A280594 -id:A280594 - cp's OEIS frontend

A280593 Natural numbers whose digits can be formed by typing adjacent keys on a 123-456-789 keypad without repeating a digit.

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 21, 23, 25, 32, 36, 41, 45, 47, 52, 54, 56, 58, 63, 65, 69, 74, 78, 85, 87, 89, 96, 98, 123, 125, 145, 147, 214, 236, 254, 256, 258, 321, 325, 365, 369, 412, 452, 456, 458, 478, 521, 523, 541, 547, 563, 569, 587, 589, 632, 652, 654, 658, 698, 741, 745, 785, 789

Offset: 1

FUNG Cheok Yin, Jan 06 2017

A subsequence of A010784. - FUNG Cheok Yin, Jul 05 2018

			The keypad is:
+---+---+---+
| 1 | 2 | 3 |
+---+---+---+
| 4 | 5 | 6 |
+---+---+---+
| 7 | 8 | 9 |
+---+---+---+
It is visibly obvious that 2589 can be formed on the keypad.

FUNG Cheok Yin, Table of n, a(n) for n = 1..653
FUNG Cheok Yin, C++ program

Cf. A010784, A280594, A280595.

g = Graph[{1 <-> 2, 1 <-> 4,
    2 <-> 1, 2 <-> 3, 2 <-> 5,
    3 <-> 2, 3 <-> 6,
    4 <-> 1, 4 <-> 5, 4 <-> 7,
    5 <-> 2, 5 <-> 4, 5 <-> 6, 5 <-> 8,
    6 <-> 3, 6 <-> 5, 6 <-> 9,
    7 <-> 4, 7 <-> 8,
    8 <-> 5, 8 <-> 7, 8 <-> 9,
    9 <-> 6, 9 <-> 8}];
f[{a_, b_}] := FindPath[g, a, b, Infinity, All]
ff = f /@ Flatten[Outer[List, r = Range[9], r], 1];
A280593 = Sort[Join[r, FromDigits /@ Flatten[ff, 1]]] (* Jean-François Alcover, Jan 06 2017 *)

A280595 Nonnegative numbers whose digits can be formed by typing adjacent keys on a 123-456-789-00X keypad without repeating a digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 21, 23, 25, 32, 36, 41, 45, 47, 52, 54, 56, 58, 63, 65, 69, 70, 74, 78, 80, 85, 87, 89, 96, 98, 123, 125, 145, 147, 214, 236, 254, 256, 258, 321, 325, 365, 369, 412, 452, 456, 458, 470, 478, 521, 523, 541, 547, 563, 569, 580, 587, 589, 632, 652, 654, 658

Offset: 1

FUNG Cheok Yin, Jan 06 2017

A subsequence of A010784. - FUNG Cheok Yin, Jul 05 2018

			The keypad is:
+---+---+---+
| 1 | 2 | 3 |
+---+---+---+
| 4 | 5 | 6 |
+---+---+---+
| 7 | 8 | 9 |
+---+---+---+
|   0   | x |
+---+---+---+
It is visibly obvious that 5807 can be formed on the keypad.

FUNG Cheok Yin, Table of n, a(n) for n = 1..1082

Cf. A010784, A280593, A280594.

g = Graph[{1 <-> 2, 1 <-> 4,
    2 <-> 1, 2 <-> 3, 2 <-> 5,
    3 <-> 2, 3 <-> 6,
    4 <-> 1, 4 <-> 5, 4 <-> 7,
    5 <-> 2, 5 <-> 4, 5 <-> 6, 5 <-> 8,
    6 <-> 3, 6 <-> 5, 6 <-> 9,
    7 <-> 0, 7 <-> 4, 7 <-> 8,
    8 <-> 0, 8 <-> 5, 8 <-> 7, 8 <-> 9,
    9 <-> 6, 9 <-> 8}];
f[{a_, b_}] := FindPath[g, a, b, Infinity, All]
ff = f /@ Flatten[Outer[List, r = Range[9], Range[0, 9]], 1];
A280595 = Sort[Join[r, FromDigits /@ Flatten[ff, 1]]] (* Jean-François Alcover, Jan 07 2017 *)

Initial 0 added by N. J. A. Sloane, Feb 05 2017

A281942 Nonnegative numbers whose digits can be formed by typing adjacent keys on a 123-456-789-0XX keypad without repeating a digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 21, 23, 25, 32, 36, 41, 45, 47, 52, 54, 56, 58, 63, 65, 69, 70, 74, 78, 85, 87, 89, 96, 98, 123, 125, 145, 147, 214, 236, 254, 256, 258, 321, 325, 365, 369, 412, 452, 456, 458, 470, 478, 521, 523, 541, 547, 563, 569, 587, 589, 632, 652, 654, 658, 698, 741

Offset: 1

FUNG Cheok Yin, Feb 02 2017

A subsequence of A010784. - FUNG Cheok Yin, Jul 05 2018

			The keypad is:
+---+---+---+
| 1 | 2 | 3 |
+---+---+---+
| 4 | 5 | 6 |
+---+---+---+
| 7 | 8 | 9 |
+---+---+---+
| 0 | x | x |
+---+---+---+
It is visibly obvious that 470 can be formed on the keypad.

FUNG Cheok Yin, Table of n, a(n) for n = 1..725

Cf. A010784, A280593, A280594, A280595, A281943, A215009.

