cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A297146 Numbers having an up-first zigzag pattern in base 10; see Comments.

Original entry on oeis.org

12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 56, 57, 58, 59, 67, 68, 69, 78, 79, 89, 120, 121, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 145
Offset: 1

Views

Author

Clark Kimberling, Jan 15 2018

Keywords

Comments

A number n having base-b digits d(m), d(m-1),..., d(0) such that d(i) != d(i+1) for 0 <= i < m shows a zigzag pattern of one or more segments, in the following sense. Writing U for up and D for down, there are two kinds of patterns: U, UD, UDU, UDUD, ... and D, DU, DUD, DUDU, ... . In the former case, we say n has an "up-first zigzag pattern in base b"; in the latter, a "down-first zigzag pattern in base b". Example: 2,4,5,3,0,1,4,2 has segments 2,4,5; 5,3,0; 0,1,4; and 4,2, so that 24530142, with pattern UDUD, has an up-first zigzag pattern in base 10, whereas 4,2,5,3,0,1,4,2 has a down-first pattern. The sequences A297146-A297148 partition the natural numbers. In the following guide, column four, "complement" means the sequence of natural numbers not in the corresponding sequences in columns 2 and 3.
***
Base up-first down-first complement
2 (none) A000975 A107907
3 A297124 A297125 A297126
4 A297128 A297129 A297130
5 A297131 A297132 A297133
6 A297134 A297135 A297136
7 A297137 A297138 A297139
8 A297140 A297141 A297142
9 A297143 A297144 A297145
10 A297146 A297147 A297148

Examples

			Base-10 digits of 59898: 5,9,8,9,8, with pattern UDUD, so that 59898 is in the sequence.

Crossrefs

Cf. A297147, A297148.

Programs

  • Mathematica
    a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
b = 10; t = Table[a[n, b], {n, 1, 10*z}];
u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297146 *)
v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297147 *)
Complement[Range[z], Union[u, v]]  (* A297148 *)

A297137 Numbers having an up-first zigzag pattern in base 7; see Comments.

Original entry on oeis.org

9, 10, 11, 12, 13, 17, 18, 19, 20, 25, 26, 27, 33, 34, 41, 63, 64, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 119, 120, 121, 123, 124, 125, 126, 127, 128, 129, 131, 132, 133, 134, 135, 136
Offset: 1

Views

Author

Clark Kimberling, Jan 15 2018

Keywords

Comments

A number n having base-b digits d(m), d(m-1),..., d(0) such that d(i) != d(i+1) for 0 <= i < m shows a zigzag pattern of one or more segments, in the following sense. Writing U for up and D for down, there are two kinds of patterns: U, UD, UDU, UDUD, ... and D, DU, DUD, DUDU, ... . In the former case, we say n has an "up-first zigzag pattern in base b"; in the latter, a "down-first zigzag pattern in base b". Example: 2,4,5,3,0,1,4,2 has segments 2,4,5; 5,3,0; 0,1,4; and 4,2, so that 24530142, with pattern UDUD, has an up-first zigzag pattern in base 10, whereas 4,2,5,3,0,1,4,2 has a down-first pattern. The sequences A297137-A297139 partition the natural numbers. See the guide at A297146.

Examples

			Base-7 digits of 4751: 1,6,5,6,5, with pattern UDUD, so that 4751 is in the sequence.

Links

Crossrefs

Cf. A297138, A297139.

Programs

  • Maple
    read("transforms") :
isA297137 := proc(n)
    local dgs,ud;
    dgs := convert(n,base,7) ;
    if nops(dgs) < 2 then
        return false;
    end if;
    ud := DIFF(dgs) ;
    if 0 in ud then
        return false;
    else
        simplify( op(-1,ud) < 0) ;
    end if;
end proc:
for n from 1 to 200 do
    if isA297137(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 18 2018
  • Mathematica
    a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
b = 7; t = Table[a[n, b], {n, 1, 10*z}];
u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297137 *)
v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297138 *)
Complement[Range[z], Union[u, v]]  (* A297139 *)

A297139 Numbers whose base-7 digits d(m), d(m-1),..., d(0) have m=0 or else d(i) = d(i+1) for some i in {0,1,...,m-1}.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 16, 24, 32, 40, 48, 49, 56, 57, 58, 59, 60, 61, 62, 65, 73, 81, 89, 97, 98, 106, 112, 113, 114, 115, 116, 117, 118, 122, 130, 138, 146, 147, 155, 163, 168, 169, 170, 171, 172, 173, 174, 179, 187, 195, 196, 204, 212, 220, 224, 225, 226
Offset: 1

Views

Author

Clark Kimberling, Jan 15 2018

Keywords

Comments

These numbers comprise the complement of the set of numbers in the union of A297137 and A297138.
Differs from A044818 first for 344 = 1001_7. 1001_7 has two runs of length 1 and is not in A044818, but it has a sequence of two consecutive zeros and is in this sequence. - R. J. Mathar, Jan 18 2018

Examples

			Base-7 digits of 4998: 2,0,4,0,0, so that 4998 is in the sequence.

Crossrefs

Cf. A297137, A297138.

Programs

  • Maple
    read("transforms") :
isA297139 := proc(n)
        local dgs,ud;
        dgs := convert(n,base,7) ;
        if nops(dgs) < 2 then
                return true;
        end if;
        if 0 in DIFF(dgs) then
                true;
        else
                false;
        end if;
end proc:
for n from 1 to 300 do
        if isA297139(n) then
                printf("%d,",n) ;
        end if;
end do: # R. J. Mathar, Jan 18 2018
  • Mathematica
    a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
b = 7; t = Table[a[n, b], {n, 1, 10*z}];
u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297137 *)
v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297138 *)
Complement[Range[z], Union[u, v]]  (* A297139 *)
Showing 1-3 of 3 results.

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