A033627 -id:A033627 - cp's OEIS frontend

A114801 2-concatenation-free sequence starting (1,2).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 121, 123, 124, 125, 126, 127, 128, 129, 131, 132, 134, 135, 136, 137, 138, 139, 141, 142, 143, 145, 146, 147, 148, 149, 151, 152, 153, 154, 156, 157, 158

Offset: 1

Jonathan Vos Post, Feb 18 2006

Starting with the terms (1,2) this sequence consists of minimum increasing terms such that no term is the concatenation of any two previous distinct terms. The next consecutive number skipped after 121 is 122 = Concatenate(1, 22). This is the analog of a 2-Stöhr sequence with concatenation (base 10) substituting for addition. A033627 "0-additive sequence: not the sum of any previous pair" is another name for the 2-Stöhr sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Stöhr Sequence.
Rémy Sigrist, PARI program for A114801

Cf. A084383, A033627.

conc[x_, y_] := FromDigits@ Flatten@ IntegerDigits[{x, y}]; L = {1, 2}; cc = {12, 21}; Do[k = 1 + Max@L; While[MemberQ[cc, k], k++]; cc = Union[cc, conc[#, k] & /@ L, conc[k, #] & /@ L]; AppendTo[L, k];, {65}]; L (* Giovanni Resta, Jun 15 2016 *)

PARI
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See Links section.
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from itertools import count, islice
def agen(): # generator of terms
    cats1, cats2, an, s = {"1", "2"}, {"12", "21"}, 3, "3"
    yield from [1, 2]
    while True:
        yield an
        cats2 |= {s + c for c in cats1} | {c + s for c in cats1}
        cats1.add(s)
        while (s:=str(an)) in cats1 or s in cats2:
            an += 1
print(list(islice(agen(), 59))) # Michael S. Branicky, Feb 01 2024

a(0) = 1, a(1) = 2, for n>2: a(n) = least k > a(n-1) such that k is not an element of {Concatenate(a(i), a(j))} for any distinct a(i) <= a(n-1) and a(j) <= a(n-1).

Data corrected by Giovanni Resta, Jun 14 2016

A114802 3-concatenation-free sequence starting (1,2).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 121, 131, 141, 151, 161, 171, 181, 191, 200, 212, 232, 242, 252, 262, 272, 282, 292, 300, 313, 323, 343, 353, 363, 373, 383, 393, 400, 414, 424, 434, 454

Offset: 1

Jonathan Vos Post, Feb 18 2006

Starting with the terms (1,2) this sequence consists of minimum increasing integer terms such that no term is the concatenation of any two or three previous distinct terms. The next consecutive numbers skipped after 121 are 122 = Concatenate(1,22) and 123 = Concatenate(1,2,3). This is the analog of a 3-Stöhr sequence with concatenation (base 10) substituting for addition. A026474 is a 3-Stöhr sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Stöhr Sequence.

Cf. A084383, A033627, A026474.

conc[w_] := Flatten[ (FromDigits /@ Flatten /@ IntegerDigits /@ (Permutations[#]) &) /@ Subsets[w, {2, 3}]]; up = 10^3; L = {1, 2, 3}; cc = conc[L]; Do[k = 1 + Max@L; While[MemberQ[cc, k], k++]; If[k > up, Break[]]; Do[cc = Union[cc, Select[ conc[{k, L[[i]], L[[j]]}], # <= up &]], {i, Length[L]}, {j, i - 1}]; AppendTo[L, k], {60}]; L (* Giovanni Resta, Jun 15 2016 *)

from itertools import islice
def incats(s, L, k):
    if s == "": return True
    if k == 0: return False
    return any(s.startswith(w) and incats(s[len(w):], L[:i]+L[i+1:], k-1) for i, w in enumerate(L))
def agen(): # generator of terms
    L, an, s = ["1", "2"], 3, "3"
    yield from [1, 2]
    while True:
        yield an
        L.append(s)
        while incats((s:=str(an)), L, 3):
            an += 1
print(list(islice(agen(), 70))) # Michael S. Branicky, Feb 01 2024

a(0) = 1, a(1) = 2, for n>2: a(n) = least k > a(n-1) such that k is not an element of {Concatenate[a(h),a(i),a(j)]} or {Concatenate[a(i),a(j)]} for any three distinct a(h), a(i), and a(j), where h, i, j < n.

