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 1-9 of 9 results.

A299957 The sum a(n) + a(n+1) always has at least one digit "1". Lexicographically first such sequence of nonnegative integers without duplicate term.

Original entry on oeis.org

0, 1, 9, 2, 8, 3, 7, 4, 6, 5, 10, 11, 20, 21, 30, 31, 40, 41, 50, 51, 49, 12, 19, 22, 29, 32, 39, 42, 58, 13, 18, 23, 28, 33, 38, 43, 48, 52, 53, 47, 14, 17, 24, 27, 34, 37, 44, 56, 15, 16, 25, 26, 35, 36, 45, 46, 54, 55, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80
Offset: 0

Views

Author

Eric Angelini, Feb 22 2018

Keywords

Comments

The sequence starts with a(0) = 0 and is always extended with the smallest integer not yet present that does not lead to a contradiction. The sequence is a permutation of the natural numbers.
Originally the sequence was defined starting with a(1) = 1 and using only positive integers. This leads to the same sequence restricted to positive indices, which yields a permutation of the positive integers. - M. F. Hasler, Feb 28 2018

Examples

			1 + 9 = 10; 9 + 2 = 11; 2 + 8 = 10; 8 + 3 = 11; 3 + 7 = 10; 7 + 4 = 11; 4 + 6 = 10; 6 + 5 = 11; etc.

Links

Crossrefs

Cf. A299952 (different constraint: a(n) + a(n+1) must be substring of concatenation of a(1..n+1)).
Cf. A299970, A299982, ..., A299988, A299969 (nonnegative analog with digit 0, 2, ..., 9), A299971, A299972, ..., A299979 (positive analog with digit 0, 2, ..., 9).
Cf. A299980, A299981, A299402, A299403, A298974, A298975, A299996, A299997, A298978, A298979 for the analog using multiplication: a(n)*a(n+1) has a digit 0, resp. 1, ..., resp. 9.

Programs

  • Mathematica
    Nest[Append[#, Block[{k = 1}, While[Nand[FreeQ[#, k], DigitCount[k + #[[-1]], 10, 1] > 0], k++]; k]] &, {1}, 98] (* Michael De Vlieger, Feb 22 2018 *)
  • PARI
    a(n, f=1, a=0, u=[a])={for(n=a+1, n, f&&if(f==1,print1(a","),write(f,n-1," "a)); for(k=u[1]+1, oo, setsearch(u, k)&&next; setsearch(Set(digits(a+k)),1)&&(a=k)&&break); u=setunion(u, [a]); u[2]==u[1]+1&&u=u[^1]); a} \\ M. F. Hasler, Feb 22 2018

Extensions

Extended to a(0) = 0 by M. F. Hasler, Feb 28 2018

A299969 Lexicographic first sequence of nonnegative integers such that a(n) + a(n+1) has a digit 9, and no term occurs twice.

Original entry on oeis.org

0, 9, 10, 19, 20, 29, 30, 39, 40, 49, 41, 8, 1, 18, 11, 28, 21, 38, 31, 48, 42, 7, 2, 17, 12, 27, 22, 37, 32, 47, 43, 6, 3, 16, 13, 26, 23, 36, 33, 46, 44, 5, 4, 15, 14, 25, 24, 35, 34, 45, 50, 59, 60, 69, 70, 79, 80, 89, 90, 99, 91, 58, 51, 68, 61, 78, 71, 88, 81, 98, 92, 57, 52, 67, 62, 77, 72, 87, 82, 97, 93, 56, 53, 66, 63, 76, 73, 86, 83, 96, 94
Offset: 0

Views

Author

M. F. Hasler and Eric Angelini, Feb 22 2018

Keywords

Comments

A permutation of the nonnegative integers.
It happens that from a(50) = 50 on, this sequence coincides with the variant A299979 (starting at 1 and having only positive terms). Indeed the two sequences have the property that the terms a(0..49) resp. A299979(1..49) exactly contain all numbers from 0 to 49, respectively 1 to 49. - M. F. Hasler, Feb 28 2018

Links

Crossrefs

Cf. A299979 (analog with positive terms), A299957 (analog with digit 1), A299970, A299982, ..., A299988 (digit 0, 2, ..., 8).
Cf. A299980, A299981, A299402, A299403, A298974, A298975, A299996, A299997, A298978, A298979 for the analog using multiplication: a(n)*a(n+1) has a digit 0, resp. 1, ..., resp. 9.

