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 1-10 of 14 results. Next

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 *)

A306111 Numbers with digits in {0,...,8} such that every other digit is strictly less than its neighbors.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 65, 70, 71, 72, 73, 74, 75, 76, 80, 81, 82, 83, 84, 85, 86, 87, 101, 102, 103, 104, 105, 106, 107, 108, 201, 202, 203, 204, 205, 206, 207, 208, 212, 213, 214, 215, 216, 217, 218, 301, 302, 303, 304, 305, 306, 307
Offset: 1

Views

Author

M. F. Hasler, Oct 05 2018

Keywords

Comments

Terms of A032864 written in base 9.

Examples

			There are 1+2+3+4+5+6+7+8 = 9*4 = 36 terms with 2 digits.
We obtain the 3-digit terms by appending to each of these the 1-digit terms starting with a digit larger than the last digit of the prefix: 10.{1..8}, 20.{1..8}, 21.{2..8}, 30.{1..8}, ..., 86.{7..8}, 87.{8}.
We obtain the 4-digit terms by appending to each of the 2 digit terms, the 2-digit terms starting with a digit larger than the last digit of the prefix: 10.{10,...,87}, 20.{10,...,87}, 21.{20,...,87}, 30.{10,...,87}, ..., 86.{70,...,87}, 87.{80..87}.
That way we obtain all terms with n digits by taking the 2-digit terms and appending to each of these the suitable subsequence of n-2 digit terms.

Crossrefs

Cf. A306105 .. A306110 and A297147: analog for bases 3..8 and 10.
Cf. A032864 and A032858 .. A032865 for other bases 3..10.

Programs

  • PARI
    A(Nmax=100,K=8,A=[0..K],i=vector(2*K,i,max(1,i-K+1)),c(T,v)=apply(t->t+T,v))={for(n=0,oo, for(k=10,K*11-1,if(k%10

Formula

a(n) = A007095(A032864(n)).
Numbers in A297147 having no digit 9: Intersection of A297147 with A007095.

A306105 Numbers with digits in {0,1,2} such that every other digit is strictly less than its neighbors.

Original entry on oeis.org

0, 1, 2, 10, 20, 21, 101, 102, 201, 202, 212, 1010, 1020, 1021, 2010, 2020, 2021, 2120, 2121, 10101, 10102, 10201, 10202, 10212, 20101, 20102, 20201, 20202, 20212, 21201, 21202, 21212, 101010, 101020, 101021, 102010, 102020, 102021, 102120, 102121
Offset: 1

Views

Author

M. F. Hasler, Oct 05 2018

Keywords

Comments

Terms of A032858 written in base 3.
There are A000045(n+2) terms with n digits (where 0 is taken to have no digits), so the first term with n digits is at index A000071(n+3). See A032858 for the proof.

Crossrefs

Cf. A306106 .. A306111 and A297147: analog for bases 3..9 and 10.
Cf. A000045 (Fibonacci), A000071(n) = Sum(k=0..n-2,A45(k)) = A000045(n)-1.
Cf. A032858 and A032859 .. A032865 for other bases 3..10.

Programs

  • PARI
    {A=[0,1,2]; F=[1,1]; for(n=0,4, F=[F[2],vecsum(F)]; for(k=1,3, T=max(k*10,21)*10^n; A=concat(A,apply(t->t+T,A[F[2]-1+if(k>2,F*[2,-1]~)..vecsum(F)-2]))));A}

Formula

a(n) = A007089(A032858(n)).

A306106 Numbers with digits in {0,1,2,3} such that every other digit is strictly less than its neighbors.

Original entry on oeis.org

0, 1, 2, 3, 10, 20, 21, 30, 31, 32, 101, 102, 103, 201, 202, 203, 212, 213, 301, 302, 303, 312, 313, 323, 1010, 1020, 1021, 1030, 1031, 1032, 2010, 2020, 2021, 2030, 2031, 2032, 2120, 2121, 2130, 2131, 2132, 3010, 3020, 3021, 3030, 3031, 3032, 3120, 3121, 3130, 3131, 3132, 3230, 3231, 3232, 10101, 10102, 10103
Offset: 1

Views

Author

M. F. Hasler, Oct 05 2018

Keywords

Comments

Terms of A032859 written in base 4.

Crossrefs

Cf. A306105 .. A306111 and A297147: analog for bases 3..9 and 10.
Cf. A032859 and A032858 .. A032865 for other bases 3..10.

Programs

  • PARI
    A(Nmax=100, K=3, A=[0..K], i=vector(2*K, i, max(1, i-K+1)), c(T, v)=apply(t->t+T, v))={for(n=0, oo, for(k=10, K*11, if(k%10

Formula

a(n) = A007090(A032859(n)).
Terms in A297147 having only digits < 4; intersection of A297147 and A007090.

