A007501 -id:A007501 - cp's OEIS frontend

A372421 Number of steps required to kill the hydra in a version of the hydra game (see comments) where the rightmost head is chopped off in each step and new heads are grown to the left.

0, 1, 3, 9, 49, 1230, 757071, 286578628063, 41063655031378934880024, 843111882268046256673111236649909091104560309, 355418823010783945962646271385485944012152784388172734299894340514265378207290093661367905

Offset: 0

Paolo Xausa and Pontus von Brömssen, Apr 30 2024

nonn

The hydra is represented as an ordered tree, initialized to a path with n edges, with the root of the tree at a terminal node of the path. At the k-th step, the leaf (head) that is reached by following the rightmost path from the root is chopped off (equivalently, for this specific hydra, the head to be chopped off is always one of the heads farthest from the root). If only the root remains, the hydra dies and the game ends. If the head chopped off was directly connected to the root, nothing more happens in this step. Otherwise, k new heads are grown from the node two levels closer to the root from the head chopped off (its grandparent).

In this version, the new heads grow to the left of all existing branches of the grandparent, while in A372101, they grow to the right.

			For n = 3, the first three steps are illustrated in the diagrams below. In these diagrams, "R" denotes the root, "o" internal nodes, "X" the head to be chopped off, and "H" other heads.
.
                     H       H H           H H H
                    /        |/             \|/
R--o--o--X  =>  R--o--X  =>  R--o--X  =>  H--R--X
                                            /
                                           H
.
After this no more heads will grow, so another 6 steps are needed to chop off the remaining heads. The total number of steps is thus a(3) = 3 + 6 = 9.

Paolo Xausa, Table of n, a(n) for n = 0..13
Wikipedia, Hydra game.

Cf. A000217, A180368, A372101.
Partial sums of A370615.
Last element in each row of A372593.
Sequences with similar recurrences: A006894, A007501.

Block[{n = 0}, NestList[++n + PolygonalNumber[#] &, 0, 11]]

a(0) = 0; for n >= 1, a(n) = a(n-1)*(a(n-1)+1)/2 + n = A000217(a(n-1)) + n.

a(n) ~ 2 * c^(2^n), where c = 1.2222440178780117503347646365410387156780573376846000146... - Pontus von Brömssen and Vaclav Kotesovec, May 09 2024

A086714 a(1) = 4, a(n) = a(n-1)*(a(n-1) - 1)/2.

Original entry on oeis.org

4, 6, 15, 105, 5460, 14903070, 111050740260915, 6166133456248548335768188155, 19010600900133834176644234577571914951562754277857057935

Offset: 1

Jon Perry, Jul 29 2003

nonn

The next two terms, a(10) and a(11), have 111 and 221 digits. - Harvey P. Dale, Jun 10 2011
Interpretation through plane geometry: Start with the a(n)-sided regular polygon, connect all the vertices to create a figure having a(n+1)=A000217(a(n)-1) edges. Repeat to obtain this sequence. - T. D. Noe, May 13 2016
Let y(1) = x1+x2+x3+x4, and define y(n+1) as the plethysm e2[y(n)], where e2 represents the second elementary symmetric function. Then a(n) is y(n) evaluated at x1=x2=x3=x4=1. - Per W. Alexandersson, Jun 06 2020
Each term is the number of coordinate planes in Euclidean space of the dimensionality of the previous term. - Shel Kaphan, Feb 06 2023

			a(2) = a(1)*(a(1)-1)/2 = 4*3/2 = 6.

Vincenzo Librandi, Table of n, a(n) for n = 1..13

Cf. A251702, A251794, A006893.

Cf. A000217, A007501, A013589, A050542, A050548, A050536, A050909.

RecurrenceTable[{a[1]==4,a[n]==(a[n-1](a[n-1]-1))/2},a[n],{n,10}] (* Harvey P. Dale, Jun 10 2011 *)

v=vector(10,i,(i==1)*4); for(i=2,10,v[i]=v[i-1]*(v[i-1]-1)/2); v

a086714(upto)={my(a217(n)=n*(n+1)/2,a=4);for(k=1,upto,print1(a,", ");a=a217(a-1))};
a086714(9) \\ Hugo Pfoertner, Sep 18 2021

Limit_{n->oo} a(n)^(1/2^n) = 1.280497808541657066685323460209089278782... (see A251794). - Vaclav Kotesovec, Feb 15 2014, updated Dec 09 2014
a(n) ~ 2 * A251794^(2^n). - Vaclav Kotesovec, Dec 09 2014
a(n+1) = binomial(a(n), 2). - Shel Kaphan, Feb 06 2023

A099179 Iterated tetrahedral numbers.

