A101544 -id:A101544 - cp's OEIS frontend

A101545 Inverse permutation to A101544.

1, 2, 3, 4, 5, 7, 8, 10, 6, 11, 13, 14, 9, 16, 17, 19, 20, 12, 22, 23, 25, 26, 15, 28, 29, 31, 32, 34, 18, 35, 37, 38, 21, 40, 41, 43, 44, 46, 24, 47, 49, 50, 27, 52, 53, 55, 56, 58, 30, 59, 61, 62, 33, 64, 65, 67, 68, 36, 70, 71, 73, 74, 39, 76, 77, 79, 80, 82, 42, 83, 85, 86

Offset: 1

Reinhard Zumkeller, Dec 06 2004

nonn

A101547(n) = a(a(n)).

Cf. A101544, A101547.

A101546 a(n) = A101544(A101544(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 13, 9, 6, 11, 7, 8, 29, 10, 18, 20, 12, 23, 25, 14, 15, 53, 16, 17, 63, 33, 19, 36, 39, 21, 41, 22, 43, 45, 24, 49, 48, 26, 27, 103, 28, 30, 113, 58, 31, 61, 32, 34, 129, 35, 69, 70, 37, 73, 75, 38, 40, 153, 78, 42, 82, 83, 44, 86, 89, 46, 91, 47, 93, 95, 50, 98

Offset: 1

Reinhard Zumkeller, Dec 06 2004

nonn

Inverse permutation to A101547.

Cf. A101544, A101547.

A132677 Period 3: repeat [1, 2, -3].

Original entry on oeis.org

1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3, 1, 2, -3

Offset: 0

Paul Curtz, Nov 15 2007

a(n) is proportional to its 6n-th differences.

Nonsimple continued fraction expansion of 1+sqrt(2/5) = 1.63245553... (see A010494). - R. J. Mathar, Mar 08 2012

Index entries for linear recurrences with constant coefficients, signature (-1,-1).

Cf. A000748, A010494, A049347, A101544.

&cat [[1, 2, -3]^^30]; // Wesley Ivan Hurt, Jul 01 2016

seq(op([1, 2, -3]), n=0..50); # Wesley Ivan Hurt, Jul 01 2016
A132677 := proc(n)
    op(1+modp(n,3),[1,2,-3]) ;
end proc: # R. J. Mathar, Mar 14 2025

PadRight[{}, 100, {1, 2, -3}] (* Wesley Ivan Hurt, Jul 01 2016 *)

a(n)=[1,2,-3][1+n%3] \\ Jaume Oliver Lafont, Mar 24 2009

G.f.: (1+3*x)/(1+x+x^2). - Jaume Oliver Lafont, Mar 24 2009
a(n) = cos(2*Pi*n/3) + 5*sin(2*Pi*n/3)/sqrt(3). - R. J. Mathar, Oct 08 2011
a(n) + a(n-1) + a(n-2) = 0 for n > 1, a(n) = a(n-3) for n > 2. - Wesley Ivan Hurt, Jul 01 2016

A309157 Rectangular array in 3 columns that solve the complementary equation c(n) = a(n) + b(2n), where a(1) = 1; see Comments.

Original entry on oeis.org

1, 2, 5, 3, 4, 12, 6, 7, 20, 8, 9, 26, 10, 11, 33, 13, 14, 41, 15, 16, 47, 17, 18, 54, 19, 21, 61, 22, 23, 68, 24, 25, 75, 27, 28, 83, 29, 30, 89, 31, 32, 96, 34, 35, 104, 36, 37, 110, 38, 39, 117, 40, 42, 124, 43, 44, 131, 45, 46, 138, 48, 49, 146, 50, 51

Offset: 1

Clark Kimberling, Jul 15 2019

Let A = (a(n)), B = (b(n)), and C = (c(n)). A unique solution (A,B,C) exists for these conditions: (1) A,B,C must partition the positive integers, and (2) A and B are defined by mex (minimal excludant, as in A067017); that is, a(n) is the least "new" positive integer, and likewise for b(n).