Programs

  • Mathematica
    Nest[Append[#, Block[{k = 1}, While[Nand[FreeQ[#, k], DigitCount[#[[-1]] + k, 10, 9] > 0], k++]; k]] &, {0}, 90] (* Michael De Vlieger, Mar 01 2018 *)
  • PARI
    a(n,f=1,d=9,a=0,u=[a])={for(n=1,n,f&&if(f==1,print1(a","),write(f,n-1," "a));for(k=u[1]+1,oo,setsearch(u,k)&&next;setsearch(Set(digits(a+k)),d)&&(a=k)&&break);u=setunion(u,[a]);u[2]==u[1]+1&&u=u[^1]);a}

A299988 Lexicographic first sequence of nonnegative integers such that a(n) + a(n+1) has a digit 8, and no term occurs twice.

Original entry on oeis.org

0, 8, 10, 18, 20, 28, 30, 38, 40, 41, 7, 1, 17, 11, 27, 21, 37, 31, 47, 33, 5, 3, 15, 13, 25, 23, 35, 43, 39, 9, 19, 29, 49, 32, 6, 2, 16, 12, 26, 22, 36, 42, 44, 4, 14, 24, 34, 46, 52, 56, 62, 66, 72, 76, 82, 86, 92, 88, 50, 48, 60, 58, 70, 68, 80, 78, 90, 91, 57, 51, 67, 61, 77, 71, 87, 81, 97, 83, 45, 53, 55, 63, 65, 73, 75, 93, 85, 95, 89, 59, 69
Offset: 0

Views

Author

M. F. Hasler and Eric Angelini, Feb 22 2018

Keywords

Comments

A permutation of the nonnegative integers.

Crossrefs

Cf. A299978 (analog with positive terms), A299957 (analog with digit 1), A299970, A299982..A299987, A299969 (digit 0, 2..7, 9).

Programs

  • PARI
    a(n,f=1,d=8,a=0,u=[a])={for(n=1,n,f&&if(f==1,print1(a","),write(f,n-1," "a));for(k=u[1]+1,oo,setsearch(u,k)&&next;setsearch(Set(digits(a+k)),d)&&(a=k)&&break);u=setunion(u,[a]);u[2]==u[1]+1&&u=u[^1]);a}

A299952 The sum a(n) + a(n+1) is a substring of the concatenation of all terms up to a(n+1). Lexicographic first sequence of positive integers without duplicate terms having this property.

Original entry on oeis.org

1, 10, 99, 11, 80, 19, 61, 30, 31, 49, 12, 2, 4, 5, 3, 6, 7, 15, 9, 13, 17, 14, 8, 16, 20, 25, 23, 22, 26, 27, 18, 34, 28, 21, 24, 29, 32, 35, 36, 44, 37, 43, 38, 33, 41, 39, 42, 40, 51, 45, 46, 47, 52, 57, 53, 56, 54, 55, 63, 59, 50, 60, 58, 64, 66, 65, 74, 48, 62, 68, 71, 77, 72, 67, 78, 70
Offset: 1

Views

Author

Eric Angelini and Lars Blomberg, Feb 22 2018

Keywords

Comments

The sequence starts with a(1) = 1 and is always extended with the smallest integer not yet present that does not lead to a contradiction.
This is probably a permutation of the natural numbers (after 10000 terms, the smallest integer not yet present is 9990).

Examples

			a(1) + a(2) = 1 + 10 = 11 and “11” is visible in [1,10]
a(2) + a(3) = 10 + 99 = 109 and “109” is visible in [10,99]
a(3) + a(4) = 99 + 11 = 110 and “110” is visible in [1,10]
a(4) + a(5) = 11 + 80 = 91 and “91” is visible in [99,11]
a(5) + a(6) = 80 + 19 = 99 and “99” is visible in [99]
a(6) + a(7) = 19 + 61 = 80 and “80” is visible in [80]
...

Links

Crossrefs

Cf. A300000.
For a different constraint on a(n)+a(n+1) (must have a digit '1'), see A299957 and A299970, A299982, ..., A299988, A299969 (nonnegative analog with digit 0, 2, ..., 9), A299971, A299972, ..., A299979 (positive analog with digit 0, 2, ..., 9).
Cf. A299980, A299981, A299402, A299403, A298974, A298975, A299996, A299997, A298978, A298979 for the analog using multiplication: a(n)*a(n+1) has a digit 0, resp. 1, ..., resp. 9.

