Reed Kelly - cp's OEIS frontend

A214646 a(n) = (a(n-2) + a(n-3))/gcd(a(n-2), a(n-3)) with a(1) = a(2) = a(3) = 1.

1, 1, 1, 2, 2, 3, 2, 5, 5, 7, 2, 12, 9, 7, 7, 16, 2, 23, 9, 25, 32, 34, 57, 33, 91, 30, 124, 121, 77, 245, 18, 46, 263, 32, 309, 295, 341, 604, 636, 945, 310, 527, 251, 27, 778, 278, 805, 528, 1083, 1333, 537, 2416, 1870, 2953, 2143, 4823, 5096, 6966, 109

Offset: 1

Reed Kelly, Jul 24 2012

nonn

A variation on A214551 with the second and third terms being added and divided by the greatest common divisor of the pair of numbers.

			a(6) = (a(4)+a(3))/gcd(a(4),a(3)) = (2+1)/1 = 3.
a(19) = (a(17)+a(16))/gcd(a(17),a(16)) = (2+16)/2 = 9.

Reed Kelly, Table of n, a(n) for n = 1..1003

Cf. A214551, A000931.

DivGCDxy[n_, x_, y_, init_] := Module[{t, a, i}, t = init;
  Do[AppendTo[t, (t[[-x]] + t[[-y]])/GCD[t[[-x]], t[[-y]]]], {n}];
  t]; DivGCDxy[100, 2, 3, {1, 1, 1}]

NAME adapted to offset and b-file. - R. J. Mathar, Jun 19 2021

A214652 a(n) = (a(n-1) + a(n-4))/gcd(a(n-1), a(n-4)) with a(1) = a(2) = a(3) = a(4) = 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 7, 12, 19, 29, 36, 4, 23, 52, 22, 13, 36, 22, 2, 15, 17, 39, 41, 56, 73, 112, 153, 209, 282, 197, 350, 559, 841, 1038, 694, 1253, 2094, 522, 608, 1861, 3955, 4477, 5085, 6946, 10901, 1398, 2161, 9107, 20008, 10703, 12864, 21971

Offset: 1

Reed Kelly, Jul 24 2012

nonn

A variation on A214551 with the first and fourth terms being added and divided by the greatest common divisor of the pair of numbers.

			a(11) = (a(10)+a(7))/gcd(a(10),a(7)) = (10+4)/gcd(10,4) = 7
a(13) = (a(12)+a(9))/gcd(a(13),a(9)) = (12+7)/gcd(12,7) = 19

Reed Kelly, Table of n, a(n) for n = 1..1004

cf. A003269, A214551

GCDxy[n_, x_, y_, init_] := Module[{t, a, i}, t = init;
  Do[AppendTo[t, (t[[-x]] + t[[-y]])/GCD[t[[-x]], t[[-y]]]], {n}];
  t]; GCDxy[100, 1, 4, {1, 1, 1, 1}]
RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==1,a[n]==(a[n-1]+a[n-4])/GCD[ a[n-1],a[n-4]]},a,{n,60}] (* Harvey P. Dale, Apr 08 2019 *)

Definition corrected by Harvey P. Dale, Apr 08 2019

NAME adapted to offset and b-file. - R. J. Mathar, Jun 19 2021

A214551 Reed Kelly's sequence: a(n) = (a(n-1) + a(n-3))/gcd(a(n-1), a(n-3)) with a(0) = a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 3, 2, 3, 2, 2, 5, 7, 9, 14, 3, 4, 9, 4, 2, 11, 15, 17, 28, 43, 60, 22, 65, 25, 47, 112, 137, 184, 37, 174, 179, 216, 65, 244, 115, 36, 70, 37, 73, 143, 180, 253, 36, 6, 259, 295, 301, 80, 75, 376, 57, 44, 105, 54, 49, 22, 38, 87, 109, 147

Offset: 0

Reed Kelly, Jul 20 2012

Like Narayana's Cows sequence A000930, except that the sums are divided by the greatest common divisor (gcd) of the prior terms.