			c(1) = a(1) + b(2) > = 1 + 3, so that
a(2) = mex{1,2} = 3;
b(2) = mex{1,2,3} = 4;
c(1) = 5.
Then c(2) = a(2) + b(4) >= 3 + 8, so that
a(3) = 6, b(3) = 7;
a(4) = 8, b(4) = 9;
c(2) = a(2) + b(4) = 3 + 9 = 12.
   n    a(n) b(n) c(n)
  --------------------
   1      1    2    5
   2      3    4   12
   3      6    7   20
   4      8    9   26
   5     10   11   33
   6     13   14   41
   7     15   16   47
   8     17   18   54
   9     19   21   61
  10     22   23   68

Clark Kimberling and Peter J. C. Moses, Complementary Equations with Advanced Subscripts, J. Int. Seq. 24 (2021) Article 21.3.3.

Cf. A326663 (3rd column),
A101544 solves c(n) = a(n) + b(n),
A326661 solves c(n) = a(n) + b(3n),
A326662 solves c(n) = a(2n) + b(2n).

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = b = c = {}; h = 1; k = 2;
Do[Do[AppendTo[a,
  mex[Flatten[{a, b, c}], Max[Last[a /. {} -> {0}], 1]]];
  AppendTo[b, mex[Flatten[{a, b, c}], Max[Last[b /. {} -> {0}], 1]]], {k}];
  AppendTo[c, a[[h Length[a]/k]] + Last[b]], {150}];
{a, b, c} // ColumnForm
a = Take[a, Length[c]]; b = Take[b, Length[c]];
Flatten[Transpose[{a, b, c}]] (* Peter J. C. Moses, Jul 04 2019 *)

A332341 Prime scale sequence (see comments).

Original entry on oeis.org

-2, -3, 5, -7, -11, -13, 31, -17, -19, -23, 59, -29, -37, -41, 107, -43, -47, -53, -61, -67, 271, -71, -73, -79, 223, -83, -89, -97, 269, -101, -103, -109, 313, -113, -127, -131, -137, -139, 647, -149, -151, -157, 457, -163, -167, -173, 503, -179, -181, -191, -193, -197, 941

Offset: 1

Ivan N. Ianakiev, Feb 10 2020

sign

Take a double-pan balance scale and name the pans "negative" and "positive". At each step, the question is: "Is there an unused prime that would balance the scale if added to the positive pan?" If the answer is positive, add that prime to the positive pan. Otherwise, add the smallest unused prime to the negative pan.

Is the number of primes in the positive pan infinite?

			2 and 3 unbalance the scale (and are negative), but 5 = 2 + 3 balances it (and is positive).

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A000040, A101544, A332787, A332788.

a[1]=-2;a[n_]:=a[n]=Module[{tab=Table[a[i],{i,1,n-1}],
totalN=Abs[Total[Select[Table[a[i],{i,1,n-1}],Negative]]],
totalP=Total[Select[Table[a[i],{i,1,n-1}],Positive]],
l=NextPrime[Last[Select[Table[a[i],{i,1,n-1}],Negative]],-1],
m=NextPrime[Abs[Last[Select[Table[a[i],{i,1,n-1}],Negative]]]]},
If[totalN==totalP,If[PrimePi[tab[[-1]]]-PrimePi[Abs[tab[[-2]]]]==1,-NextPrime[tab[[-1]]],
If[FreeQ[Abs[tab],m],-m,While[!FreeQ[Abs[tab],m],m=NextPrime[m]];-m]],
If[PrimeQ[totalN-totalP]&&FreeQ[Abs[tab],totalN-totalP],totalN-totalP,
If[FreeQ[Abs[tab],Abs[l]],l,While[!FreeQ[Abs[tab],Abs[l]],l=NextPrime[l,-1]];l]]]];a/@Range[53]

from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    used, d, nextp = set(), 0, 2
    while True:
        if d > 0 and d not in used and isprime(d):
            used.add(d); yield d; d = 0
        while nextp in used:
            nextp = nextprime(nextp)
        used.add(nextp); yield -nextp; d += nextp
print(list(islice(agen(), 53))) # Michael S. Branicky, May 12 2022

A080427 a(1)=1 and, for n>1, a(n) is the smallest positive integer such that the absolute difference |a(n)-a(n-1)| has not occurred previously.

