A007501 -id:A007501 - cp's OEIS frontend

A100012 Let h(k) = a(k)((145a(k)^3)-(280a(k)^2)+(179a(k))-38)/6, then a(n) = h(a(n-1)) for n >= 1 and a(0) =2.

2, 120, 4930988840, 14287387711051307292599794275187472361080

Offset: 0

Jonathan Vos Post, Nov 17 2004

The next term has 163 digits.

Cf. A092182, A007501, A099179, A000332, A006564.

NestList[#/6*(145#^3-280#^2+179#-38)&,2,3] (* Harvey P. Dale, Apr 09 2015 *)

A159860 The maximum length of a string of identical characters which can be reduced to one character in "n" nested substitution operations, e.g. replace(string, substring, character) such that all shorter strings will also reduce to one character.

Original entry on oeis.org

2, 4, 10, 40, 460, 53590, 718052410, 128899816953780640, 4153790702679538920955222740373360, 4313494300416744426870901874924164733839903365825579313972159982440

Offset: 1

Russell Harper (russell.harper(AT)springboardnetworks.com), Apr 24 2009

The ideal substring length is related to A007501. It can be shown these are equivalent problems.
For n = 1, the ideal substring length is 2.
For n > 1:
n = 2, term 0 of A007501, substring length = 2
n = 3, term 1 of A007501, substring length = 3
n = 4, term 2 of A007501, substring length = 6
etc.
This has applications in text processing operations in computer languages where recursions or loops may not be possible (e.g. standard SQL). To remove extra spaces, one might be tempted to nest several replace operations but use the same substring length, or perhaps double or halve at each step, both of which will not clear as effectively as using substring lengths as indicated in A007501.

			To illustrate, suppose we have a string of repeating Xs.
n = 1: replace(string, "XX", "X"), the longest string which will reduce to "X" is "XX"
n = 2: replace(replace(string, "XX", "X"), "XX", "X") will reduce up to 4 Xs to "X"
n = 3: replace(replace(replace(string, "XXX", "X"), "XX", "X"), "XX", "X") up to 10 Xs
n = 4: replace(replace(replace(replace(string, "XXXXXX", "X"), "XXX", "X"), "XX", "X"), "XX", "X") up to 40 Xs
etc.

2, followed by A007501

a:= proc(n) option remember; a(n-1)*(a(n-1)+6)/4 end: a(1):=2:
seq(a(n), n=1..10);  # Alois P. Heinz, Oct 11 2024

NestList[-Floor[(#+3)/2]^2+(#+3)*Floor[(#+3)/2]-2&,2,9] (* Shenghui Yang, Oct 11 2024 *)

// q is this sequence, p is A007501
set q = 2
output q
repeat
set p = q / 2 + 1
set q = p * (p + 1) - 2
output q
end repeat

Given substring length p as indicated in A007501, sequence is p(p+1)-2.

a(n) = a(n-1)*(a(n-1) + 6)/4. - N. Sato, Feb 01 2010

Missing a(9) inserted by Alois P. Heinz, Oct 11 2024

A285634 a(1) = 4, a(n) = Product_{d|a(n-1)} d.

Original entry on oeis.org

4, 8, 64, 2097152, 3450873173395281893717377931138512726225554486085193277581262111899648

Offset: 1

Ilya Gutkovskiy, Apr 23 2017

Iterating the product-of-divisors function.
The next term is too large to include.
Let a(n) = Product_{d|a(n-1)} d, with a(1) = p^k, p is a prime, k >= 0 and b(n) = b(n-1)*(b(n-1) + 1)/2, with b(1) = k, then a(n) = p^b(n).
The next term has 8067 digits. - Harvey P. Dale, Apr 18 2019

			a(1) = 4;
a(2) = 8 because 4 has 3 divisors {1, 2, 4} and 1*2*4 = 8;
a(3) = 64 because 64 has 7 divisors {1, 2, 4, 8, 16, 32, 64} and 1*2*4*8*16*32*64 = 2097152, etc.
...
a(6) = 2^26796;
a(7) = 2^359026206;
a(8) = 2^64449908476890321;
a(9) = 2^2076895351339769460477611370186681, etc.

Eric Weisstein's World of Mathematics, Divisor Product

Cf. A000005, A000079, A007497, A007501, A007955.

