A080096 -id:A080096 - cp's OEIS frontend

A055186 Cumulative counting sequence: method A (adjective-before-noun)-pairs with first term 0.

0, 1, 0, 2, 0, 1, 1, 3, 0, 3, 1, 1, 2, 4, 0, 5, 1, 2, 2, 2, 3, 5, 0, 6, 1, 5, 2, 3, 3, 1, 4, 1, 5, 6, 0, 9, 1, 6, 2, 5, 3, 2, 4, 4, 5, 1, 6, 7, 0, 11, 1, 8, 2, 6, 3, 4, 4, 6, 5, 4, 6, 0, 7, 0, 8, 1, 9, 10, 0, 13, 1, 9, 2, 7, 3, 7, 4, 7, 5, 7, 6, 2, 7, 2, 8, 2, 9, 0, 10, 1, 11, 12, 0, 15, 1, 13, 2, 8, 3, 8, 4

Offset: 1

Clark Kimberling, Apr 27 2000

Start with 0; at n-th step, write down what is in the sequence so far.
"Look and Say" how many times each integer (in increasing order), <= max {existing terms} appears in the sequence. Then concatenate.  Sequence's graph looks roughly like that of A080096.
For the original version, where "increasing order..." is "order of 1st appearance", see A217760.  The conjecture formerly placed here applies to A217760. - Clark Kimberling, Mar 24 2013

			Write 0, thus having 1 0, thus having 2 0's and 1 1, thus having 3 0's and 3 1's and 1 2, etc. 0; 1,0; 2,0,1,1; 3,0,3,1,1,2; ...

Zak Seidov, Table of n, a(n) for n = 1..1019 (the first 22 steps)

Cf. A005150. For other versions see A051120, A079668, A079686.
Cf. A055168-A055185 (method B) and A055187-A055191 (method A).
Cf. A217760.

s={0};Do[ta=Table[{Count[s, # ], # }&/@Range[0,Max[s]]]; s=Flatten[{s,ta}],{22}];s (* Zak Seidov, Oct 23 2009 *)

Conjectures: a(n) < 2*sqrt(n); limit as n goes to infinity Max( a(k) : 1<=k<=n)/sqrt(n) exist = 2. - Benoit Cloitre, Jan 28 2003

Edited by N. J. A. Sloane, Jan 17 2009 at the suggestion of R. J. Mathar
Removed a conjecture. - Clark Kimberling, Oct 24 2009
Entries changed to match b-file. - N. J. A. Sloane, Oct 04 2010

A077623 a(1)=1, a(2)=2, a(3)=4, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

Original entry on oeis.org

1, 2, 4, 1, 5, 0, 6, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 10, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 6, 4, 7, 3, 8, 2, 9, 1, 10, 0, 11, 1, 10, 2, 9, 3, 8, 4, 7, 5, 6, 6

Offset: 1

Benoit Cloitre, Dec 02 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A077653, A079623, A079624, A080096, A088226.

a077623 n = a077623_list !! (n-1)
a077623_list = 1 : 2 : 4 : zipWith3 (\u v w -> abs (w - v - u))
               a077623_list (tail a077623_list) (drop 2 a077623_list)
-- Reinhard Zumkeller, Oct 11 2014

m:=120;
A077623:=[n le 3 select 2^(n-1) else Abs(Self(n-1) - Self(n-2) - Self(n-3)): n in [1..m+5]];
[A077623[n]: n in [1..m]]; // G. C. Greubel, Sep 11 2024

a[n_]:= a[n]= If[n<4, 2^(n-1), Abs[a[n-1] -a[n-2] -a[n-3]]];
Table[a[n], {n,120}] (* G. C. Greubel, Sep 11 2024 *)

@CachedFunction
def a(n): # a = A077623
    if n<4: return 2^(n-1)
    else: return abs(a(n-1)-a(n-2)-a(n-3))
[a(n) for n in range(1,121)] # G. C. Greubel, Sep 11 2024

a(n)/sqrt(n) is bounded. More precisely, let M(n) = Max ( a(k) : 1<=k<=n ); then M(n) = floor(sqrt(n+29)) for n>=4