Corrected and edited by Giovanni Resta, Jun 15 2016

A119789 T(n, k) = floor(log_{goldenratio}(Fibonacci(n)*Fibonacci(k))), with T(n, k) = 0 for n < 3, T(n, 0) = n-2 for n > 2, triangle read by rows.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 3, 3, 3, 4, 5, 6, 4, 4, 4, 5, 6, 7, 8, 5, 5, 5, 6, 7, 8, 9, 10, 6, 6, 6, 7, 8, 9, 10, 11, 12, 7, 7, 7, 8, 9, 10, 11, 12, 13, 14, 8, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16

Offset: 0

Roger L. Bagula, Jul 30 2006

			Triangle begins as:
  0;
  0, 0;
  0, 0, 0;
  1, 1, 1, 2;
  2, 2, 2, 3, 4;
  3, 3, 3, 4, 5, 6;
  4, 4, 4, 5, 6, 7, 8;
  5, 5, 5, 6, 7, 8, 9, 10;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000045, A001477, A016789, A033627, A035517, A064455, A140090.

A119789:= func< n,k | n le 2 select 0 else k le 1 select n-2 else n+k-4 >;
[A119789(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 17 2022

f[n_, k_]= If[n<3, 0, If[k==0, n-2, Floor[Log[GoldenRatio, Fibonacci[n]*Fibonacci[k]]]]];
Table[f[n, k], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
T[n_, k_]:= T[n, k]= If[n<3, 0, If[k<2, n-2, n+k-4]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 17 2022 *)

def A119789(n,k):
    if (n<3): return 0
    elif (k<2): return n-2
    else: return n+k-4
flatten([[A119789(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 17 2022

T(n, k) = floor(log_{goldenratio}(Fibonacci(n)*Fibonacci(k))), with T(n, k) = 0 for n < 3, T(n, 0) = n-2 for n > 2.
From G. C. Greubel, Dec 17 2022: (Start)
T(n, k) = n+k-4, with T(n, k) = 0 for n < 3, T(n, 0) = n-2 for n >= 3.
T(n, n) = 2*T(n, 0).
T(2*n, n) = 0*[n<2] + A016789(n-2)*[n>1].
T(2*n, n+1) = 3*A001477(n-1), for n > 0.
T(2*n, n-1) = A033627(n) - [n=1].
T(3*n, n) = n*[n<2] + 4*A000027(n-2)*[n>1].
Sum_{k=0..n} T(n, k) = 0*[n<2] + A140090(n-2)*[n>1].
Sum_{k=0..n} (-1)^k * T(n, k) = 0*[n<2] + (-1)^n*A064455(n-2)*[n>1]. (End)

Edited by G. C. Greubel, Dec 17 2022

A244749 0-additive sequence: a(n) is the smallest number larger than a(n-1) that is not the sum of any subset of earlier terms, starting with initial values {2, 5}.

Original entry on oeis.org

2, 5, 6, 9, 10, 28, 29, 85, 86, 256, 257, 769, 770, 2308, 2309, 6925, 6926, 20776, 20777, 62329, 62330, 186988, 186989, 560965, 560966, 1682896, 1682897, 5048689, 5048690, 15146068, 15146069, 45438205, 45438206, 136314616, 136314617, 408943849, 408943850, 1226831548, 1226831549

Offset: 1

N. J. A. Sloane and Robert G. Wilson v, Jul 05 2014

This sequence differs from A003664.

			The numbers 11-27 are not in the sequence since some combination of the previous terms add to it. example 17=2+5+10.
The number 28 however is a term since no combination of the previous terms cannot be found which sum to 28.

R. K. Guy, "s-Additive sequences," preprint, 1994.

Steven R. Finch, Are 0-additive sequences always regular?, Amer. Math. Monthly, 99 (1992), 671-673.
Index entries for linear recurrences with constant coefficients, signature (-1,3,3).

Cf. A003662, A003663, A005408, A026471, A026474, A033627, A051039, A051040, A244151, A244750.