Programs

  • Mathematica
    Nest[Function[a, Append[a, Block[{k = 1, d}, While[Nand[FreeQ[a, k], SequenceCount[Flatten@ IntegerDigits[Append[a, k]], IntegerDigits[a[[-1]] + k]] > 0], k++]; k]]], {1}, 75] (* Michael De Vlieger, Feb 22 2018 *)
  • PARI
    a(n,show=1,a=1,s=a,u=[a],t,m)={for(n=2,n, show&&print1(a","); for(k=u[1]+1,oo, setsearch(u,k)&&next;m=Mod(a+k,10^#Str(a+k));t=s*10^#Str(k)+k; until(k>=t\=10,t==m&&(a=k)&&break(2)));s=s*10^#Str(a)+a;u=setunion(u,[a]); u[2]==u[1]+1&&u=u[^1]);a} \\ M. F. Hasler, Feb 22 2018

A299983 Lexicographic first sequence of nonnegative integers such that a(n) + a(n+1) has a digit 3, and no term occurs twice.

Original entry on oeis.org

0, 3, 10, 13, 17, 6, 7, 16, 14, 9, 4, 19, 11, 2, 1, 12, 18, 5, 8, 15, 20, 23, 30, 33, 40, 43, 50, 53, 60, 63, 67, 26, 27, 36, 37, 46, 47, 56, 57, 66, 64, 29, 24, 39, 34, 49, 44, 59, 54, 69, 61, 22, 21, 32, 31, 42, 41, 52, 51, 62, 68, 25, 28, 35, 38, 45, 48, 55, 58, 65, 70, 73, 80, 83, 90, 93, 100, 103, 110, 113, 117, 76, 77, 86, 87, 96, 97, 106, 107
Offset: 0

Views

Author

M. F. Hasler and Eric Angelini, Feb 22 2018

Keywords

Comments

A permutation of the nonnegative integers.

Crossrefs

Cf. A299973 (analog with positive terms), A299957 (analog with digit 1), A299970, A299982, ..., A299988, A299969 (digit 0, 2, ..., 9).

Programs

  • PARI
    a(n,f=1,d=3,a=0,u=[a])={for(n=1,n,f&&if(f==1,print1(a","),write(f,n-1," "a));for(k=u[1]+1,oo,setsearch(u,k)&&next;setsearch(Set(digits(a+k)),d)&&(a=k)&&break);u=setunion(u,[a]);u[2]==u[1]+1&&u=u[^1]);a}

A299987 Lexicographic first sequence of nonnegative integers such that a(n) + a(n+1) has a digit 7, and no term occurs twice.

Original entry on oeis.org

0, 7, 10, 17, 20, 27, 30, 37, 33, 4, 3, 14, 13, 24, 23, 34, 36, 1, 6, 11, 16, 21, 26, 31, 39, 8, 9, 18, 19, 28, 29, 38, 32, 5, 2, 15, 12, 25, 22, 35, 40, 47, 50, 57, 60, 67, 70, 77, 80, 87, 83, 44, 43, 54, 53, 64, 63, 74, 73, 84, 86, 41, 46, 51, 56, 61, 66, 71, 76, 81, 89, 48, 49, 58, 59, 68, 69, 78, 79, 88, 82, 45, 42, 55, 52, 65, 62, 75, 72, 85, 90
Offset: 0

Views

Author

M. F. Hasler and Eric Angelini, Feb 22 2018

Keywords

Comments

A permutation of the nonnegative integers.

Crossrefs

Cf. A299977 (analog with positive terms), A299957 (analog with digit 1), A299970, A299982..A299988, A299969 (digit 0, 2..8, 9).

Programs

  • PARI
    a(n,f=1,d=7,a=0,u=[a])={for(n=1,n,f&&if(f==1,print1(a","),write(f,n-1," "a));for(k=u[1]+1,oo,setsearch(u,k)&&next;setsearch(Set(digits(a+k)),d)&&(a=k)&&break);u=setunion(u,[a]);u[2]==u[1]+1&&u=u[^1]);a}

A299984 Lexicographic first sequence of nonnegative integers such that a(n) + a(n+1) has a digit 4, and no term occurs twice.