RecurrenceTable[{a[1] == 4, a[n] == Sqrt[a[n - 1]]^DivisorSigma[0, a[n - 1]]}, a, {n, 5}]
NestList[Times@@Divisors[#]&,4,4] (* Harvey P. Dale, Apr 18 2019 *)

a(1) = 4, a(n) = a(n-1)^(A000005(a(n-1))/2).

a(n) = 2^A007501(n-1).

A296374 a(0) = 3; a(n) = a(n-1)(a(n-1)^2 - 3a(n-1) + 4)/2.

Original entry on oeis.org

3, 6, 66, 137346, 1295413937737986, 1086915296274625337063297033180803022465442306

Offset: 0

Ilya Gutkovskiy, Dec 11 2017

nonn

The next term is too large to include.

			a(0) = 3;
a(1) = 6 and 6 is the 3rd triangular number;
a(2) = 66 and 66 is the 6th hexagonal number;
a(3) = 137346 and 137346 is the 66th 66-gonal number, etc.

Index to sequences related to polygonal numbers

Cf. A001146, A007501, A057145, A060354, A097419, A099137, A099147, A099153.

RecurrenceTable[{a[0] == 3, a[n] == a[n - 1] (a[n - 1]^2 - 3 a[n - 1] + 4)/2}, a[n], {n, 5}]

a(0) = 3; a(n) = [x^a(n-1)] x*(1 - 2*x + 4*x^2)/(1 - x)^4.

a(0) = 3; a(n) = a(n-1)! * [x^a(n-1)] exp(x)*x*(1 + x^2/2).

A341463 a(n) = (-1)^(n+1) * (3^n+1)/2.

Original entry on oeis.org

-1, 2, -5, 14, -41, 122, -365, 1094, -3281, 9842, -29525, 88574, -265721, 797162, -2391485, 7174454, -21523361, 64570082, -193710245, 581130734, -1743392201, 5230176602, -15690529805, 47071589414, -141214768241, 423644304722, -1270932914165, 3812798742494, -11438396227481, 34315188682442

Offset: 0

Jack W Grahl, Feb 12 2021

Shown by Tutte (he erroneously gave the negative of this sequence) to be the value of the function g(X_n), where X_n is the graph with one vertex and n loops, and g() is the extension to all graphs of the function f(G) defined on trivalent graphs by f(G) =(-1)^n.Q(G), where 2n is the number of vertices of G, and Q(G) is the number of spanning subgraphs of G such that every vertex of G is incident with 2 edges, and obeying the recursions discussed by Tutte in the article.

This sequence is given in balanced ternary representation as (-1), 1(-1), (-1)11, 1(-1)(-1)(-1), (-1)1111, 1(-1)(-1)(-1)(-1)(-1), etc.

W. T. Tutte, Some polynomials associated with graphs, Combinatorics, Proceedings of the British Combinatorial Conference. Vol. 13. Cambridge Univ. Press London, 1973.

Index entries for linear recurrences with constant coefficients, signature (-4,-3).

Absolute values give A007501.

Cf. A076040.

def a(n):
    return (-1)**(n+1) * (3 ** n + 1) // 2

a(n) = -4*a(n-1) - 3*a(n-2) for n > 1.

G.f.: -(1 + 2*x)/(1 + 4*x + 3*x^2). - Stefano Spezia, Feb 13 2021

cp's OEIS Frontend

A100012 Let h(k) = a(k)*((145*a(k)^3)-(280*a(k)^2)+(179*a(k))-38)/6, then a(n) = h(a(n-1)) for n >= 1 and a(0) =2.

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A159860 The maximum length of a string of identical characters which can be reduced to one character in "n" nested substitution operations, e.g. replace(string, substring, character) such that all shorter strings will also reduce to one character.

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A285634 a(1) = 4, a(n) = Product_{d|a(n-1)} d.

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A296374 a(0) = 3; a(n) = a(n-1)*(a(n-1)^2 - 3*a(n-1) + 4)/2.

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A341463 a(n) = (-1)^(n+1) * (3^n+1)/2.

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A100012 Let h(k) = a(k)((145a(k)^3)-(280a(k)^2)+(179a(k))-38)/6, then a(n) = h(a(n-1)) for n >= 1 and a(0) =2.

A296374 a(0) = 3; a(n) = a(n-1)(a(n-1)^2 - 3a(n-1) + 4)/2.