It is a strong conjecture that 8 and 10 are missing from this sequence, but it would be nice to have a proof! See A214321 for the conjectured values. [I have often referred to this as "Reed Kelly's sequence" in talks.] - N. J. A. Sloane, Feb 18 2017

			a(14)=9, a(16)=3, therefore a(17)=(9+3)/gcd(9,3) = 12/3 = 4.
a(24)=28, a(26)=60, therefore a(27)=(28+60)/gcd(28,60) = 88/4 = 22.

T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Benoit Cloitre, Graph of a(n)^(1/n) for n=1 up to 381817
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.

Similar to A000930. Cf. A341312, A341313, which are also similar.
Cf. also A214320, A214321, A214322, A214323 (gcd's), A219898 (records), A214324, A214325, A214330, A214331, A214809, A227836, A227837.
Starting with a(2) = 3 gives A214626. - Reinhard Zumkeller, Jul 23 2012

a214551 n = a214551_list !! n
a214551_list = 1 : 1 : 1 : zipWith f a214551_list (drop 2 a214551_list)
   where f u v = (u + v) `div` gcd u v
-- Reinhard Zumkeller, Jul 23 2012

a:= proc(n) a(n):= `if`(n<3, 1, (a(n-1)+a(n-3))/igcd(a(n-1), a(n-3))) end:
seq(a(n), n=0..100); # Alois P. Heinz, Oct 18 2012

t = {1, 1, 1}; Do[AppendTo[t, (t[[-1]] + t[[-3]])/GCD[t[[-1]], t[[-3]]]], {100}]
f[l_List] := Append[l, (l[[-1]] + l[[-3]])/GCD[l[[-1]], l[[-3]]]]; Nest[f, {1, 1, 1}, 62] (* Robert G. Wilson v, Jul 23 2012 *)
RecurrenceTable[{a[0]==a[1]==a[2]==1,a[n]==(a[n-1]+a[n-3])/GCD[ a[n-1], a[n-3]]},a,{n,70}] (* Harvey P. Dale, May 06 2014 *)

first(n)=my(v=vector(n+1)); for(i=1,min(n,3),v[i]=1); for(i=4,#v, v[i]=(v[i-1]+v[i-3])/gcd(v[n-1],v[i-3])); v \\ Charles R Greathouse IV, Jun 21 2017

use bignum;
my @seq = (1, 1, 1);
print "1 1\n2 1\n3 1\n";
for ( my $i = 3; $i < 400; $i++ )
{
    my $next = ( $seq[$i-1] + $seq[$i-3] ) /
        gcd( $seq[$i-1], $seq[$i-3] );
    my $ind = $i+1;
    print "$ind $next\n";
    push( @seq, $next );
}
sub gcd {
    my ($x, $y) = @_;
    ($x, $y) = ($y, $x % $y) while $y;
    return $x;
}

from math import gcd
def aupton(nn):
    alst = [1, 1, 1]
    for n in range(3, nn+1):
        alst.append((alst[n-1] + alst[n-3])//gcd(alst[n-1], alst[n-3]))
    return alst
print(aupton(64)) # Michael S. Branicky, Mar 28 2022

def A214551Rec():
    x, y, z = 1, 1, 1
    yield x
    while True:
        x, y, z =  y, z, (z + x)//gcd(z, x)
        yield x
A214551 = A214551Rec();
print([next(A214551) for  in range(65)])  # _Peter Luschny, Oct 18 2012

It appears that, very roughly, a(n) ~ constant*exp(0.123...*n). - N. J. A. Sloane, Sep 07 2012. See next comment for more precise estimate.
If a(n)^(1/n) converges the limit should be near 1.126 (see link). - Benoit Cloitre, Nov 08 2015
Robert G. Wilson v reports that at around 10^7 terms a(n)^(1/n) is about exp(1/8.4). - N. J. A. Sloane, May 05 2021

A157199 The terms of this sequence are the first several terms of tcW(r,r-1,r+1), where r=2,3,4,.... Informally, the function tcW is like the multi-color Van der Waerden function W, except that the second parameter determines the number of colors found in the target subsequence. See links for definition.

