Srikanth K S - cp's OEIS frontend

User: Srikanth K S

Srikanth K S has authored 4 sequences.

A174686 Number of equivalence classes of 3 X 3 matrices filled with n colors so that no two rotations are identical.

120, 4860, 65280, 487500, 2517480, 10084200, 33546240, 96840360, 249975000, 589446660, 1289882880, 2651032020, 5165127240, 9610650000, 17179607040, 29646614160, 49589350200, 80671305420, 127999200000, 198568990620, 301816016040, 450286556280, 660449894400

Offset: 2

Srikanth K S, Mar 27 2010

nonn

Each class contains a set of 4 matrices so that all of them can be obtained by successive rotation but no two are identical.

a(n) = (n^9 - n^5)/4 \\ Michel Marcus, Mar 04 2013

a(n) = (n^9 - n^(floor(9/2) + 1))/4. More generally for any m X m matrix f(n,m) = (n^(m^2) - n^(m^2/2))/4 if m is even, and f(n,m) = (n^(m^2) - n^(floor(m^2/2)+1))/4 if m is odd.

More terms from Michel Marcus, Mar 04 2013

A160492 a(n) = number of solutions to an equation x_1 + ... + x_j =0 with 1<=j<=n satisfying -n<=x_i<=n (1<=i<=j).

Original entry on oeis.org

1, 6, 45, 560, 9795, 223524, 6284089, 210208560, 8156750283, 360297117070, 17853149451841, 980844453593160, 59179098916735213, 3890176308574524934, 276750779199166606705, 21185250061147839785120, 1736385140876356212244563, 151719500906542020597450498

Offset: 1

Srikanth K S, May 15 2009

nonn

The number of variables in the equation can be from 1 to n and each variable can have a value of -n to n. See A286928 for the case of exactly n variables. - Andrew Howroyd, May 16 2017

			From _Andrew Howroyd_, May 16 2017 (Start)
Case n=3:
1 variable: {0} is only solution.
2 variables: {-3,3}, {-2,2}, {-1,1}, {0,0}, {1,-1}, {2,-2}, {3,-3}.
3 variables: {-3 0 3}x6, {-3 1 2}x6, {-2 -1 3}x6, {-2 0 2}x6,
             {-2 1 1}x3, {-1 -1 2}x3, {-1 0 1}x6, {0 0 0}x1
In the above, {-3 0 3}x6 means that the values can be expanded to 6 solutions by considering different orderings.
In total there are 1 + 7 + 37 = 45 solutions so a(3)=45.
(End)

Cf. A286928.

zerocompositionswithzero[p_] := Module[{united = {}, i, zerosums = {}, count = 0}, For[i = 1, i <= p, i = i + 1, united = Union[united, Tuples[Table[x, {x, -p, p}], i]] ]; For[i = 1, i <= Length[united], i = i + 1, If[Sum[united[[i, j]], {j, 1, Length[united[[i]]]}] == 0, zerosums = Append[zerosums, united[[i]]]; count = count + 1;]; ]; Return[{count, zerosums}]; ];

\\ nr compositions of r with max value m into exactly k parts.
compositions(r,m,k)=sum(i=0,floor((r-k)/m),(-1)^i*binomial(r-1-i*m, k-1)*binomial(k, i));
a(n)=sum(v=1,n,compositions(v*(n+1),2*n+1,v));  \\ Andrew Howroyd, May 16 2017

from sympy import binomial
def C(r, m, k): return sum([(-1)**i*binomial(r - 1 - i*m, k - 1)*binomial(k, i) for i in range(int((r - k)/m) + 1)])
def a(n): return sum([C(v*(n + 1), 2*n + 1, v) for v in range(1, n + 1)]) # Indranil Ghosh, May 16 2017, after the PARI program by Andrew Howroyd

a(n) = Sum_{k=1..n} Sum_{i=0..floor(k/2)} (-1)^i*binomial(k*(n+1)-i*(2*n+1)-1, k-1)*binomial(k, i). - Andrew Howroyd, May 16 2017

Name clarified and a(6)-a(18) from Andrew Howroyd, May 16 2017

A161823 Among any n lines on the plane, there exists a pair at an angle not more than a(n) degrees.

Original entry on oeis.org

90, 60, 45, 36, 30, 26, 23, 20, 18, 17, 15, 14, 13, 12, 12, 11, 10, 10, 9, 9, 9, 8, 8, 8, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 2

Srikanth K S, Jun 20 2009

nonn

a(n) = ceiling(180/n) for n > 1.

A161171 Erroneous version of A035008.

Original entry on oeis.org

0, 0, 12, 48, 174, 238, 318, 414, 526, 654, 798, 958, 1134, 1326, 1534

Offset: 1

Srikanth K S, Jun 04 2009

dead

Previous name was: Number of moves possible by a knight on an n X n chessboard.

a(n)= 8 (n - 4)^2 + (n - 2) 16 + 46 + (n - 2) 24 for n > 4

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User: Srikanth K S

A174686 Number of equivalence classes of 3 X 3 matrices filled with n colors so that no two rotations are identical.

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A160492 a(n) = number of solutions to an equation x_1 + ... + x_j =0 with 1<=j<=n satisfying -n<=x_i<=n (1<=i<=j).

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A161823 Among any n lines on the plane, there exists a pair at an angle not more than a(n) degrees.

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A161171 Erroneous version of A035008.

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