A077653 a(1)=1, a(2)=2, a(3)=2, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

Original entry on oeis.org

1, 2, 2, 1, 3, 0, 4, 1, 3, 2, 2, 3, 1, 4, 0, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 10, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 6, 4, 7

Offset: 1

Benoit Cloitre, Dec 02 2002

nonn

Conjecture : let z(1)=x; z(2)=y; z(3)= z; z(n)=abs(z(n-1)-z(n-2)-z(n-3)) if z(n) is unbounded (i.e. x,y,z are such that z(n) doesn't reach a cycle of length 2), then there are 2 integers n(x,y,z) and w(x,y,z) such that M(n) = floor(sqrt(n+w(x,y,z))) for n>n(,x,y,z) where M(n) = Max ( a(k) : 1<=k<=n ). As example : w(1,2,2)=9 n(1,2,2)=4; w(1,2,4)=29 n(1,2,4)=4; w(1,2,8)=157 n(1,2,8)=9

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A077623, A079623, A079624, A080096, A088226.

a077653 n = a077653_list !! (n-1)
a077653_list = 1 : 2 : 2 : zipWith3 (\u v w -> abs (w - v - u))
               a077653_list (tail a077653_list) (drop 2 a077653_list)
-- Reinhard Zumkeller, Oct 11 2014

m:=120;
A077653:=[n le 3 select Floor((n+2)/2) else Abs(Self(n-1) - Self(n-2) - Self(n-3)): n in [1..m+5]];
[A077653[n]: n in [1..m]]; // G. C. Greubel, Sep 11 2024

nxt[{a_,b_,c_}]:={b,c,Abs[c-b-a]}; NestList[nxt,{1,2,2},110][[All,1]] (* Harvey P. Dale, Sep 01 2020 *)

@CachedFunction
def a(n): # a = A077653
    if n<4: return int((n+2)//2)
    else: return abs(a(n-1)-a(n-2)-a(n-3))
[a(n) for n in range(1,101)] # G. C. Greubel, Sep 11 2024

a(n)/sqrt(n) is bounded. More precisely, let M(n) = Max ( a(k) : 1<=k<=n ); then M(n)= floor(sqrt(n+9)) for n>4

A079623 a(1) = a(2) = 1, a(3)=4, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

Original entry on oeis.org

1, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 10, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 6, 4, 7, 3, 8, 2, 9, 1, 10, 0, 11, 1, 10, 2, 9, 3, 8

Offset: 1

Benoit Cloitre, Jan 30 2003

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A077623, A077653, A078108, A079624, A080096, A088226.

a079623 n = a079623_list !! (n-1)
a079623_list = 1 : 1 : 4 : zipWith3 (\u v w -> abs (w - v - u))
               a079623_list (tail a079623_list) (drop 2 a079623_list)
-- Reinhard Zumkeller, Oct 11 2014

m:=120;
A079623:=[n le 3 select 4^Floor((n-1)/2) else Abs(Self(n-1) - Self(n-2) - Self(n-3)): n in [1..m+5]];
[A079623[n]: n in [1..m]]; // G. C. Greubel, Sep 11 2024

nxt[{a_,b_,c_}]:={b,c,Abs[c-b-a]}; NestList[nxt,{1,1,4},110][[All,1]] (* Harvey P. Dale, Aug 12 2020 *)

@CachedFunction
def a(n): # a = A079623
    if n<4: return 4^((n-1)//2)
    else: return abs(a(n-1)-a(n-2)-a(n-3))
[a(n) for n in range(1,101)] # G. C. Greubel, Sep 11 2024

a(n*(n-10)) = 0.

Max( a(k) : 1<=k<=n) = floor(sqrt(n+24)).

