A047345 -id:A047345 - cp's OEIS frontend

A030123 Most likely total for a roll of n 6-sided dice, choosing the smallest if there is a choice.

0, 1, 7, 10, 14, 17, 21, 24, 28, 31, 35, 38, 42, 45, 49, 52, 56, 59, 63, 66, 70, 73, 77, 80, 84, 87, 91, 94, 98, 101, 105, 108, 112, 115, 119, 122, 126, 129, 133, 136, 140, 143, 147, 150, 154, 157, 161, 164, 168, 171, 175, 178, 182, 185, 189, 192

Offset: 0

Eric W. Weisstein

In fact ceiling(7n/2) is just as likely as floor(7n/2), so sequence could equally well be A047345. - Henry Bottomley, Jan 19 2001. a(1) is the only exception to this rule. - Dmitry Kamenetsky, Nov 03 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (corrected by Sean A. Irvine, Jan 18 2019)
Eric Weisstein's World of Mathematics, Dice.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A047355.

I:=[7,10,14]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..60]]; // Vincenzo Librandi, Oct 19 2013

A030123:=n->floor(7*n/2): seq(A030123(n), n=2..100); # Wesley Ivan Hurt, Jan 23 2017

CoefficientList[Series[-(3 x^2 - 3 x - 7)/((x - 1)^2 (x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 19 2013 *)

a(n)=7*n\2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = floor(7*n/2) for n >= 2.
From Colin Barker, Jun 09 2013: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n >= 5.
G.f.: x - x^2 * (3*x^2-3*x-7) / ((x-1)^2*(x+1)). (End)

a(0) and a(1) added by Dmitry Kamenetsky, Nov 03 2017

A302928 Maximum number of 4's possible in an infinite Minesweeper grid with n mines.

Original entry on oeis.org

0, 0, 0, 2, 2, 4, 6, 6, 7, 8, 10, 12, 12, 13, 14, 16, 18, 18, 19, 21, 22, 24, 24, 26, 28, 30, 30, 31, 33, 34, 36, 36

Offset: 1

Dmitry Kamenetsky, Apr 16 2018

Question: what is the maximum value possible for (a(n) - n)? The current record is 5, which occurs at n=31.

There is no maximum. We can take k copies of the 17-mine configuration and place them sufficiently far apart to get 18k fours. So a(17k) - 17k >= 18k - 17k = k can get arbitrarily large. A possibly more interesting question is to ask about the maximum ratio a(n)/n. - Yevhenii Diomidov, Jan 19 2022

Dmitry Kamenetsky, The first 32 solutions
Wikipedia, Minesweeper.

Cf. A047345, A302929, A302930, A302931.

A302929 Maximum number of 5's possible in an infinite Minesweeper grid with n mines.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 4, 4, 5, 6, 6, 7, 8, 10, 10, 11, 12, 13, 14, 15, 16, 16, 17, 18, 20, 20, 22, 22, 23, 24, 26

Offset: 1

Dmitry Kamenetsky, Apr 16 2018

Dmitry Kamenetsky, The first 32 solutions
Wikipedia, Minesweeper.

Cf. A047345, A302928, A302930, A302931.

A302930 Maximum number of 6's possible in an infinite Minesweeper grid with n mines.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18

Offset: 1

Dmitry Kamenetsky, Apr 16 2018

Dmitry Kamenetsky, The first 32 solutions
Wikipedia, Minesweeper.

Cf. A047345, A302928, A302929, A302931.

A302931 Maximum number of 7's possible in an infinite Minesweeper grid with n mines.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10

Offset: 1

Dmitry Kamenetsky, Apr 16 2018

Dmitry Kamenetsky, The first 32 solutions
Wikipedia, Minesweeper.

Cf. A047345, A302928, A302929, A302930.

A140869 Triangle read by rows where T(m,n) = floor((2mn+m+n-2)/2), m >= n >= 1.

Original entry on oeis.org

1, 2, 5, 4, 7, 11, 5, 10, 14, 19, 7, 12, 18, 23, 29, 8, 15, 21, 28, 34, 41, 10, 17, 25, 32, 40, 47, 55, 11, 20, 28, 37, 45, 54, 62, 71, 13, 22, 32, 41, 51, 60, 70, 79, 89, 14, 25, 35, 46, 56, 67, 77, 88, 98, 109, 16, 27, 39, 50, 62, 73, 85, 96, 108, 119, 131, 17, 30, 42, 55, 67, 80, 92, 105, 117, 130, 142, 155

Offset: 1

Vincenzo Librandi, Jan 16 2009

Conjecture: If h does not belong to the sequence, then 4*h+5 is prime. - Vincenzo Librandi, Nov 18 2012

First column: A001651; second column: A047215; third column: A047345. - Vincenzo Librandi, Nov 18 2012

			Triangle begins:
1;
2,  5;
4,  7,  11;
5,  10, 14, 19;
7,  12, 18, 23, 29;
8,  15, 21, 28, 34, 41;
10, 17, 25, 32, 40, 47, 55; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A002327, A153085, A001651, A047215, A047345.