Original entry on oeis.org

2, 4, 20, 1540, 609896980, 37811003218473324699257860, 9009555207802177724984164589516456320805205201729086740415363658290866918420

Offset: 1

Jonathan Vos Post, Nov 15 2004

The next term, a(8), has 228 digits. - Harvey P. Dale, Dec 18 2012

			a(2) = Tet(2) = the 2nd tetrahedral number = 2*(2+1)*(2+2)/6 = 4;
a(3) = Tet(Tet(2)) = the 4th tetrahedral number = 4*(4+1)*(4+2)/6 = 20;
a(4) = Tet(Tet(Tet(2))) = the 20th tetrahedral number = 20*(20+1)*(20+2)/6 = 1540.

Eric Weisstein's World of Mathematics, Tetrahedral Number.

Cf. A007501, A000292.

NestList[(#(#+1)(#+2))/6&,2,6] (* Harvey P. Dale, Dec 18 2012 *)

a(n)= A000292(a(n-1)). - R. J. Mathar, Jun 09 2008

Corrected and extended by R. J. Mathar, Jun 09 2008

A283681 Unique sequence with a(1)=1, a(2)=2, representing an array read by antidiagonals in which the i-th row is this sequence itself multiplied by i.

Original entry on oeis.org

1, 2, 2, 2, 4, 3, 2, 4, 6, 4, 4, 4, 6, 8, 5, 3, 8, 6, 8, 10, 6, 2, 6, 12, 8, 10, 12, 7, 4, 4, 9, 16, 10, 12, 14, 8, 6, 8, 6, 12, 20, 12, 14, 16, 9, 4, 12, 12, 8, 15, 24, 14, 16, 18, 10, 4, 8, 18, 16, 10, 18, 28, 16, 18, 20, 11, 4, 8, 12, 24, 20, 12, 21, 32, 18, 20, 22, 12, 6, 8

Offset: 1

Ivan Neretin, Mar 14 2017

Any integer greater than 1 appears infinitely many times.
In particular, any n appears at the position (n^2 + n)/2. For prime n > 2, this is its first appearance; for composite n, it is not the first.
2 appears at the positions 2, 3, 4, 7, 22, 232, 26797, ... (A007501(n) + 1).
When the sequence is considered as an array, any prime n appears only in the first row (infinitely many times) and in the first column (once).

			The sequence begins: 1, 2, 2, 2, 4, 3, 2, 4, 6, 4, ...
It represents a rectangular array read by downward antidiagonals. The first row of the array is this very sequence itself. The second row is this sequence multiplied by 2, and so on:
  1  2  2  2  4  3 ...
  2  4  4  4  8  ...
  3  6  6  6  ...
  4  8  8  ...
  5 10  ...
  6 ...
  ...

Ivan Neretin, Table of n, a(n) for n = 1..26796

Cf. A007501 (number of terms produced by the Mathematica code after n iterations).

Cf. A283682, A283683.

Nest[Flatten@Table[#[[n - i]]*i, {n, Length[#] + 1}, {i, n - 1}] &, {1, 2}, 4]

a((n^2+n)/2)=n.

A099129 Let T(n) be the n-th triangular number n*(n+1)/2; then a(n) = n-th iteration T(T(T(...(n)))).

Original entry on oeis.org

0, 1, 6, 231, 1186570, 347357071281165, 2076895351339769460477611370186681, 143892868802856286225154411591351342616163027795335641150249224655238508171

Offset: 0

Jonathan Vos Post, Nov 14 2004

The next term, a(8), has 162 digits. - Harvey P. Dale, May 29 2013

			a(3) = 231 because we can write the 3-time iterated expression on T(3), the triangular number sequence n*(n+1)/2, namely: T(T(T(3))) = 231.

Alois P. Heinz, Table of n, a(n) for n = 0..10

Cf. A000217, A007501, A058009 (analog with primes), A097547.

a:= n-> (t-> (t@@n)(n))(j-> j*(j+1)/2):
seq(a(n), n=0..7);  # Alois P. Heinz, Sep 29 2023

Table[Nest[(#(#+1))/2&,n,n],{n,8}] (* Harvey P. Dale, May 29 2013 *)

a(n) = my(k = n); for (j=1, n, k = k*(k+1)/2;); k; \\ Michel Marcus, Jan 01 2017

a(n) = A000217^n(n).

