A132131 -id:A132131 - cp's OEIS frontend

A132133 Number of n-digit "Punctual Birds" (cf. A131881).

1, 9, 45, 270, 2104, 16941, 142245, 1226146

Offset: 0

Graeme McRae, Aug 11 2007

The number of n-digit "Early Birds" is 9*10^(n-1) - A132133(n), which is 0, 45, 630, 6896, 73059, 757755, 7773854, ... for n = 1, 2, ...

Here a(0) = 1 corresponds to the number 0 which is punctual in the sense that it does not occur before position 0, and which may be considered to have 0 digits for convenience: The index of 10^n in A131881 is then Sum_{k=0..n} a(k). - M. F. Hasler, Oct 25 2019

			a(2) = 45 because there are 45 2-digit Punctual Birds (10, 11, 13-20, 22, 24-30, 33, 35-40, 44, 46-50, 55, 57-60, 66, 68-70, 77, 79, 80, 88 and 90)

Cf. A033307, A117804, A116700, A131881, A132131, A132132.

Extended to a(0) = 1 by M. F. Hasler, Oct 25 2019

A083655 Numbers which do not appear prematurely in the binary Champernowne word (A030190).

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 16, 32, 36, 64, 128, 136, 256, 512, 528, 1024, 2048, 2080, 4096, 8192, 8256, 16384, 32768, 32896, 65536, 131072, 131328, 262144, 524288, 524800, 1048576, 2097152, 2098176, 4194304, 8388608, 8390656, 16777216, 33554432

Offset: 0

Reinhard Zumkeller, May 01 2003

In other words, numbers k whose binary expansion first appears in A030190 at its expected place, i.e., n appears first starting at position A296349(n). - N. J. A. Sloane, Dec 17 2017

a(n) are the Base 2 "Punctual Bird" numbers: write the nonnegative integers, base 2, in a string 011011100101110111.... Sequence gives numbers which do not occur in the string ahead of their natural place. - Graeme McRae, Aug 11 2007

Graeme McRae, Aug 11 2007, Table of n, a(n) for n = 0..113 - Corrected by _Rémy Sigrist_, Jun 14 2020
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-8).

A000079 is a subsequence.
Cf. A030190, A030304, A007088, A131881, A132131.
For the complement, the "Early Bird" numbers, see A296365.

LinearRecurrence[{0,0,6,0,0,-8},{0,1,2,4,8,10,16,32,36},50] (* Harvey P. Dale, Aug 19 2020 *)

a(n)= if (n<=2, n, my (m=n\3); if (n%3==0, 2^(2*m), n%3==1, 2^(2*m+1), 2^m + 2^(2*m+1)))  \\ Rémy Sigrist, Jun 14 2020

concat(0, Vec(x*(1 + 2*x + 4*x^2 + 2*x^3 - 2*x^4 - 8*x^5 - 8*x^6 - 8*x^7) / ((1 - 2*x^3)*(1 - 4*x^3)) + O(x^40))) \\ Colin Barker, Jun 14 2020

a(n) = 2^((2*n+1)\3) + (n%3==2)<<(n\3) - (n<3) \\ Charles R Greathouse IV, Dec 16 2022

A083653(a(n))=a(n), A083654(a(n))=1.
a(0)=0, a(1)=1, a(2)=2; then for n>=1, a(3n)=2^(2n), a(3n+1)=2^(2n+1), a(3n+2)=2^(2n+1)+2^n. - Graeme McRae, Aug 11 2007
From Colin Barker, Jun 14 2020: (Start)
G.f.: x*(1 + 2*x + 4*x^2 + 2*x^3 - 2*x^4 - 8*x^5 - 8*x^6 - 8*x^7) / ((1 - 2*x^3)*(1 - 4*x^3)).
a(n) = 6*a(n-3) - 8*a(n-6) for n>8. (End)
a(n) = 2^floor(2*(n+2)/3-1) + (floor((n+1)/3)-floor(n/3))*2^(floor(n/3)) - floor(5/(n+3)). - Alan Michael Gómez Calderón, Dec 15 2022

More terms from Graeme McRae, Aug 11 2007

A132132 "Early Bird" numbers n such that n-1 is not an Early Bird (cf. A116700).

Original entry on oeis.org

12, 21, 23, 31, 34, 41, 45, 51, 56, 61, 67, 71, 78, 81, 89, 91, 101, 110, 121, 131, 141, 151, 161, 171, 181, 191, 201, 210, 231, 241, 251, 261, 271, 281, 291, 301, 310, 341, 351, 361, 371, 381, 391, 401, 410, 451, 461, 471, 481, 491, 501, 510, 561, 571, 581

Offset: 1

Graeme McRae, Aug 11 2007

Punctual Birds (A131881) are all numbers k with A132131(n) <= k < A132132(n) for some n Early Birds (A116700) are all numbers k with A132132(n) <= k < A132131(n+1) for some n

			a(1)=12 because 12 is the first Early Bird.
a(2)=21 because 21 is the second Early Bird.
a(3)=23 because 23 is the third Early Bird.
a(4)=31 because 31 is the fourth Early Bird.
a(5)=34 because 31-32 are Early Birds and 33 is not an Early Bird.

Graeme McRae, Table of n, a(n) for n = 1..27348

Cf. A033307, A117804, A116700, A131881, A132131, A132133.

A132134 Base 3 "Punctual Bird" numbers: write the natural numbers, base 3, in a string 12101112202122100101102... Sequence gives numbers which do not occur in the string ahead of their natural place.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 11, 15, 18, 27, 29, 30, 32, 33, 35, 42, 45, 54, 60, 81, 83, 86, 87, 89, 92, 95, 96, 98, 99, 101, 104, 105, 107, 123, 126, 135, 141, 153, 162, 243, 245, 248, 249, 251, 252, 254, 257, 258, 260, 261, 263, 266, 267, 269, 275, 276, 278, 285, 287, 288

Offset: 1

Graeme McRae, Aug 11 2007

			a(5)=6 because 6 (20, base 3) is the fifth number that appears first in its "natural" place in the string of concatenated base-3 numbers.

Cf. A033307, A117804, A116700, A131881, A132131, A132132, A132133, A132135.

cp's OEIS Frontend

A132133 Number of n-digit "Punctual Birds" (cf. A131881).

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A083655 Numbers which do not appear prematurely in the binary Champernowne word (A030190).

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A132132 "Early Bird" numbers n such that n-1 is not an Early Bird (cf. A116700).

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A132134 Base 3 "Punctual Bird" numbers: write the natural numbers, base 3, in a string 12101112202122100101102... Sequence gives numbers which do not occur in the string ahead of their natural place.

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