[Floor(((2*n*k+n+k-2)/2)): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 18 2012

Flatten[Table[Floor[(2*n*m+m+n-2)/2],{n,1,10},{m,n}]] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2012 *)

A238290 a(n+1) = a(n) + 6 + 2(n - 2floor(n/2)) for n > 0, a(0) = 0.

Original entry on oeis.org

0, 8, 14, 22, 28, 36, 42, 50, 56, 64, 70, 78, 84, 92, 98, 106, 112, 120, 126, 134, 140, 148, 154, 162, 168, 176, 182, 190, 196, 204, 210, 218, 224, 232, 238, 246, 252, 260, 266, 274, 280, 288, 294, 302, 308, 316, 322, 330, 336, 344, 350, 358, 364, 372, 378

Offset: 0

Jose Eduardo Blazek, Feb 22 2014

			G.f.: 8*x + 14*x^2 + 22*x^3 + 28*x^4 + 36*x^5 + 42*x^6 + 50*x^7 + 56*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1).

Cf. A047345.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(2*x*(4+3*x)/((1-x)*(1-x^2)))); // G. C. Greubel, Aug 07 2018

CoefficientList[Series[2 x (4 + 3 x)/((1 - x) (1 - x^2)), {x, 0, 80}], x] (* Vincenzo Librandi, Feb 26 2014 *)
Table[(14 n - (-1)^n + 1)/2, {n, 0, 60}] (* Bruno Berselli, Feb 26 2014 *)

x='x+O('x^50); concat([0], Vec(2*x*(4+3*x)/((1-x)*(1-x^2)))) \\ G. C. Greubel, Aug 07 2018

a(n) = floor((2*n+1)/4) + 2*floor((2*n+1)/2) + 3*floor(3*(2*n+1)/4).
a(n) = 8*floor((n+1)/2) + 6*floor(n/2).
G.f.: 2 * x * (4 + 3*x) / ((1 - x) * (1 - x^2)). - Michael Somos, Feb 24 2014
a(n) = 2*A047345(n+1) = (14*n - (-1)^n + 1)/2. - Bruno Berselli, Feb 26 2014
E.g.f.: 7*exp(x)*x + sinh(x). - Stefano Spezia, May 15 2021

A047291 Numbers that are congruent to {0, 1, 4, 6} mod 7.

Original entry on oeis.org

0, 1, 4, 6, 7, 8, 11, 13, 14, 15, 18, 20, 21, 22, 25, 27, 28, 29, 32, 34, 35, 36, 39, 41, 42, 43, 46, 48, 49, 50, 53, 55, 56, 57, 60, 62, 63, 64, 67, 69, 70, 71, 74, 76, 77, 78, 81, 83, 84, 85, 88, 90, 91, 92, 95, 97, 98, 99, 102, 104, 105, 106, 109, 111

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047336, A047345.

I:=[0, 1, 4, 6, 7]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, Apr 26 2012

A047291:=n->(-13-(-1)^n+(3-I)*(-I)^n+(3+I)*I^n+14*n)/8: seq(A047291(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Select[Range[0,120], MemberQ[{0,1,4,6}, Mod[#,7]]&] (* Vincenzo Librandi, Apr 26 2012 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,4,6,7},100] (* G. C. Greubel, Jun 01 2016 *)

x='x+O('x^100); concat(0, Vec(x^2*(1+3*x+2*x^2+x^3)/((1-x)^2*(1+x)*(1+x^2)))) \\ Altug Alkan, Dec 24 2015

From Colin Barker, Mar 13 2012: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
G.f.: x^2*(1 + 3*x + 2*x^2 + x^3)/((1-x)^2*(1+x)*(1+x^2)). (End)
a(n) = (-13 - (-1)^n + (3-i)*(-i)^n + (3+i)*i^n + 14*n)/8 where i=sqrt(-1). - Colin Barker, May 14 2012
a(2k) = A047336(k), a(2k-1) = A047345(k). - Wesley Ivan Hurt, Jun 01 2016
E.g.f.: (4 - sin(x) + 3*cos(x) + (7*x - 6)*sinh(x) + 7*(x - 1)*cosh(x))/4. - Ilya Gutkovskiy, Jun 01 2016

A047293 Numbers that are congruent to {0, 2, 4, 6} mod 7.