Original entry on oeis.org

0, 4, 10, 14, 20, 21, 3, 1, 13, 11, 23, 17, 7, 27, 15, 9, 5, 19, 22, 2, 12, 28, 6, 8, 16, 18, 24, 25, 29, 35, 39, 45, 49, 55, 59, 65, 69, 71, 33, 31, 43, 41, 53, 51, 63, 61, 73, 67, 37, 47, 57, 77, 64, 30, 34, 40, 44, 50, 54, 60, 74, 66, 38, 26, 48, 36, 58, 46, 68, 56, 78, 62, 32, 42, 52, 72, 70, 75, 79, 85, 89, 95, 99, 105, 109, 115, 119, 121, 83
Offset: 0

Views

Author

M. F. Hasler and Eric Angelini, Feb 22 2018

Keywords

Comments

A permutation of the nonnegative integers.

Crossrefs

Cf. A299974 (analog with positive terms), A299957 (analog with digit 1), A299970, A299982, ..., A299988, A299969 (digit 0, 2, ..., 8, 9).

Programs

  • PARI
    a(n,f=1,d=4,a=0,u=[a])={for(n=1,n,f&&if(f==1,print1(a","),write(f,n-1," "a));for(k=u[1]+1,oo,setsearch(u,k)&&next;setsearch(Set(digits(a+k)),d)&&(a=k)&&break);u=setunion(u,[a]);u[2]==u[1]+1&&u=u[^1]);a}

A299985 Lexicographic first sequence of nonnegative integers such that a(n) + a(n+1) has a digit 5, and no term occurs twice.

Original entry on oeis.org

0, 5, 10, 15, 20, 25, 26, 9, 6, 19, 16, 29, 21, 4, 1, 14, 11, 24, 27, 8, 7, 18, 17, 28, 22, 3, 2, 13, 12, 23, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 76, 39, 36, 49, 46, 59, 56, 69, 66, 79, 71, 34, 31, 44, 41, 54, 51, 64, 61, 74, 77, 38, 37, 48, 47, 58, 57, 68, 67, 78, 72, 33, 32, 43, 42, 53, 52, 63, 62, 73, 80, 85, 90, 95, 100, 105, 110, 115, 120
Offset: 0

Views

Author

M. F. Hasler and Eric Angelini, Feb 22 2018

Keywords

Comments

A permutation of the nonnegative integers.

Crossrefs

Cf. A299975 (analog with positive terms), A299957 (analog with digit 1), A299970, A299982..A299988, A299969 (digit 0, 2..8, 9).

Programs

  • PARI
    a(n,f=1,d=5,a=0,u=[a])={for(n=1,n,f&&if(f==1,print1(a","),write(f,n-1," "a));for(k=u[1]+1,oo,setsearch(u,k)&&next;setsearch(Set(digits(a+k)),d)&&(a=k)&&break);u=setunion(u,[a]);u[2]==u[1]+1&&u=u[^1]);a}

A299986 Lexicographic first sequence of nonnegative integers such that a(n) + a(n+1) has a digit 6, and no term occurs twice.

Original entry on oeis.org

0, 6, 10, 16, 20, 26, 30, 31, 5, 1, 15, 11, 25, 21, 35, 27, 9, 7, 19, 17, 29, 32, 4, 2, 14, 12, 24, 22, 34, 28, 8, 18, 38, 23, 3, 13, 33, 36, 40, 46, 50, 56, 60, 66, 70, 76, 80, 81, 45, 41, 55, 51, 65, 61, 75, 71, 85, 77, 39, 37, 49, 47, 59, 57, 69, 67, 79, 82, 44, 42, 54, 52, 64, 62, 74, 72, 84, 78, 48, 58, 68, 88, 73, 43, 53, 63, 83, 86, 90, 96, 100
Offset: 0

Views

Author

M. F. Hasler and Eric Angelini, Feb 22 2018

Keywords

Comments

A permutation of the nonnegative integers.

Crossrefs

Cf. A299976 (analog with positive terms), A299957 (analog with digit 1), A299970, A299982..A299988, A299969 (digit 0, 2..8, 9).

Programs

  • PARI
    a(n,f=1,d=6,a=0,u=[a])={for(n=1,n,f&&if(f==1,print1(a","),write(f,n-1," "a));for(k=u[1]+1,oo,setsearch(u,k)&&next;setsearch(Set(digits(a+k)),d)&&(a=k)&&break);u=setunion(u,[a]);u[2]==u[1]+1&&u=u[^1]);a}
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