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 1-7 of 7 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

Views

Author

Clark Kimberling

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, ...

Links

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

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.

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.

A306110 Numbers with digits in {0,...,7} such that every other digit is strictly less than its neighbors.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 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, 101, 102, 103, 104, 105, 106, 107, 201, 202, 203, 204, 205, 206, 207, 212, 213, 214, 215, 216, 217, 301, 302, 303, 304, 305, 306, 307, 312, 313, 314, 315, 316, 317, 323, 324, 325
Offset: 1

Views

Author

M. F. Hasler, Oct 05 2018

Keywords

Comments

Terms of A032863 written in base 8.

Crossrefs

Cf. A306105 .. A306111 and A297147: analog for bases 3..9 and 10.
Cf. A032863 and A032858 .. A032865 for other bases 3..10.

Programs

  • PARI
    A(Nmax=100,K=7,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) = A007094(A032863(n)).

A306107 Numbers with digits in {0,1,2,3,4} such that every other digit is strictly less than its neighbors.

Original entry on oeis.org

0, 1, 2, 3, 4, 10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 101, 102, 103, 104, 201, 202, 203, 204, 212, 213, 214, 301, 302, 303, 304, 312, 313, 314, 323, 324, 401, 402, 403, 404, 412, 413, 414, 423, 424, 434, 1010, 1020, 1021, 1030, 1031, 1032, 1040, 1041, 1042, 1043, 2010, 2020, 2021, 2030, 2031, 2032, 2040, 2041
Offset: 1

Views

Author

M. F. Hasler, Oct 05 2018

Keywords

Comments

Terms of A032860 written in base 5.

Crossrefs

Cf. A306105 .. A306111 and A297147: analog for bases 3..9 and 10.
Cf. A032860 and A032858 .. A032865 for other bases 3..10.

Programs

  • PARI
    A(Nmax=100, K=4, 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) = A007091(A032860(n)).

A306108 Numbers with digits in {0,...,5} such that every other digit is strictly less than its neighbors.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, 52, 53, 54, 101, 102, 103, 104, 105, 201, 202, 203, 204, 205, 212, 213, 214, 215, 301, 302, 303, 304, 305, 312, 313, 314, 315, 323, 324, 325, 401, 402, 403, 404, 405, 412, 413, 414, 415, 423, 424, 425, 434, 435, 501, 502, 503, 504, 505, 512, 513, 514
Offset: 1

Views

Author

M. F. Hasler, Oct 05 2018

Keywords

Comments

Terms of A032861 written in base 6.

Crossrefs

Cf. A306105 .. A306111 and A297147: analog for bases 3..9 and 10.
Cf. A032862 and A032858 .. A032865 for other bases 3..10.

Programs

  • PARI
    A(Nmax=100, K=5, 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) = A007092(A032861(n)).

A306109 Numbers with digits in {0,...,6} such that every other digit is strictly less than its neighbors.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 65, 101, 102, 103, 104, 105, 106, 201, 202, 203, 204, 205, 206, 212, 213, 214, 215, 216, 301, 302, 303, 304, 305, 306, 312, 313, 314, 315, 316, 323, 324, 325, 326, 401, 402, 403, 404, 405, 406, 412, 413, 414, 415
Offset: 1

Views

Author

M. F. Hasler, Oct 05 2018

Keywords

Comments

Terms of A032862 written in base 7.

Crossrefs

Cf. A306105 .. A306111 and A297147: analog for bases 3..9 and 10.
Cf. A032863 and A032858 .. A032865 for other bases 3..10.

Programs

  • PARI
    A(Nmax=100, K=6, 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) = A007093(A032862(n)).
Showing 1-7 of 7 results.

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