A007501 -id:A007501 - cp's OEIS frontend

A099137 Iterated pentagonal numbers starting with 2.

2, 5, 35, 1820, 4967690, 37016913420305, 2055377818749516111922729385, 6336866966711278058881411525857304983854251425952352645

Offset: 0

Jonathan Vos Post, Nov 14 2004

Eric Weisstein's World of Mathematics, Pentagonal Number

Cf. A007501, A000326.

NestList[PolygonalNumber[5,#]&,2,8] (* Harvey P. Dale, Sep 07 2024 *)

a(n)=if(n==1,2,a(n-1)*(3*a(n-1)-1)/2); \\ Joerg Arndt, Mar 05 2023

Let P(n) = n*(3*n-1)/2, a(1) = 2 and a(n) = P(a(n-1)) for n >= 2.

Edited by Giovanni Resta, Jun 17 2016

Edited by Joerg Arndt, Mar 05 2023

A099185 Iterated octahedral numbers, starting at a(0) = 2.

Original entry on oeis.org

2, 6, 146, 2074806, 5954444157018557346, 140744820294208035204656447906095566299588102457814757606

Offset: 0

Jonathan Vos Post, Nov 15 2004

For additional terms see b-file.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 50, 1996.
Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1952.

Vincenzo Librandi, Table of n, a(n) for n = 0..7
Eric Weisstein's World of Mathematics, Octahedral Number.

Cf. A007501, A005900.

RecurrenceTable[{a[0]==2, a[n+1]==(2 a[n]^3 + a[n])/3}, a, {n, 6}] (* Vincenzo Librandi, Sep 16 2018 *)

a(0) = 2, a(n+1) = oct(a(n)) = (2*a(n)^3 + a(n))/3.

Edited by Jianing Song, Sep 16 2018

A099186 a(0) = 1, a(1) = 12, a(n+1) = A006564(a(n)).

Original entry on oeis.org

1, 12, 3972, 156624027132, 9605393649115032262909140007773492, 2215570185245038872814956021479696635481690776469126111743280383568619886232750745465064720338596867052

Offset: 0

Jonathan Vos Post, Nov 15 2004

			a(2) = A006564(a(1)) = A006564(12) = 12*(5*12^2 - 5*12 + 2)/2 = 3972.

H. K. Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc. 131 (2003), 65-75. MR 1929024

Cf. A007501, A006564.

Definition and comments condensed by R. J. Mathar, Sep 09 2009

A099187 Let D(n) = n(9n^2-9*n+2)/2 then a(k+1) = D(a(k)) and a(0) = 1.

Original entry on oeis.org

1, 20, 34220, 180318314012420, 26383476911029432816173777932463879690054620

Offset: 1

Jonathan Vos Post, Nov 15 2004

Michel Marcus, Table of n, a(n) for n = 0..6
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.

Cf. A007501, A006566.

Dod[n_]:= n*(9*n^2-9*n+2)/2;
a[n_]:= If[n==0, Dod[1], If[n==1, Dod[2], Dod[a[n-1]]]];
Table[a[n], {n, 0, 4}] (* G. C. Greubel, Mar 22 2019 *)

dod(n) = n*(9*n^2-9*n+2)/2;
a(n) = if (n==0, 1, if (n==1, dod(2), dod(a(n-1)))); \\ Michel Marcus, Dec 14 2015

Let D(n) = n*(9*n^2-9*n+2)/2 then a(k+1) = D(a(k)) and a(0) = 1.

A100010 a(0) = 2 and a(n) = f(a(n-1)) where f(n) = n^2(3n^2-4*n+2).

Original entry on oeis.org

2, 24, 941184, 2354066797535483525627904

Offset: 0

Jonathan Vos Post, Nov 16 2004

Previous name was: Iterated hyperdiamond numbers, starting with 24-cell(2) = 24. Hyperdiamond numbers, figurate numbers based on the 4-dimensional 24-cell, have the formula 24-cell(n) = n^2*(3*n^2-4*n+2). This sequence is the hyperdiamond number of the hyperdiamond number of ... of 2.
The next term has 98 digits.
This need not start at 24-cell(2) = 24. For example, starting at a(0) = 3, which is not a hyperdiamond number, we have a(1) = 24-cell(3) = 3^2*((3*3^2)-(4*3)+2) = 153; and a(2) = 24-cell(24-cell(3)) = 24-cell(153) = 153^2*((3*153^2)-(4*153)+2) = 1629664353; and a(3) = 24-cell(24-cell(24-cell(3))) = 24-cell(1629664353) = 21159914972910583843562449776792301953.

			a(0) = 2 is the seed for this instance of the more general recurrence;
a(1) = 24-cell(2) = 2^2*(3*2^2-4*2+2) = 24;
a(2) = 24-cell(24-cell(2)) = 24-cell(24) = 24^2*(3*24^2-4*24+2) = 941184.