Original entry on oeis.org

9, 13, 22, 26, 44, 50, 25, 28

Offset: 2

Reed Kelly, Feb 25 2009

If W(r,k) is the standard multi-color Van der Waerden function with r colors and a required monochrome arithmetic subsequence of length k, then tcW(r,1,k) = W(r,k). In tcW(r,1,k), the 1 would indicate a monochrome subsequence. For tcW(r,2,k) an arithmetic subsequence of length k in 1 OR 2 colors would match the criteria. For tcW(r,3,k) an arithmetic subsequence of length k in 1, 2, or 3 colors suffices.

			a(2) = tcW(2,1,3) = W(2,3) = 9. If {1,...,9} is colored in 2 colors, then a 3-term arithmetic subsequence exists in 1 color (monochrome).
a(3) = tcW(3,2,4) = 13. If {1,...,13} is colored in 3 colors, then a 4-term arithmetic subsequence exists in at most 2 colors.

Reed Kelly, On a Generalization of Ramsey Theory, 2009.
Reed Kelly, Code for Computing Tuple-chromatic Ramsey Numbers and Tuple-chromatic Van der Waerden Numbers

Another part of the tcW function: A157102. The 2-color Van der Waerden numbers: A005346, W(2, k). Multi-color Van der Waerden numbers with 3-term monochrome arithmetic subsequences A135415, W(r, 3).

a(r) = tcW(r,r-1,r+1).

A157102 Tuple-chromatic Van der Waerden numbers.

Original entry on oeis.org

3, 7, 7, 21, 11, 43, 15, 25, 19, 111, 23, 157, 27, 43, 31, 273, 35, 343, 39, 61, 43, 507, 47, 121, 51, 79, 55, 813, 59, 931, 63, 97, 67, 171, 71, 1333, 75, 115, 79, 1641, 83, 1807, 87, 133, 91, 2163, 95, 337, 99, 151, 103, 2757, 107, 271, 111, 169, 115, 3423, 119, 3661

Offset: 2

Reed Kelly, Feb 22 2009, Feb 25 2009

nonn

See links for definition. Specifically, the terms of this sequence are the first several terms of tcW(r,r-1,r), where r=2,3,4,.... Informally, the function tcW is like the multi-color Van der Waerden function W, except that the second parameter determines the number of colors found in the target subsequence. If W(r,k) is the standard multi-color Van der Waerden function with r colors and a required monochrome arithmetic subsequence of length k, then tcW(r,1,k) = W(r,k). In tcW(r,1,k), the 1 would indicate a monochrome subsequence. For tcW(r,2,k) an arithmetic subsequence of length k in 1 OR 2 colors would match the criteria. For tcW(r,3,k) an arithmetic subsequence of length k in 1, 2, or 3 colors suffices.

a(r) = tcW(r,r-1,r).

			a(2) = tcW(2,1,2) = W(2,2) = 3. If {1,2,3} is colored in 2 colors, then a 2 term arithmetic subsequence exists in 1 color (monochrome).
a(3) = tcW(3,2,3) = 7. If {1,...,7} is colored in 3 colors, then a 3 term arithmetic subsequence exists that is colored in at most 2 colors.
a(2) = (2-1)(2) + 1 = 3 a(15) = (15-1)(3) + 1 = 43.

Reed Kelly, On a Generalization of Ramsey Theory, 2009.
Reed Kelly, Code for Computing Tuple-chromatic Ramsey Numbers and Tuple-chromatic Van der Waerden Numbers

The 2-color Van der Waerden numbers: A005346, W(2, k). Multi-color Van der Waerden numbers with 3 term monochrome arithmetic subsequences A135415, W(r, 3).

Table[(x - 1) * (FactorInteger[x])[[1]][[1]] + 1, {x, 2, 100}]

a(n) = (n-1)*(smallest prime factor of n) + 1.

A140516 For definition see comments lines.

Original entry on oeis.org

1, 3, 6, 8, 9, 13, 16, 17, 19, 20, 21, 26, 31, 32, 36, 42, 43, 45, 49, 50, 52, 53, 54, 58, 62, 65, 66, 67, 71, 78, 80, 82, 85, 87, 89, 90, 97, 98, 103, 106, 108, 112, 113, 114, 116, 120, 122, 123, 124, 128, 129, 134, 135, 139, 141, 143, 145, 147, 148, 153, 155, 157, 161

Offset: 1

Reed Kelly, Jul 01 2008

nonn

Inspired by Van der Waerden's Theorem, the positive integers are partitioned into two disjoint sequences, A and B, so that they obey the following rules:
Rule 1: For a given n, let A(n) and B(n) be the subsets of A and B with terms less than or equal to n. Any maximal arithmetic subsequence of A(n) must have at least as many terms as any maximal arithmetic subsequence of B(n).
Rule 2: Construct the sequences by starting at 1 and always favoring to add terms to B, unless it violates Rule 1. Add terms to A if they cannot be added to B.
The sequence listed here is the A sequence.