Initial 0 added by N. J. A. Sloane, Feb 05 2017

A281943 Nonnegative numbers whose digits can be formed by typing adjacent keys on a 789-456-123-00X keypad without repeating a digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 20, 21, 23, 25, 32, 36, 41, 45, 47, 52, 54, 56, 58, 63, 65, 69, 74, 78, 85, 87, 89, 96, 98, 102, 120, 123, 125, 145, 147, 201, 210, 214, 236, 254, 256, 258, 320, 321, 325, 365, 369, 410, 412, 452, 456, 458, 478, 520, 521, 523, 541, 547, 563, 569, 587

Offset: 1

FUNG Cheok Yin, Feb 02 2017

A subsequence of A010784. - FUNG Cheok Yin, Jul 05 2018

			The keypad is:
+---+---+---+
| 7 | 8 | 9 |
+---+---+---+
| 4 | 5 | 6 |
+---+---+---+
| 1 | 2 | 3 |
+---+---+---+
|   0   | x |
+---+---+---+
It is visibly obvious that 5210 can be formed on the keypad.

FUNG Cheok Yin, Table of n, a(n) for n = 1..1082

Cf. A010784, A280593, A280594, A280595, A281942, A215009.

Initial 0 added by N. J. A. Sloane, Feb 05 2017

A327692 Number of length-n phone numbers that can be dialed by a chess knight on a 0-9 keypad that starts on any number and takes n-1 steps.

Original entry on oeis.org

10, 20, 46, 104, 240, 544, 1256, 2848, 6576, 14912, 34432, 78080, 180288, 408832, 944000, 2140672, 4942848, 11208704, 25881088, 58689536, 135515136, 307302400, 709566464, 1609056256, 3715338240, 8425127936, 19453763584

Offset: 1

Derek Lim, Sep 22 2019

nonn

The keypad is of the form:
+---+---+---+
| 1 | 2 | 3 |
+---+---+---+
| 4 | 5 | 6 |
+---+---+---+
| 7 | 8 | 9 |
+---+---+---+
| * | 0 | # |
+---+---+---+

			For n = 1 the a(1) = 10 numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
For n = 2 the a(2) = 20 numbers are 04, 06, 16, 18, 27, 29, 34, 38, 43, 49, 40, 61, 67, 60, 72, 76, 81, 83, 92, 94.

Derek Lim, Table of n, a(n) for n = 1..2500

Cf. A280594, A169696.

def number_dialable(N):
    reach = ((4,6),(6,8),(7,9),(4,8),(3,9,0),(),(1,7,0),(2,6),(1,3),(2,4))
    M = [[0] * 10 for _ in range(N)]
    M[0] = [1]*10
    for step in range(1,N):
        for tile in range(10):
            for nxt in reach[tile]:
                M[step][nxt] += M[step-1][tile]
    return [sum(row) for row in M]

Conjectures from Colin Barker, Oct 01 2019: (Start)
G.f.: 2*x*(5 + 10*x - 7*x^2 - 8*x^3 + 2*x^4) / (1 - 6*x^2 + 4*x^4).
a(n) = 6*a(n-2) - 4*a(n-4) for n>6. (End)
Comments from Francesca Arici, Apr 17 2024: (Start)
The recursive formula a(n) = 6*a(n-2) - 4*a(n-4) also holds for n=6.
It can be proved using results from graph theory. Indeed, if we consider the directed graph associated to the knight dialler problem, then a(n) equals the number of paths in the graph of length n-1 in the graph. This number can be expressed in terms of the grand sum of powers of the incidence matrix A(i,j) of the graph.
Moreover, the matrix A is diagonalizable over the reals, with one zero eigenvalue, say L(0)=0. Combining this with the formula for the grand sum of a diagonalizable matrix in term of its eigenvalues, the above conjecture reduces to checking an algebraic condition on the nonzero eigenvalues L(1), ..., L(8) of A. (End)

A322511 Nonnegative numbers whose digits can be formed by typing adjacent keys on a 123-456-789 keypad without repeating a digit and without changing direction.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 21, 23, 25, 32, 36, 41, 45, 47, 52, 54, 56, 58, 63, 65, 69, 74, 78, 85, 87, 89, 96, 98, 123, 147, 258, 321, 369, 456, 654, 741, 789, 852, 963, 987

Offset: 1

Jason Jewell, Dec 13 2018

Subsequence of A280593.

Diagonal moves are not allowed.

			The keypad is:
+---+---+---+
| 1 | 2 | 3 |
+---+---+---+
| 4 | 5 | 6 |
+---+---+---+
| 7 | 8 | 9 |
+---+---+---+
258 can be formed in a straight line, while 256 cannot, even though 256 is formed with adjacent digits. Therefore 258 is a part of this sequence while 256 is not.

Cf. A280593, A280594, A280595, A281942.

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A280593 Natural numbers whose digits can be formed by typing adjacent keys on a 123-456-789 keypad without repeating a digit.

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A280595 Nonnegative numbers whose digits can be formed by typing adjacent keys on a 123-456-789-00X keypad without repeating a digit.

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A281942 Nonnegative numbers whose digits can be formed by typing adjacent keys on a 123-456-789-0XX keypad without repeating a digit.

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A281943 Nonnegative numbers whose digits can be formed by typing adjacent keys on a 789-456-123-00X keypad without repeating a digit.

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A327692 Number of length-n phone numbers that can be dialed by a chess knight on a 0-9 keypad that starts on any number and takes n-1 steps.

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A322511 Nonnegative numbers whose digits can be formed by typing adjacent keys on a 123-456-789 keypad without repeating a digit and without changing direction.

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