Original entry on oeis.org

1, 1, 2, 4, 1, 5, 10, 1, 7, 14, 1, 9, 19, 1, 12, 24, 1, 15, 30, 1, 17, 34, 1, 20, 40, 1, 22, 44, 1, 25, 50, 1, 27, 54, 1, 29, 59, 1, 32, 64, 1, 35, 70, 1, 37, 74, 1, 39, 79, 1, 42, 84, 1, 45, 90, 1, 47, 94, 1, 49, 99, 1, 52, 104, 1, 55, 110, 1, 57, 114, 1, 60, 120, 1, 62, 124, 1, 65, 130

Offset: 1

John W. Layman, Feb 19 2003

nonn

It appears (1) that a(3n+2)=1 for n=1,2,3,... and (2) that the sequence {a(3n+3)-a(3n)}={3,2,2,3,3,2,3,2,3,2,2,3,3,2,2,3,3,2,...} consists only of 2's and 3's and that the sequence of the lengths of runs of consecutive 3's in {a(3n+3)-a(3n)} is given by {1,2,1,1,2,2,2,1,...}=A026465.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A026465, A067992, A101544.

{ my(s=0, v=1, d); for (n=1, 79, print1 (v, ", "); for (w=1, oo, if (!bittest(s, d=abs(v-w)), s+=2^d; v=w; break))) } \\ Rémy Sigrist, Apr 12 2020

It appears that abs(a(n+2)-a(n+1)) = A101544(n) for any n > 0. - Rémy Sigrist, Apr 12 2020

A351298 Concatenation of the lexicographically earliest 6-term closed circuits formed on a square grid by distinct segments of length a(n) at right angle.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 8, 9, 10, 15, 18, 11, 12, 13, 14, 24, 26, 16, 17, 19, 20, 35, 37, 21, 22, 23, 25, 44, 47, 27, 28, 29, 30, 56, 58, 31, 32, 33, 34, 64, 66, 36, 38, 39, 40, 75, 78, 41, 42, 43, 45, 84, 87, 46, 48, 49, 50, 95, 98, 51, 52, 53, 54, 104, 106, 55, 57, 59, 60, 114, 117, 61, 62, 63, 65, 124, 127, 67

Offset: 1

Eric Angelini and Carole Dubois, Feb 07 2022

			[1, 2, 3, 5, 4, 7]  is a closed circuit on a square grid formed by going 1 cell up (North), 2 cells to the right (East), 3 cells up again (North), 5 cells to the right again (East), 4 cells down (South) and 7 cells to the left (West); the next smallest such circuit is given by [6, 8, 9, 10, 15, 18] as all the terms of the final sequence must be distinct; the next circuit is [11, 12, 13, 14, 24, 26], etc. Concatenating all circuits gives the sequence.

Carole Dubois, Table of n, a(n) for n = 1..1002
Eric Angelini, More Manhattan distinct distances, Feb. 7th, 2022, personal blog.

Cf. A101544 (where the array has 3 columns; there are 6 columns here: if we label them a, b, c, d, e, f the terms a + c = e and b + d = f).

cp's OEIS Frontend

A101545 Inverse permutation to A101544.

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A101546 a(n) = A101544(A101544(n)).

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A132677 Period 3: repeat [1, 2, -3].

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A309157 Rectangular array in 3 columns that solve the complementary equation c(n) = a(n) + b(2n), where a(1) = 1; see Comments.

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Mathematica

A332341 Prime scale sequence (see comments).

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Python

A080427 a(1)=1 and, for n>1, a(n) is the smallest positive integer such that the absolute difference |a(n)-a(n-1)| has not occurred previously.

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Author

Keywords

Comments

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Crossrefs

Programs

PARI

Formula

A351298 Concatenation of the lexicographically earliest 6-term closed circuits formed on a square grid by distinct segments of length a(n) at right angle.

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