The sequence grows like O(n^2^n*1/2^n). This can be derived from the growth O(n^2*1/2) of the triangle sum by iteration. - Hieronymus Fischer, Jan 21 2006

Offset changed to 1 by Georg Fischer, Jun 20 2022

a(0)=0 prepended by Alois P. Heinz, Sep 29 2023

A108225 a(0) = 0, a(1) = 2; for n >= 2, a(n) = (a(n-1) + a(n-2))*(a(n-1) - a(n-2) + 1)/2.

Original entry on oeis.org

0, 2, 3, 5, 12, 68, 2280, 2598062, 3374961778893, 5695183504492614029263280, 16217557574922386301420536972254869595782763547562

Offset: 0

N. J. A. Sloane, Jun 16 2005

nonn

From a posting by Antreas P. Hatzipolakis to the Yahoo news group "Hyacinthos", circa Jun 10 2005.
The next term has 99 digits. - Harvey P. Dale, Jun 09 2011
a(n) for n>0 gives the rank of the unlabeled binary rooted tree, among those with n+1 leaves, that has the largest rank according to the bijection of Colijn and Plazzotta (2018) between unlabeled binary rooted trees and positive integers. - Noah A Rosenberg, Jun 03 2022

Robert Israel, Table of n, a(n) for n = 0..14
C. Colijn and G. Plazzotta, A metric on phylogenetic tree shapes, Syst. Biol., 67 (2018), 113-126.
Luc Devroye, Michael R. Doboli, Noah A. Rosenberg, and Stephan Wagner, Tree height and the asymptotic mean of the Colijn-Plazzotta rank of unlabeled binary rooted trees, arXiv:2409.18956 [math.CO], 2024. See p. 3.
N. A. Rosenberg, On the Colijn-Plazzotta numbering scheme for unlabeled binary rooted trees, Discr. Appl. Math., 291 (2021), 88-98.
J.S. Seneschal, Iteration of Semi-Complete Graphs

First differences give A103410.
Cf. A006894.
Cf. A000217, A007501, A002658, A072638.

F:=proc(n) option remember; if n <= 1 then RETURN(2*n) fi; (F(n-1)+F(n-2))*(F(n-1)-F(n-2)+1)/2; end;
a[ -2]:=-2:a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=binomial(a[n-1]+2,2) od: seq(a[n]+2, n=-2..8); # Zerinvary Lajos, Jun 08 2007

RecurrenceTable[{a[0]==0,a[1]==2,a[n]==(a[n-1]+a[n-2])(a[n-1]- a[n-2]+1)/2},a[n],{n,15}] (* Harvey P. Dale, Jun 09 2011 *)

Conjecture: a(n) = A006894(n) + 1. - R. J. Mathar, Apr 23 2007
From J.S. Seneschal, Jul 17 2025 (Start)
a(n) = A000217(a(n)) - A072638(n) = A072638(n-1) + 2.
a(n) = A002658(n-1) + a(n-1) for n > 1. (End)

A129440 a(0)=0, a(1)=1, a(2)=5 and for n>2: a(n) = a(n-1)(a(n-1) + 1)(2*a(n-1) + 1)/6.

Original entry on oeis.org

0, 1, 5, 55, 56980, 61667666167030, 78172010815921069181209893626754427513955

Offset: 0

Reinhard Zumkeller, Apr 15 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..8

Cf. A000330, A007501, A251702.

[0,1] cat [n le 1 select 5 else Self(n-1)*(Self(n-1)+1)*(2*Self(n-1)+1)/6: n in [1..8]]; // G. C. Greubel, Feb 06 2024

Flatten[{0, 1, RecurrenceTable[{a[2] == 5, a[n] == a[n-1]*(a[n-1] + 1)*(2*a[n-1] + 1)/6}, a[n], {n, 8}]}] (* Vaclav Kotesovec, Dec 17 2014 *)
Join[{0,1},NestList[(#(#+1)(2#+1))/6&,5,5]] (* Harvey P. Dale, Sep 13 2022 *)

def a(n): # a = A129440
    if n<3: return (0,1,5)[n]
    else: return a(n-1)*(a(n-1)+1)*(2*a(n-1)+1)/6
[a(n) for n in range(9)] # G. C. Greubel, Feb 06 2024

a(n) = A000330(if n<=2 then n else a(n)).

a(n) ~ sqrt(3) * c^(3^n), where c = 1.13701835838072682283814038264701129587627956851233106833915157... . - Vaclav Kotesovec, Dec 17 2014

A145272 First differences of successive iterated triangular numbers with seed 2.