H. S. M. Coxeter, Regular Polytopes, 3rd ed. New York: Dover, 1973.

Michel Marcus, Table of n, a(n) for n = 0..5
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000.

Cf. A092181, A007501, A099179, A000332.

f(n) = n^2*(3*n^2-4*n+2); \\ A092181
a(n) = if (n==0, 2, f(a(n-1))); \\ Michel Marcus, Dec 14 2015

a(0) = 2; hyperdiamond numbers, figurate numbers based on the 4-dimensional 24-cell, have the formula 24-cell(n) = n^2*(3*n^2-4*n+2). a(1) = 24-cell(2) = 24. a(2) = 24-cell(24-cell(2)) = 941184. For k>1, a(k+1) = 24-cell(a(k)).

New name from Joerg Arndt, Feb 23 2022

A100682 Floor of 4th root of pentatope numbers.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 30, 31, 31

Offset: 0

Jonathan Vos Post, Dec 06 2004

Conjecture: a(n) = floor((n - 3/2)/24^(1/4)) for n not in {0, 1, 6, 17, 2403, 5318}. - Charles R Greathouse IV, May 01 2012

			a(3) = 1 because floor((3*4*5*6/24)^(1/4)) = floor(15^(1/4)) = floor(1.96798967) = 1.

J. H. Conway and R. K. Guy, The Book of Numbers, pp. 55-57, Copernicus Press, NY, 1996.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Pentatope Number

Cf. A000332, A100009, A007501, A099179.

[Floor(Binomial(n+3, 4)^(1/4)): n in [3..70]]; // Vincenzo Librandi, Dec 14 2015

a:= n-> floor(binomial(n+3, 4)^(1/4)):
seq(a(n), n=0..70);  # Alois P. Heinz, Dec 14 2015

a(n)=binomial(n+3,4)^(1/4)\1 \\ Charles R Greathouse IV, May 01 2012

a(n)=sqrtnint(binomial(n+3,4),4) \\ Charles R Greathouse IV, Dec 14 2015

from math import comb
from sympy import integer_nthroot
def A100682(n): return integer_nthroot(comb(n+3,4),4)[0] # Chai Wah Wu, Oct 02 2024

a(n) = floor((A000332(n+3))^(1/4)) = floor(Ptop(n)^(1/4)) = floor(C(n+3, 4)^1/4) = floor((n * (n+1) * (n+2) * (n+3)/4!)^(1/4)).

a(n) = 0.4518... * n + O(1). - Charles R Greathouse IV, Dec 14 2015

A117632 Number of 1's required to build n using {+,T} and parentheses, where T(i) = i*(i+1)/2.

Original entry on oeis.org

1, 2, 2, 3, 4, 2, 3, 4, 4, 3, 4, 4, 5, 6, 4, 5, 6, 6, 7, 6, 2, 3, 4, 4, 5, 6, 4, 3, 4, 5, 5, 6, 6, 5, 6, 4, 5, 6, 6, 7, 8, 4, 5, 6, 4, 5, 6, 6, 5, 6, 6, 7, 8, 8, 3, 4, 5, 5, 6, 7, 5, 6, 6, 7, 6, 4, 5, 6, 6, 7, 8, 6, 7, 8, 8, 5, 6, 4, 5, 6, 6, 7, 6, 6, 7, 8, 6, 7, 8, 8, 5, 6, 7, 7, 8, 9, 7, 8, 6, 7, 8, 8

Offset: 1

Jonathan Vos Post, Apr 08 2006

nonn

This problem has the optimal substructure property.

			a(1) = 1 because "1" has a single 1.
a(2) = 2 because "1+1" has two 1's.
a(3) = 2 because 3 = T(1+1) has two 1's.
a(6) = 2 because 6 = T(T(1+1)).
a(10) = 3 because 10 = T(T(1+1)+1).
a(12) = 4 because 12 = T(T(1+1)) + T(T(1+1)).
a(15) = 4 because 15 = T(T(1+1)+1+1).
a(21) = 2 because 21 = T(T(T(1+1))).
a(28) = 3 because 28 = T(T(T(1+1))+1).
a(55) = 3 because 55 = T(T(T(1+1)+1)).

W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, 1971.
R. K. Guy, Unsolved Problems Number Theory, Sect. F26.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, December 1971. [Annotated scanned copy]
R. K. Guy, Some suspiciously simple sequences, Amer. Math. Monthly 93 (1986), 186-190; 94 (1987), 965; 96 (1989), 905.
Ed Pegg, Jr., Integer Complexity.
Eric Weisstein's World of Mathematics, Integer Complexity.
Wikipedia, Optimal substructure.