			A gets 1, because B cannot yet have an AS of length 1.
B gets 2, because we favor adding to B and A already has an AS of length 1.
A gets 3, because B cannot yet form an AS of length 2.
B gets 4, because A now has an AS of length 2.
B gets 5, because 2,4,5 does not contain an AS of length 3.
A gets 6, because 2,4,5,6 would contain an AS of length 3.

CheckMaxASFromEnd[A_,x_,m_] := Module[{i,n=Length[A],d,k}, If[ m>n+1, Return[False];,]; If[ And[ m<=n+1, n < 2], Return[True];,]; For[ i=n, i>0, i--, d = x-A[[i]]; bot = x-(m-1)*d; If[ bot <= 0, Break[];,]; For[ k=x-2*d, k>=x-(m-1)*d, k -= d, If[ MemberQ[A,k], If[ k == x-(m-1)*d, Return[True];,];, Break[]; ]; ]; ]; False ];
ASRace[n_] := Module[{i,m=0,A={},B={}}, For[i=1,i <= n, i++, If[ CheckMaxASFromEnd[B,i,m+1], If[ CheckMaxASFromEnd[A,i,m+1], m++, ]; A = Append[A,i];, B = Append[B,i]; ]; ]; {A,B} ];

A128118 At the n-th iteration the sequence of integers from n down to 1 are appended to the sequence a(n) times.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1, 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 9, 8, 7, 6, 5, 4, 3

Offset: 1

Reed Kelly, Feb 15 2007

			a(4) = 2 so 4,3,2,1,4,3,2,1 is added to the end of the sequence at the fourth iteration.

Reed Kelly, Feb 15 2007, Table of n, a(n) for n = 1..555

Cf. A128117, A001462, A002260.

crr[n_] := Module[ { A = {1,2,1,2,1}, i, j }, For [ i = 3, i <= n, i++, A = Join[ A, Flatten[ Table[ Range[i,1,-1], {A[[i]]} ]]]; ]; A ]

A128117 At the n-th iteration the sequence of integers from 1 to n are appended to the sequence a(n) times.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4

Offset: 1

Reed Kelly, Feb 15 2007

			a(5) = 2 so 1,2,3,4,5,1,2,3,4,5 is added to the end of the sequence at the fifth iteration.

Rémy Sigrist, Table of n, a(n) for n = 1..10258 (first 371 terms from Reed Kelly)

Cf. A001462.

cr[n_] := Module[ { A = {1,1,2}, i, j }, For [ i = 3, i <= n, i++, A = Join[ A, Flatten[ Table[ Range[i], {A[[i]]} ]]]; ]; A ]

a = vector(101, n, 1); m=0; for (n=1, 10, for (t=1, a[n], for (k=1, n, print1 (a[m++]=k", ")))) \\ Rémy Sigrist, Feb 09 2022

A123403 Combining the conditional divide-by-two concept from Collatz sequences with Pascal's triangle, we can arrive at a new kind of triangle. Start with an initial row of just 4. To compute subsequent rows, start by appending a zero to the beginning and end of the previous row. Like Pascal's triangle, add adjacent terms of the previous row to create each of the subsequent terms. The only change is that each term is divided by two if it is even. Then take the center of this triangle. In other words, take the n-th term from the (2n)th row.

Original entry on oeis.org

4, 2, 3, 5, 9, 15, 27, 25, 47, 89, 107, 119, 241, 545, 699, 1471, 3313, 4288, 15661, 31739, 30813, 35143, 92101, 123614, 384815, 788429, 1532363, 2995379, 6281191, 13569969, 16900339, 26062940, 28141406, 57780803, 122540851, 263162577

Offset: 1

Reed Kelly, Oct 14 2006

Reed Kelly, Collatz-Pascal Triangle

Cf. A123402.

