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 1-4 of 4 results.

A032858 Numbers whose base-3 representation Sum_{i=0..m} d(i)*3^i has d(m) > d(m-1) < d(m-2) > ...

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 10, 11, 19, 20, 23, 30, 33, 34, 57, 60, 61, 69, 70, 91, 92, 100, 101, 104, 172, 173, 181, 182, 185, 208, 209, 212, 273, 276, 277, 300, 303, 304, 312, 313, 516, 519, 520, 543, 546, 547, 555, 556, 624, 627, 628, 636, 637
Offset: 1

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Author

Keywords

Comments

Every other base-3 digit must be strictly less than its neighbors. - M. F. Hasler, Oct 05 2018
The terms can be generated in the following way: if A(n) are the terms with n digits in base 3, the terms with n+2 digits are obtained by prefixing them with '10' and with '20', and prefixing '21' to those starting with a digit '2'. It is easy to prove that #A(n) = A000045(n+2), since from the above we have #A(n+2) = 2*#A(n) + #A(n-1) = #A(n) + #A(n+1). (The #A(n-1) numbers starting with '2' are #A(n-2) numbers prefixed with '20' and #A(n-3) prefixed with '21'.) - M. F. Hasler, Oct 05 2018

Examples

			The base-3 representation of the initial terms is 0, 1, 2, 10, 20, 21, 101, 102, 201, 202, 212, 1010, 1020, 1021, 2010, 2020, 2021, 2120, 2121, 10101, 10102, ...
		

Crossrefs

Cf. A032859 .. A032865 for base-4 .. 10 variants.
Cf. A000975 (or A056830 in binary) for the base-2 analog.
Cf. A306105 for these terms written in base 3.

Programs

  • Mathematica
    sdQ[n_]:=Module[{s=Sign[Differences[IntegerDigits[n, 3]]]}, s==PadRight[{}, Length[s], {-1, 1}]]; Select[Range[0, 700], sdQ] (* Vincenzo Librandi, Oct 06 2018 *)
  • PARI
    is(n,b=3)=!for(i=2,#n=digits(n,b),(n[i-1]-n[i])*(-1)^i>0||return) \\ M. F. Hasler, Oct 05 2018

Formula

a(A000071(n+3)) = floor(3^(n+1)/8) = A033113(n). - M. F. Hasler, Oct 05 2018

Extensions

Definition edited, cross-references and a(1) = 0 inserted by M. F. Hasler, Oct 05 2018

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.

A032863 Numbers whose base-8 representation Sum_{i=0..m} d(i)*8^i has d(m) > d(m-1) < d(m-2) > ...

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 16, 17, 24, 25, 26, 32, 33, 34, 35, 40, 41, 42, 43, 44, 48, 49, 50, 51, 52, 53, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 129, 130, 131, 132, 133, 134, 135, 138, 139, 140, 141, 142, 143, 193, 194, 195
Offset: 1

Views

Author

Keywords

Comments

Base-8 digits must be strictly alternating in size: every other digit must be strictly less than its neighbor(s). Also: numbers whose base-8 expansion, considered as a decimal number, is in A032865 = the base-10 variant of this sequence. - M. F. Hasler, Oct 05 2018

Examples

			From _M. F. Hasler_, Oct 05 2018: (Start)
The base-8 representation of 7, 8, 16, 17, 24, 25, 26, 32, 33 is 7, 10, 20, 21, 30, 31, 32, 40, 41.
Numbers 61, 62, 65, 66, ..., 70, 71, 129, 130, ... have the base-8 expansion 76, 77, 101, 102, ..., 106, 107, 201, 202, ... (End)
		

Crossrefs

Cf. A032858, A032859, A032860, A032861, A032862, this sequence, A032864, A032865 for bases 3 to 10.

Programs

  • Mathematica
     sdQ[n_]:=Module[{s=Sign[Differences[IntegerDigits[n, 8]]]}, s==PadRight[{}, Length[s], {-1, 1}]]; Select[Range[0, 700], sdQ] (* Vincenzo Librandi, Oct 06 2018 *)
  • PARI
    is(n)=!for(i=2,#n=digits(n,8),(n[i-1]-n[i])*(-1)^i>0||return) \\ M. F. Hasler, Oct 05 2018

Extensions

a(1) = 0 added by Vincenzo Librandi, Oct 06 2018
Showing 1-4 of 4 results.