Original entry on oeis.org

1, 3, 15, 210, 26565, 358999410, 64449908117864115, 2076895351339769396027702893296360, 2156747150208372213435450937462080290024600343143329179374709804540

Offset: 1

Greg Huber, Oct 06 2008

Also a triangular number -- follows from the definition of A007501.

			a(1)=T(2)-2.
a(2)=T(3)-3.
a(3)=T(6)-6.

First differences of successive terms of sequence A007501.

Cf. A000217.

a(n) = T(x(n-1))-x(n-1) where x(n)=T(x(n-1)) and T(x)=x*(x+1)/2.

A241241 If x is in the sequence then so are x^2 and x(x+1)/2.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 9, 10, 16, 21, 36, 45, 55, 81, 100, 136, 231, 256, 441, 666, 1035, 1296, 1540, 2025, 3025, 3321, 5050, 6561, 9316, 10000, 18496, 26796, 32896, 53361, 65536, 97461, 194481, 222111, 443556, 536130, 840456, 1071225, 1186570, 1679616, 2051325

Offset: 1

Reinhard Zumkeller, Apr 17 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000290, A000217.

Subsequence of A005214; some subsequences: A001146, A007501, A011764, A176594, A173501, A050909.

import Data.Set (singleton, deleteFindMin, insert)
a241241 n = a241241_list !! (n-1)
a241241_list = 0 : 1 : f (singleton 2) where
   f s = m : f (insert (a000290 m) $ insert (a000217 m) s')
         where (m, s') = deleteFindMin s

Nest[Flatten[{#,#^2,(#(#+1))/2}]&,{0,1,2},5]//Union (* Harvey P. Dale, Aug 12 2016 *)

Initial 0 and 1 prepended by Jon Perry, Apr 17 2014

A059406 a(n) = a(n-1)*(3a(n-1) + 1)/2 with a(1) = 1.

Original entry on oeis.org

1, 2, 7, 77, 8932, 119675402, 21483302825630107, 692298450446589820159203790062227, 718915716736124071145150312487360788973317227637596552328693330407

Offset: 1

Robert G. Wilson v, Jan 29 2001

nonn

a(1) = 1, a(n+1) = sum of a(n) numbers beginning with a(n) + 1: a(2) = 2, a(3) = 3+4 = 7, a(4) = 8+9+...13+14 = 77. - Amarnath Murthy, Sep 07 2005

			a(3) = 7 so a(4) = 7*(3*7 + 1)/2 = 7*22/2 = 77.

Harry J. Smith, Table of n, a(n) for n = 1..12

Cf. A000326.

f[n_Integer] := n(3n + 1)/2; NestList[f, 2, 8]

{ a=2/3; for (n = 1, 12, write("b059406.txt", n, " ", a=a*(3*a + 1)/2); ) } \\ Harry J. Smith, Jun 26 2009

a(n) = A007501(n)/3 = A005449(a(n-1)). - Henry Bottomley, Oct 03 2001

cp's OEIS Frontend

A372421 Number of steps required to kill the hydra in a version of the hydra game (see comments) where the rightmost head is chopped off in each step and new heads are grown to the left.

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A086714 a(1) = 4, a(n) = a(n-1)*(a(n-1) - 1)/2.

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A099179 Iterated tetrahedral numbers.

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A283681 Unique sequence with a(1)=1, a(2)=2, representing an array read by antidiagonals in which the i-th row is this sequence itself multiplied by i.

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A099129 Let T(n) be the n-th triangular number n*(n+1)/2; then a(n) = n-th iteration T(T(T(...(n)))).

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A108225 a(0) = 0, a(1) = 2; for n >= 2, a(n) = (a(n-1) + a(n-2))*(a(n-1) - a(n-2) + 1)/2.

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A129440 a(0)=0, a(1)=1, a(2)=5 and for n>2: a(n) = a(n-1)*(a(n-1) + 1)*(2*a(n-1) + 1)/6.

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A129440 a(0)=0, a(1)=1, a(2)=5 and for n>2: a(n) = a(n-1)(a(n-1) + 1)(2*a(n-1) + 1)/6.