A079624 a(1) = a(2) = 1, a(3) = 6, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

Original entry on oeis.org

1, 1, 6, 4, 3, 7, 0, 10, 3, 7, 6, 4, 9, 1, 12, 2, 11, 3, 10, 4, 9, 5, 8, 6, 7, 7, 6, 8, 5, 9, 4, 10, 3, 11, 2, 12, 1, 13, 0, 14, 1, 13, 2, 12, 3, 11, 4, 10, 5, 9, 6, 8, 7, 7, 8, 6, 9, 5, 10, 4, 11, 3, 12, 2, 13, 1, 14, 0, 15, 1, 14, 2, 13, 3, 12, 4, 11, 5, 10, 6, 9, 7, 8, 8, 7, 9, 6, 10, 5, 11, 4, 12, 3

Offset: 1

Benoit Cloitre, Jan 30 2003

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A077623, A077653, A078108, A079623, A080096, A088226.

a079624 n = a079624_list !! (n-1)
a079624_list = 1 : 1 : 6 : zipWith3 (\u v w -> abs (w - v - u))
               a079624_list (tail a079624_list) (drop 2 a079624_list)
-- Reinhard Zumkeller, Oct 11 2014

m:=120;
A079624:=[n le 3 select 6^Floor((n-1)/2) else Abs(Self(n-1) - Self(n-2) - Self(n-3)): n in [1..m+5]];
[A079624[n]: n in [1..m]]; // G. C. Greubel, Sep 11 2024

a[n_]:= a[n]= If[n<4, 6^Floor[(n-1)/2], Abs[a[n-1] -a[n-2] -a[n-3]]];
Table[a[n], {n,100}] (* G. C. Greubel, Sep 11 2024 *)

@CachedFunction
def a(n): # a = A079624
    if n<4: return 6^((n-1)//2)
    else: return abs(a(n-1)-a(n-2)-a(n-3))
[a(n) for n in range(1,101)] # G. C. Greubel, Sep 11 2024

For n >= 5, a(n^2 + 24*n - 13) = 0.

For n >= 38, Max( a(k) : 1<=k<=n) = floor(sqrt(n+156)).

A088226 a(1)=0, a(2)=0, a(3)=1; for n>3, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 1, 1, 2, 0, 3, 1, 2, 2, 1, 3, 0, 4, 1, 3, 2, 2, 3, 1, 4, 0, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 10, 1, 9, 2

Offset: 1

Benoit Cloitre, Nov 04 2003

nonn

Conjecture: a(n) = m/2 where m is the smallest even distance from n to a square. - Ralf Stephan, Sep 23 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A077623, A077653, A079623, A079624, A080096.

a088226 n = a088226_list !! (n-1)
a088226_list = 0 : 0 : 1 : zipWith3 (\u v w -> abs (w - v - u))
               a088226_list (tail a088226_list) (drop 2 a088226_list)
-- Reinhard Zumkeller, Oct 11 2014

m:=120;
A088226:=[n le 3 select Floor((n-1)/2) else Abs(Self(n-1) - Self(n-2) - Self(n-3)): n in [1..m+5]];
[A088226[n]: n in [1..m]]; // G. C. Greubel, Sep 11 2024

RecurrenceTable[{a[1]==a[2]==0,a[3]==1,a[n]==Abs[a[n-1]-a[n-2]-a[n-3]]},a,{n,110}] (* Harvey P. Dale, Apr 13 2012 *)

a(n)=t=sqrtint(n);if((n-t*t)%2==0,(n-t*t)/2,((t+1)^2-n)/2) \\ Ralf Stephan, Sep 23 2013

@CachedFunction
def a(n): # a = A088226
    if n<4: return int((n-1)//2)
    else: return abs(a(n-1)-a(n-2)-a(n-3))
[a(n) for n in range(1,101)] # G. C. Greubel, Sep 11 2024

a(k^2 + 2*m + 2) = k-m and a(k^2 + 2*m + 1) = m, for k >= 0 and 0 <= m <= k.

cp's OEIS Frontend

A055186 Cumulative counting sequence: method A (adjective-before-noun)-pairs with first term 0.

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A077623 a(1)=1, a(2)=2, a(3)=4, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

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A077653 a(1)=1, a(2)=2, a(3)=2, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

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A079623 a(1) = a(2) = 1, a(3)=4, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

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A079624 a(1) = a(2) = 1, a(3) = 6, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

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A088226 a(1)=0, a(2)=0, a(3)=1; for n>3, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

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