A032859 Numbers whose base-4 representation Sum_{i=0..m} d(i)*4^i has d(m) > d(m-1) < d(m-2) > ...

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 9, 12, 13, 14, 17, 18, 19, 33, 34, 35, 38, 39, 49, 50, 51, 54, 55, 59, 68, 72, 73, 76, 77, 78, 132, 136, 137, 140, 141, 142, 152, 153, 156, 157, 158, 196, 200, 201, 204, 205, 206, 216, 217, 220, 221, 222, 236, 237, 238
Offset: 1

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A007090, A306106.
Cf. A032858..A032865 for bases 3..10.
Cf. A306106..A306111 and A297147 for bases 3..9 and 10.

Programs

  • Mathematica
    Join[{0},Select[Range[250],(Sign/@Differences[IntegerDigits[#,4]]) == PadRight[ {},IntegerLength[#,4]-1,{-1,1}]&]] (* Harvey P. Dale, Sep 18 2022 *)

Extensions

a(1)=0 inserted by Georg Fischer, Dec 18 2020

A032864 Numbers whose base-9 representation Sum_{i=0..m} d(i)*9^i has d(m) > d(m-1) < d(m-2) > ...

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 18, 19, 27, 28, 29, 36, 37, 38, 39, 45, 46, 47, 48, 49, 54, 55, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 163, 164, 165, 166, 167, 168, 169, 170, 173
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A007095.
Different from A032888.
Cf. A032858..A032865 for bases 3..10.
Cf. A306106..A306111 and A297147 for bases 3..9 and 10.

Extensions

a(1)=0 inserted by Georg Fischer, Dec 18 2020

A032860 Numbers whose base-5 representation Sum_{i=0..m} d(i)*5^i has d(m) > d(m-1) < d(m-2) > ...

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 11, 15, 16, 17, 20, 21, 22, 23, 26, 27, 28, 29, 51, 52, 53, 54, 57, 58, 59, 76, 77, 78, 79, 82, 83, 84, 88, 89, 101, 102, 103, 104, 107, 108, 109, 113, 114, 119, 130, 135, 136, 140, 141, 142, 145, 146, 147, 148, 255
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A007091.
Cf. A032858..A032865 for bases 3..10.
Cf. A306106..A306111 and A297147 for bases 3..9 and 10.

Extensions

a(1)=0 inserted by Georg Fischer, Dec 18 2020

A032862 Numbers whose base-7 representation Sum_{i=0..m} d(i)*7^i has d(m) > d(m-1) < d(m-2) > ...

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 14, 15, 21, 22, 23, 28, 29, 30, 31, 35, 36, 37, 38, 39, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 99, 100, 101, 102, 103, 104, 107, 108, 109, 110, 111, 148, 149, 150, 151, 152, 153, 156, 157, 158, 159, 160, 164
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A007093.
Cf. A032858..A032865 for bases 3..10.
Cf. A306106..A306111 and A297147 for bases 3..9 and 10.

Extensions

a(1)=0 inserted by Georg Fischer, Dec 18 2020

A297148 Numbers whose base-10 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, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 100, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 133, 144, 155, 166, 177, 188, 199, 200, 211, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 233, 244, 255, 266, 277, 288, 299, 300
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 A297146 and A297147.
Differs from A044821 first for 1001, which is in this sequence but not in A044821. - R. J. Mathar, Jan 17 2018

Examples

			Base-10 digits of 65536: 6,5,5,3,6, so that 65536 is in the sequence.

Crossrefs

Cf. A297146, A297147.

Programs

  • Maple
    read("transforms") :
isA297148 := proc(n)
    local dgs,ud;
    dgs := convert(n,base,10) ;
    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 isA297148(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 = 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 *)

A032861 Numbers whose base-6 representation Sum_{i=0..m} d(i)*6^i has d(m) > d(m-1) < d(m-2) > ...

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 12, 13, 18, 19, 20, 24, 25, 26, 27, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 73, 74, 75, 76, 77, 80, 81, 82, 83, 109, 110, 111, 112, 113, 116, 117, 118, 119, 123, 124, 125, 145, 146, 147, 148, 149, 152, 153, 154, 155
Offset: 1

Views

Author

Clark Kimberling

Keywords

Crossrefs

Cf. A007092.
Cf. A032858..A032865 for bases 3..10.
Cf. A306106..A306111 and A297147 for bases 3..9 and 10.

Programs

  • Mathematica
    sdQ[n_]:=Module[{s=Sign[Differences[IntegerDigits[n,6]]]},s==PadRight[{}, Length[ s],{-1,1}]]; Select[Range[0,200],sdQ] (* Harvey P. Dale, Dec 15 2017 *)

Extensions

a(1)=0 inserted by Georg Fischer, Dec 18 2020
Showing 1-10 of 14 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.