A080927 -id:A080927 - cp's OEIS frontend

A080900 a(1)=1; for n>1, a(n)=a(n-1)-2 if n is already in the sequence, a(n)=a(n-1)+5 otherwise.

1, 6, 11, 16, 21, 19, 24, 29, 34, 39, 37, 42, 47, 52, 57, 55, 60, 65, 63, 68, 66, 71, 76, 74, 79, 84, 89, 94, 92, 97, 102, 107, 112, 110, 115, 120, 118, 123, 121, 126, 131, 129, 134, 139, 144, 149, 147, 152, 157, 162, 167, 165, 170, 175, 173, 178, 176

Offset: 1

N. J. A. Sloane and Benoit Cloitre, Apr 01 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000 (terms 1..1000 from Ivan Neretin)
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
Benoit Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See p. 2.

Cf. A080912, A080913, A080904, A080914, A080578, A080919, A080922, A080927.

Cf. A080901 (starting value = 2), A080905 (run lengths of first differences).

Fold[Append[#1, #1[[-1]] + If[MemberQ[#1, #2], -2, 5]] &, {1}, Range[2, 57]] (* Ivan Neretin, Mar 03 2016 *)

up_to = 1001;
A080900list(up_to_n) = { my(xs=Map(), v=vector(up_to_n)); mapput(xs,1,1); v[1] = 1; for(n=2,up_to_n, v[n] = v[n-1]+if(mapisdefined(xs,n), -2, +5); mapput(xs,v[n],n)); (v); };
v080900 = A080900list(up_to);
A080900(n) = v080900[n]; \\ Antti Karttunen, Jan 22 2020

Perhaps this is asymptotic to c_0*n*(1 + c_1/log n + ...), with c_0 near 2 ?

A080924 Jacobsthal gap sequence.

Original entry on oeis.org

0, 1, 3, 1, 15, 1, 63, 1, 255, 1, 1023, 1, 4095, 1, 16383, 1, 65535, 1, 262143, 1, 1048575, 1, 4194303, 1, 16777215, 1, 67108863, 1, 268435455, 1, 1073741823, 1, 4294967295, 1, 17179869183, 1, 68719476735, 1, 274877906943, 1, 1099511627775, 1

Offset: 0

Paul Barry, Feb 26 2003

Inverse binomial transform of A080925
From Peter Bala, Dec 26 2012: (Start)
Let F(x) = product {n >= 0} (1 - x^(3*n+1))/(1 - x^(3*n+2)). This sequence is the simple continued fraction expansion of the real number F(1/4) = 0.79761 68651 30459 16010 ... = 1/(1 + 1/(3 + 1/(1 + 1/(15 + 1/(1 + 1/(63 + 1/(1 + 1/(255 + ...)))))))). See A111317. (End)
Also, the decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 3", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Apr 19 2017

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (-1,4,4).

Cf. A001045, A002450, A080926, A080927, A111317.

CoefficientList[Series[x (1 + 4 x) / ((1 + x) (1 + 2 x) (1 - 2 x)), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 05 2013 *)
LinearRecurrence[{-1, 4, 4}, {0, 1, 3}, 42] (* Jean-François Alcover, Sep 21 2017 *)

a(2n) = 3*A001045(2n) = 3*A002450(n) = 4^n-1, a(2n+1)=1.
a(n) = (2^n-2*(-1)^n+(-2)^n)/2.
G.f.: x*(1+4*x)/((1+x)*(1+2*x)*(1-2*x)).
E.g.f.: (exp(2*x)-2*exp(-x)+exp(-2*x))/2.

A080905 Sequence of run lengths in first differences of A080900.

Original entry on oeis.org

4, 1, 4, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 4, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 4, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 4, 1, 2, 1, 1, 1, 2

Offset: 1

N. J. A. Sloane, Apr 01 2003

nonn

Even after 400000 terms there is no clear pattern here.

			A080900 begins 1, 6, 11, 16, 21, 19, 24, 29, 34, 39, 37, 42, 47, 52, 57, 55, 60, 65, ..., the differences are 5, 5, 5, 5, -2, 5, 5, 5, 5, -2, ... with runs of lengths 4, 1, 4, 1, ...

Antti Karttunen, Table of n, a(n) for n = 1..102526

Cf. A080900, A080922, A080927.

up_to = 20000;
A080905list(up_to_n) = { my(xs=Map(), v, d, ds=vector(up_to_n)); mapput(xs,1,1); v = 1; for(n=2,1+up_to_n, v += (d=if(mapisdefined(xs,n), -2, +5)); mapput(xs,v,n); ds[n-1] = d); my(runlens=List([]), rl=1); for(i=2,#ds,if(ds[i]==ds[i-1],rl++, listput(runlens,rl);rl=1)); Vec(runlens); };
v080905 = A080905list(up_to);
A080905(n) = v080905[n]; \\ Antti Karttunen, Feb 24 2020

cp's OEIS Frontend

A080900 a(1)=1; for n>1, a(n)=a(n-1)-2 if n is already in the sequence, a(n)=a(n-1)+5 otherwise.

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A080924 Jacobsthal gap sequence.

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A080905 Sequence of run lengths in first differences of A080900.

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A080922 Records in A080905.

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