See also A023361 = number of compositions into sums of triangular numbers, A053614 = numbers that are not the sum of triangular numbers. Iterated triangular numbers: A050536, A050542, A050548, A050909, A007501.

Cf. A000217, A005245, A005520, A003313, A076142, A076091, A061373, A005421, A023361, A053614, A064097, A025280, A003037, A099129, A050536, A050542, A050548, A050909, A007501.

a:= proc(n) option remember; local m; m:= floor (sqrt (n*2));
      if n<3 then n
    elif n=m*(m+1)/2 then a(m)
    else min (seq (a(i)+a(n-i), i=1..floor(n/2)))
      fi
    end:
seq (a(n), n=1..110);  # Alois P. Heinz, Jan 05 2011

a[n_] := a[n] = Module[{m = Floor[Sqrt[n*2]]}, If[n < 3, n, If[n == m*(m + 1)/2, a[m], Min[Table[a[i] + a[n - i], {i, 1, Floor[n/2]}]]]]];
Array[a, 110] (* Jean-François Alcover, Jun 02 2018, from Maple *)

I do not know how many of these entries have been proved to be minimal. - N. J. A. Sloane, Apr 15 2006

Corrected and extended by Alois P. Heinz, Jan 05 2011

A275342 2-adic valuation of iterated triangular numbers, starting with 2.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 3, 2, 1, 0, 0, 2, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 2, 1, 0, 2, 1, 0, 0, 2, 1, 0, 0, 0, 3, 2, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 0, 3, 2, 1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 0, 2, 1, 0, 0, 4, 3, 2, 1, 0, 0, 4, 3, 2, 1, 0, 2, 1, 0, 1, 0, 0, 0, 1

Offset: 1

Jeffrey Shallit, Jul 23 2016

nonn

			t(1) = 2, t(2) = 3, t(3) = 6, t(4) = 21, so a(1) = 1, a(2) = 0, a(3) = 1, a(4) = 0.

Cf. A007501, A007814, A117872.

Writing t(1) = 2, t(n+1) = t(n)(t(n)+1)/2, the sequence is nu_2 (t(n)), where nu_2 (x) is the exponent of the highest power of 2 dividing x.

a(n) = A007814(A007501(n)). - Michel Marcus, Jul 23 2016

A096662 Least nontrivial n-tuply triangular number.

Original entry on oeis.org

3, 6, 21, 231, 26796, 359026206, 64449908476890321, 2076895351339769460477611370186681, 2156747150208372213435450937462082366919951682912789656986079991221

Offset: 1

Robert G. Wilson v, Jul 02 2004

nonn

Cf. A000217, A002817, A064322, A066370.

f[n_] := n(n + 1)/2; Table[ Nest[f, 2, n], {n, 10}]

a(n)=A007501(n). [From R. J. Mathar, Sep 04 2008]

A100011 a(n+1) = a(n)((261a(n)^3)-(504a(n)^2)+(283a(n))-38)/2 for n > 0, a(0) = 2.

Original entry on oeis.org

2, 600, 16858418928600, 10540904502382779165253193559218987587063928137089696600

Offset: 0

Jonathan Vos Post, Nov 16 2004

The next term (a(4)) has 223 digits and a(5) has 891 digits. - Harvey P. Dale, Dec 22 2024

Cf. A092183, A007501, A099179, A000332.

NestList[# (261 #^3 - 504 #^2 + 283 # - 38)/2 &, 2, 5] (* Harvey P. Dale, Dec 22 2024 *)

cp's OEIS Frontend

A099137 Iterated pentagonal numbers starting with 2.

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A099185 Iterated octahedral numbers, starting at a(0) = 2.

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A099186 a(0) = 1, a(1) = 12, a(n+1) = A006564(a(n)).

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A099187 Let D(n) = n*(9*n^2-9*n+2)/2 then a(k+1) = D(a(k)) and a(0) = 1.

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A100010 a(0) = 2 and a(n) = f(a(n-1)) where f(n) = n^2*(3*n^2-4*n+2).

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A100682 Floor of 4th root of pentatope numbers.

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A117632 Number of 1's required to build n using {+,T} and parentheses, where T(i) = i*(i+1)/2.

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A099187 Let D(n) = n(9n^2-9*n+2)/2 then a(k+1) = D(a(k)) and a(0) = 1.

A100010 a(0) = 2 and a(n) = f(a(n-1)) where f(n) = n^2(3n^2-4*n+2).

A100011 a(n+1) = a(n)((261a(n)^3)-(504a(n)^2)+(283a(n))-38)/2 for n > 0, a(0) = 2.