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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.

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A126718 a(n) is the number of nonnegative integers k less than 10^n such that the decimal representation of k lacks the digits 1,2,3, at least one of digits 4,5, at least one of digits 6,7 and at least one of digits 8,9.

Original entry on oeis.org

7, 43, 235, 1171, 5467, 24403, 105595, 447091, 1864027, 7686163, 31440955, 127865011, 517788187, 2090186323, 8417944315, 33843570931, 135890057947, 545108340883, 2185079263675, 8754257900851, 35058860433307, 140360940805843, 561820285607035
Offset: 1

Views

Author

Aleksandar M. Janjic and Milan Janjic, Feb 13 2007

Keywords

Links

Crossrefs

Cf. A125910, A125945, A125946, A125947, A125948, A125880, A125630, A125987, A125904, A125858, A125909, A125908, A126646, A126645, A126644, A126643, A126642, A126641, A126640, A126639, A126635, A126634, A126633, A126632, A126631, A126628, A126627.

Programs

  • Magma
    [8*4^n-12*3^n+6*2^n-1: n in [1..30]]; // Vincenzo Librandi, May 31 2011
  • Maple
    a:=n->8*4^n-12*3^n+6*2^n-1;
  • Mathematica
    LinearRecurrence[{10,-35,50,-24},{7, 43, 235, 1171},23] (* James C. McMahon, Dec 27 2024 *)
  • PARI
    Vec(-x*(24*x^3-50*x^2+27*x-7) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)) + O(x^100)) \\ Colin Barker, Feb 22 2015

Formula

a(n) = 8*4^n - 12*3^n + 6*2^n - 1.
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4). - Colin Barker, Feb 22 2015
G.f.: -x*(24*x^3 - 50*x^2 + 27*x - 7) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)). - Colin Barker, Feb 22 2015

A255166 Difference after n generations between the total number of single toothpicks in the I-toothpick structure of A160164 and the total number of ON cells in the "Ulam-Warburton" two-dimensional cellular automaton of A147562.

Original entry on oeis.org

0, 1, 1, 5, 1, 5, 9, 21, 1, 5, 9, 21, 9, 29, 49, 77, 1, 5, 9, 21, 9, 29, 49, 77, 9, 29, 49, 85, 57, 141, 209, 261, 1, 5, 9, 21, 9, 29, 49, 77, 9, 29, 49, 85, 57, 141, 209, 261, 9, 29, 49, 85, 57, 141, 209, 269, 57, 141, 217, 333, 289, 597, 785, 845, 1, 5, 9, 21, 9, 29, 49, 77, 9, 29, 49, 85, 57, 141, 209, 261, 9, 29, 49, 85
Offset: 0

Views

Author

Omar E. Pol, Feb 15 2015

Keywords

Examples

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
0;
1;
1,5;
1,5,9,21;
1,5,9,21,9,29,49,77;
1,5,9,21,9,29,49,77,9,29,49,85,57,141,209,261;
1,5,9,21,9,29,49,77,9,29,49,85,57,141,209,261,9,29,49,85,57,141,209,269,57,141,217,333,289,597,785,845;
...
It appears that the right border gives [0, 1] together with A126645. The right border gives the largest difference between both C.A. in every period.
Also, written the positive terms as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
1;
5,1;
5,9,21,1;
5,9,21,9,29,49,77,1;
5,9,21,9,29,49,77,9,29,49,85,57,141,209,261,1;
5,9,21,9,29,49,77,9,29,49,85,57,141,209,261,9,29,49,85,57,141,209,269,57,141,217,333,289,597,785,845,1;
...
The right border gives A000012 according with the illustrations as shown below. In this triangle the right border gives the smallest difference between both C.A. in every period.
For example: after 8 generations the structures look like this:
.
.                                      O
.                                    O O O
.                                  O   O   O
.    _ _ _ _ _ _ _ _             O O O O O O O
.     |_ _|   |_ _|            O   O   O   O   O
.     | |_|_ _|_| |          O O O   O O O   O O O
      |_|_|_ _|_|_|        O   O   O   O   O   O   O
.     |   | | |   |      O O O O O O O O O O O O O O O
.     |_ _|_|_|_ _|        O   O   O   O   O   O   O
.     | |_|_ _|_| |          O O O   O O O   O O O
.     |_|_|   |_|_|            O   O   O   O   O
.    _|_ _|_ _|_ _|_             O O O O O O O
.                                  O   O   O
.     86 toothpicks                  O O O
.                                      O
.
.                                 85 ON cells
.
a(8) = 1 because the I-toothpick structure contains 86 single toothpicks and the "Ulam-Warburton" two-dimensional cellular automaton has 85 ON cells, so the difference of the number of elements between both structures is equal to 86 - 85 = 1.

Links

Crossrefs

Cf. A126645, A139250, A147562, A160164, A169707, A170903.

Formula

a(n) = A160164(n) - A147562(n).

A258800 The number of zeroless decimal numbers whose digital sum is n.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, 518145, 1035269, 2068498, 4132920, 8257696, 16499120, 32965728, 65866496, 131603200, 262947072, 525375999, 1049716729, 2097364960, 4190597000, 8372936304, 16729373488, 33425781248
Offset: 0

Views

Author

Robert G. Wilson v, Jun 10 2015

Keywords

Comments

If you were to include decimal numbers that contain any number of zeros, then a(n) would be infinity. If on the other hand, you limit the number of zeros to some number, then a(n) is finite.

Examples

			a(0) = 0 since there exists no decimal number lacking a zero whose digital sum is zero.
a(1) = 1 since there exists only one zeroless decimal number whose digital sum is one and that number is 1.
a(2) = 2 since there exist only two zeroless decimal numbers whose digital sum is two and they are 2 & 11.
a(3) = 4 since there exist only four zeroless decimal numbers whose digital sum is three and they are 3, 21, 12 & 111.
a(4) = 8 since there exist only eight zeroless decimal numbers whose digital sum is four and they are 4, 31, 13, 22, 211, 121, 112 & 1111.

Crossrefs

Cf. A104144.
Cf. A125630, A125858, A125858, A125880, A125897, A125904, A125908, A125909, A125910, A125945, A125946, A125947, A125948, A126627, A126628, A126629, A126631, A126632, A126633, A126634, A126635, A126639, A126640, A126641, A126642, A126643, A126644, A126645, A126646, A126718.
Cf. A211072.

Programs

  • Mathematica
    CoefficientList[ Series[-1 + 1/(1 - x (1 + x + x^2) (1 + x^3 + x^6)), {x, 0, 36}], x]

Formula

a(n) = A104144(n+8) for n>0.
G.f.: -(x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9)/(-1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9) = -1 + 1/(1-x(1 + x + x^2)(1 + x^3 + x^6)).
Showing 1-3 of 3 results.

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