Original entry on oeis.org

0, 2, 4, 6, 7, 9, 11, 13, 14, 16, 18, 20, 21, 23, 25, 27, 28, 30, 32, 34, 35, 37, 39, 41, 42, 44, 46, 48, 49, 51, 53, 55, 56, 58, 60, 62, 63, 65, 67, 69, 70, 72, 74, 76, 77, 79, 81, 83, 84, 86, 88, 90, 91, 93, 95, 97, 98, 100, 102, 104, 105, 107, 109, 111

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047276, A047345.

I:=[0, 2, 4, 6, 7]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, Apr 26 2012

A047293:=n->2*n-2-floor((n-1)/4): seq(A047293(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Select[Range[0,100],MemberQ[{0,2,4,6},Mod[#,7]]&] (* Vincenzo Librandi, Apr 26 2012 *)
LinearRecurrence[{1,0,0,1,-1},{0,2,4,6,7},80] (* Harvey P. Dale, Jun 21 2019 *)

A047293(n)=n*7\4-1  \\ M. F. Hasler, Apr 27 2012

a(n) = floor(ceiling((7n + 2)/2)/2).
a(n) = 2n-2-floor((n-1)/4). - Gary Detlefs, Mar 27 2010
From Colin Barker, Mar 13 2012: (Start)
a(n) = a(n-1)+a(n-4)-a(n-5) for n>5.
G.f.: x^2*(2+2*x+2*x^2+x^3)/((1-x)^2*(1+x)*(1+x^2)). (End)
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = (14n-11+i^(2n)+(1-i)*i^(-n)+(1+i)*i^n)/8 where i=sqrt(-1).
a(2n) = A047276(n), a(2n-1) = A047345(n). (End)

A047297 Numbers that are congruent to {0, 3, 4, 6} mod 7.

Original entry on oeis.org

0, 3, 4, 6, 7, 10, 11, 13, 14, 17, 18, 20, 21, 24, 25, 27, 28, 31, 32, 34, 35, 38, 39, 41, 42, 45, 46, 48, 49, 52, 53, 55, 56, 59, 60, 62, 63, 66, 67, 69, 70, 73, 74, 76, 77, 80, 81, 83, 84, 87, 88, 90, 91, 94, 95, 97, 98, 101, 102, 104, 105, 108, 109, 111

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047280, A047345.

[n : n in [0..150] | n mod 7 in [0, 3, 4, 6]]; // Wesley Ivan Hurt, Jun 02 2016

A047297:=n->(14*n-9+3*I^(2*n)-(1+I)*I^(-n)-(1-I)*I^n)/8: seq(A047297(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016

Table[(14n-9+3*I^(2n)-(1+I)*I^(-n)-(1-I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, Jun 02 2016 *)

G.f.: x^2*(3+x+2*x^2+x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011
From Wesley Ivan Hurt, Jun 02 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14*n-9+3*i^(2*n)-(1+i)*i^(-n)-(1-i)*i^n)/8 where i=sqrt(-1).
a(2k) = A047280(k), a(2k-1) = A047345(k). (End)

cp's OEIS Frontend

A030123 Most likely total for a roll of n 6-sided dice, choosing the smallest if there is a choice.

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A302928 Maximum number of 4's possible in an infinite Minesweeper grid with n mines.

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A302929 Maximum number of 5's possible in an infinite Minesweeper grid with n mines.

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A302930 Maximum number of 6's possible in an infinite Minesweeper grid with n mines.

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A302931 Maximum number of 7's possible in an infinite Minesweeper grid with n mines.

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A140869 Triangle read by rows where T(m,n) = floor((2mn+m+n-2)/2), m >= n >= 1.

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A238290 a(n+1) = a(n) + 6 + 2*(n - 2*floor(n/2)) for n > 0, a(0) = 0.

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A047291 Numbers that are congruent to {0, 1, 4, 6} mod 7.

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Magma

Maple

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A047293 Numbers that are congruent to {0, 2, 4, 6} mod 7.

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A238290 a(n+1) = a(n) + 6 + 2(n - 2floor(n/2)) for n > 0, a(0) = 0.