A030655 -id:A030655 - cp's OEIS frontend

A386014 Distinct values occurring in first differences of A030655(n), listed in order of first appearance.

22, 832, 202, 89302, 2002, 8993002, 20002, 899930002, 200002, 89999300002, 2000002, 8999993000002, 20000002, 899999930000002, 200000002, 89999999300000002, 2000000002, 8999999993000000002, 20000000002, 899999999930000000002, 200000000002, 89999999999300000000002, 2000000000002

Offset: 1

Carin Maria Sakin, Jul 14 2025

A030655(t) = 2*t-1||2*t = (2*t-1)*10^k + 2*t, where || indicates digit concatenation and 2*t has digit length k.
Difference A030655(t+1) - A030655(t) = 2*(10^k+1) when 2*t and 2*t+2 are both digit length k and this is the odd n case in the formulas below.
2*t and 2*t+2 differ in length only at t = j(k) = 5*10^(k-1)-1 with 2*t+2 then having length k+1, and this is the even n case in the formulas below.

			At t = j(3) = 499, difference A030655(t+1) - A030655(t) = 9991000 - 997998 = 8993002 which is term a(6).

Index entries for linear recurrences with constant coefficients, signature (1,110,-110,-1000,1000).

Cf. A007376, A030655, A057147, A061084.

a := proc(n)
    if type(n, even) then
        k := n/2;
        return 9*100^k - 7*10^k + 2;
    else
        k := (n + 1)/2;
        return 2*(10^k + 1);
    end if;
end proc:
seq(a(n), n=1..12);

a[n_] := Which[
  EvenQ[n], Module[{k = n/2}, 9*100^k - 7*10^k + 2],
  OddQ[n], Module[{k = (n + 1)/2}, 2*(10^k + 1)]
];
Table[a[n], {n, 1, 12}]

a(n) = if(n%2==1, 2*10^((n+1)/2)+2, 9*10^n - 7*10^(n/2) + 2) \\ David A. Corneth, Jul 17 2025

def a(n):
    k = (n + 1)//2
    if n % 2 == 0:
        return 9 * 100**k - 7 * 10**k + 2
    else:
        return 2 * (10**k + 1)

a(n) = 2*(10^k + 1) for odd n, where k = (n+1)/2.

a(n) = 9*100^k - 7*10^k + 2 for even n, where k = n/2.

More terms from Michel Marcus, Jul 16 2025

More terms from David A. Corneth, Jul 17 2025

A030656 Pair up the numbers.

Original entry on oeis.org

1, 23, 45, 67, 89, 1011, 1213, 1415, 1617, 1819, 2021, 2223, 2425, 2627, 2829, 3031, 3233, 3435, 3637, 3839, 4041, 4243, 4445, 4647, 4849, 5051, 5253, 5455, 5657, 5859, 6061, 6263, 6465, 6667, 6869, 7071, 7273, 7475, 7677, 7879, 8081, 8283, 8485, 8687, 8889

Offset: 0

Maghraoui Abdelkader

Michel Marcus, Table of n, a(n) for n = 0..5000

Cf. A030655.

Subsequence of A005408.

[Seqint(Intseq(n+1) cat Intseq(n)): n in [0..86 by 2]];  // Bruno Berselli, Jun 18 2011

seq[n_] := FromDigits[Flatten[IntegerDigits /@ #]] & /@ Partition[Range[0, (n + 1)*2], 2];
seq[44] (* Harvey P. Dale, Jan 24 2012, edited by Robert P. P. McKone, Jan 23 2021 *)

a(n)=eval(Str(2*n,2*n+1)) \\ Charles R Greathouse IV, Jul 21 2016

def a(n): return int(str(2*n)+str(2*n+1))
print([a(n) for n in range(45)]) # Michael S. Branicky, Jan 23 2021

More terms from Erich Friedman

A098080 Nontrivial slowest increasing sequence whose succession of digits is that of the nonnegative integers.

Original entry on oeis.org

0, 12, 34, 56, 78, 910, 1112, 1314, 1516, 1718, 1920, 2122, 2324, 2526, 2728, 2930, 3132, 3334, 3536, 3738, 3940, 4142, 4344, 4546, 4748, 4950, 5152, 5354, 5556, 5758, 5960, 6162, 6364, 6566, 6768, 6970, 7172, 7374, 7576, 7778, 7980, 8182, 8384, 8586, 8788, 8990, 9192, 9394, 9596, 9798, 99100, 101102

Offset: 0

Alexandre Wajnberg & Eric Angelini, Sep 13 2004

Beginning with 1, 23, 45, etc. gives a similar sequence which however grows more quickly. Other sequences can be generated by varying the "template" of the succession of digits (such as the decimal expansion of Pi, e, and so on).

1, 23, 45, 67, 89, 101, 112, 131, 415, 1617, 1819, 2021, 2223, ..., 9899, 10010, 110210, 310410, 510610, 710810, 911011, 1112113, ... does grow faster, but what about 1, 23, 45, 67, 89, 101, 112, 131, 415, 1617, ..., (2k)(2k+1), ...? The claim of "slowest" requires that after a(1), the smallest possible option must always be used (9899->10010 instead of 9899->100101). - Danny Rorabaugh, Nov 27 2015

Cf. A030655.

jd[{a_,b_}]:=Module[{ida=IntegerDigits[a],idb=IntegerDigits[b]}, FromDigits[ Join[ida,idb]]]; Join[{0},jd/@Partition[Range[110],2]] (* Harvey P. Dale, Feb 20 2012 *)

Write down the sequence of nonnegative integers and consider its succession of digits. Divide up into chunks of minimal length (and not beginning with 0) so that chunks are increasing numbers in order to form the slowest ever increasing sequence of slices (disregarding the number of digits) of the succession of the digits of the whole numbers.

A248556 Concatenate (3n-2,3n-1,3n).

Original entry on oeis.org

123, 456, 789, 101112, 131415, 161718, 192021, 222324, 252627, 282930, 313233, 343536, 373839, 404142, 434445, 464748, 495051, 525354, 555657, 585960, 616263, 646566, 676869, 707172, 737475, 767778, 798081, 828384, 858687, 888990, 919293, 949596, 979899

Offset: 1

Charles Duvall, Oct 08 2014

Cf. A030655, A091331.

[Seqint(Intseq(n+1) cat Intseq(n) cat Intseq(n-1)): n in [2..100 by 3]]; // Vincenzo Librandi, Oct 21 2014

conc3:= (a,b,c) -> c + 10^(1+ilog10(c))*(b+10^(1+ilog10(b))*a):
seq(conc3(3*k-2,3*k-1,3*k), k=1..100); # Robert Israel, Oct 23 2014

FromDigits[Flatten[IntegerDigits/@{#}]]&/@Partition[Range[100], 3] (* Vincenzo Librandi, Oct 21 2014 *)

a(n)=eval(Str(3*n-2,3*n-1,3*n)) \\ M. F. Hasler, Oct 22 2014

More terms from Vincenzo Librandi, Oct 21 2014

cp's OEIS Frontend

A386014 Distinct values occurring in first differences of A030655(n), listed in order of first appearance.

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A030656 Pair up the numbers.

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A098080 Nontrivial slowest increasing sequence whose succession of digits is that of the nonnegative integers.

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A248556 Concatenate (3n-2,3n-1,3n).

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Magma

Maple

Mathematica

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Extensions