Cf. A060469 - A060472.

f[s_List] := f[n] = Block[{k = s[[-1]] + 1, ss = Union[ Plus @@@ Subsets[s]]}, While[ MemberQ[ss, k], k++]; Append[s, k]]; Nest[ f[#] &, {2, 5}, 20] (* or *)
b = LinearRecurrence[{4, -3}, {9, 28}, 18]; Join[{2, 5, 6}, Riffle[b, b + 1]]
Join[{2, 5, 6},LinearRecurrence[{-1, 3, 3},{9, 10, 28},36]] (* Ray Chandler, Aug 03 2015 *)

Vec(x*(7*x^5+14*x^4+6*x^3-5*x^2-7*x-2)/((x+1)*(3*x^2-1)) + O(x^100)) \\ Colin Barker, Jul 11 2014

a(2n) = 4a(2n - 2) - 3a(2n - 4) and a(2n +1) = a(2n) +1, for n>2.
a(n) = -a(n-1) + 3*a(n-2) + 3*a(n-3) for n>6. - Colin Barker, Jul 11 2014
G.f.: x*(7*x^5+14*x^4+6*x^3-5*x^2-7*x-2) / ((x+1)*(3*x^2-1)). - Colin Barker, Jul 11 2014

A123031 Array read by antidiagonals: row i (i>=0) contains those positive integers n >= 2 for which the multiset { n mod k : k=2,3,...,n } contains exactly one copy of i.

Original entry on oeis.org

2, 3, 3, 5, 4, 5, 7, 6, 6, 7, 9, 8, 7, 8, 11, 11, 10, 9, 9, 12, 13, 13, 12, 11, 10, 13, 14, 17, 15, 14, 13, 12, 12, 15, 18, 19, 17, 16, 15, 14, 13, 14, 19, 20, 23, 19, 18, 17, 16, 15, 15, 16, 21, 24, 29, 21, 20, 19, 18, 17, 16, 17, 20, 25, 30, 31, 23, 22, 21, 20, 19, 18, 18, 21, 22

Offset: 1

Jared B. Ricks (jaredricks(AT)yahoo.com), Sep 24 2006

In other words, for i >= 1, the i-th row contains all numbers n>2i such that n-i does not have divisors d with i < d < n-i. If p is the smallest prime divisor of n-i then (n-i)/p <= i.

Alternatively, the i-th row (i>=1) consists of 2i+1 and positive integers n>2i+1 such that the smallest prime divisor of n-i is greater than or equal to (n-i)/i = n/i - 1.

			For example, the 0th row obviously contains all prime numbers.
The first few rows of the array are
0) 2, 3, 5, 7, 11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,
1) 3, 4, 6, 8, 12,14,18,20,24,30,32,38,42,44,48,54,60,62,68,72,74,80,84,90,98,
2) 5, 6, 7, 9, 13,15,19,21,25,31,33,39,43,45,49,55,61,63,69,73,75,81,85,91,99,
3) 7, 8, 9, 10,12,14,16,20,22,26,32,34,40,44,46,50,56,62,64,70,74,76,82,86,92,
4) 9, 10,11,12,13,15,17,21,23,27,33,35,41,45,47,51,57,63,65,71,75,77,83,87,93,
5) 11,12,13,14,15,16,18,20,22,24,28,30,34,36,42,46,48,52,58,64,66,72,76,78,84,
6) 13,14,15,16,17,18,19,21,23,25,29,31,35,37,43,47,49,53,59,65,67,73,77,79,85,
...

Rows: A000040, A008864, ...; columns: A004280, A051755, ...; diagonal starting with 2: A033627.

Additional comments from Max Alekseyev, Sep 26 2006

cp's OEIS Frontend

A114801 2-concatenation-free sequence starting (1,2).

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A114802 3-concatenation-free sequence starting (1,2).

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A119789 T(n, k) = floor(log_{goldenratio}(Fibonacci(n)*Fibonacci(k))), with T(n, k) = 0 for n < 3, T(n, 0) = n-2 for n > 2, triangle read by rows.

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A244749 0-additive sequence: a(n) is the smallest number larger than a(n-1) that is not the sum of any subset of earlier terms, starting with initial values {2, 5}.

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Mathematica

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A123031 Array read by antidiagonals: row i (i>=0) contains those positive integers n >= 2 for which the multiset { n mod k : k=2,3,...,n } contains exactly one copy of i.

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