(*Returns the center row of the CPT*) CollatzPascalCenter[init_, n_] := Module[{CPT, CENTER, ROWA, ROWB, a, i, j}, If[ListQ[init], CPT = {init}, CPT = {{0, 4, 0}}]; CENTER = {4}; For[i = 1, i < n, i++, ROWA = CPT[[i]]; ROWB = {0}; For[j = 1, j < Length[ROWA], j++, a = ROWA[[j]] + ROWA[[j + 1]]; a = a/(2 - Mod[a, 2]); If[And[EvenQ[Length[ROWA]], (j == Length[ROWA]/2)], CENTER = Append[CENTER, a],]; ROWB = Append[ROWB, a];]; ROWB = Append[ROWB, 0]; CPT = Append[CPT, ROWB];]; CENTER] CollatzPascalCenter[,200]

Define a(n, m) for integers m, n: a(0, 0)=4, a(n, m) := 0 for m<0 and n

A123402 Combining the conditional divide-by-two concept from Collatz sequences with Pascal's triangle, one can construct a new kind of triangle. Start with an initial row of just 4. To compute subsequent rows, start by appending a zero to the beginning and end of the previous row. Like Pascal's triangle, add adjacent terms of the previous row to create each of the subsequent terms. The only change is that each new term is divided by two if it is even.

Original entry on oeis.org

4, 2, 2, 1, 2, 1, 1, 3, 3, 1, 1, 2, 3, 2, 1, 1, 3, 5, 5, 3, 1, 1, 2, 4, 5, 4, 2, 1, 1, 3, 3, 9, 9, 3, 3, 1, 1, 2, 3, 6, 9, 6, 3, 2, 1, 1, 3, 5, 9, 15, 15, 9, 5, 3, 1, 1, 2, 4, 7, 12, 15, 12, 7, 4, 2, 1, 1, 3, 3, 11, 19, 27, 27, 19, 11, 3, 3, 1, 1, 2, 3, 7, 15, 23, 27, 23, 15, 7, 3, 2, 1, 1, 3, 5, 5, 11, 19

Offset: 0

Reed Kelly, Oct 14 2006

			For the row starting with (1,2,4,5,8,...) the subsequent row is computed as follows: 0+1->1, 1+2->3, (2+4)/2->3, 4+5->9, 5+8->13...

Reed Kelly, Collatz-Pascal Triangle
Reed Kelly, Collatz-Pascal Triangle, Oct 12 2006 [Local copy]

Cf. A007318, A069202.

CollatzPascalTriangle[init_, n_] := Module[{CPT, ROWA, ROWB, a, i, j}, If[ListQ[init], ROWA = init, ROWA = {4}]; CPT = {ROWA}; ROWA = Flatten[{0, ROWA, 0}]; For[i = 1, i < n, i++, ROWB = {0}; For[j = 1, j < Length[ROWA], j++, a = ROWA[[j]] + ROWA[[j + 1]]; a = a/(2 - Mod[a, 2]); ROWB = Append[ROWB, a];]; CPT = Append[CPT, Rest[ROWB]]; ROWA = Append[ROWB, 0]]; CPT] Flatten[ CollatzPascalTriangle[{4},20]]

Define a(n, m) for integers m, n: a(0, 0)=4, a(n, m) := 0 for m<0 and n

cp's OEIS Frontend

User: Reed Kelly

A214646 a(n) = (a(n-2) + a(n-3))/gcd(a(n-2), a(n-3)) with a(1) = a(2) = a(3) = 1.

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A214652 a(n) = (a(n-1) + a(n-4))/gcd(a(n-1), a(n-4)) with a(1) = a(2) = a(3) = a(4) = 1.

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A214551 Reed Kelly's sequence: a(n) = (a(n-1) + a(n-3))/gcd(a(n-1), a(n-3)) with a(0) = a(1) = a(2) = 1.

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A157102 Tuple-chromatic Van der Waerden numbers.

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A140516 For definition see comments lines.

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A128118 At the n-th iteration the sequence of integers from n down to 1 are appended to the sequence a(n) times.

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A128117 At the n-th iteration the sequence of integers from 1 to n are appended to the sequence a(n) times.

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