730, A212337.

[(3^n-1)*(n+1)/4: n in [1..30]]; // Vincenzo Librandi, Aug 31 2016

LinearRecurrence[{8, -22, 24, -9}, {1, 6, 26, 100}, 30] (* Vincenzo Librandi, Aug 31 2016 *)
Table[(3^n - 1)(n + 1)/4, {n, 0, 39}] (* Alonso del Arte, Jan 19 2020 *)

first(m)=vector(m,i, (3^i-1)*(i+1)/4); /* Anders Hellström, Aug 08 2015 */

G.f.: -x*(-1 + 2*x) / ( (3*x - 1)^2*(x - 1)^2 ).
a(n) = A212337(n - 1) - 2*A212337(n - 2).
a(n) = Sum_{k = 1..n} A027907(n, 2k - 1)*k . - J. Conrad, Aug 30 2016
a(n) = Sum_{k = 0..(n - 1)} binomial(n + 1, k + 2)*A001792(k). - Ivaylo Kortezov, Jan 21 2020
E.g.f.: exp(x)*(exp(2*x)*(1 + 3*x) - x - 1)/4. - Stefano Spezia, May 14 2024

A378727 The total number of fires in a rooted undirected infinite 4-ary tree with a self-loop at the root, when the chip-firing process starts with (4^n-1)/3 chips at the root.

Original entry on oeis.org

0, 1, 10, 67, 380, 1973, 9710, 46119, 213600, 970905, 4349650, 19262731, 84507460, 367855997, 1590728630, 6840133103, 29269406760, 124713124449, 529394487450, 2239745908435, 9447655468300, 39745309211461, 166799986198910, 698474942207927, 2918999758480880, 12176398992520233, 50707195804467810

Offset: 1

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

Each vertex of this tree has degree 5. If a vertex has at least 5 chips, the vertex fires, and one chip is sent to each neighbor. The root sends 1 chip to each of its four 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 A014916.
For binary trees, the corresponding sequence is A045618.
For ternary trees, the corresponding sequence is A212337.
For 5-ary trees, the corresponding sequence is A378728.
a(2k-1) is divisible by 10.

Yifan Xie, Table of n, a(n) for n = 1..2000
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. 15.
Wikipedia, Chip-firing game.
Index entries for linear recurrences with constant coefficients, signature (10,-33,40,-16).

Cf. A045618, A014916, A212337, A378728.

Table[((3 n - 5) 4^n + 3 n + 5)/27, {n, 30}]

a(n) = ((3*n - 5)*4^n + 3*n + 5)/27.

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.

Original entry on oeis.org

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

cp's OEIS Frontend

A261064 a(n) = (3^n-1)*(n+1)/4.

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A378727 The total number of fires in a rooted undirected infinite 4-ary tree with a self-loop at the root, when the chip-firing process starts with (4^n-1)/3 chips at the root.

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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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