A210000
Number of unimodular 2 X 2 matrices having all terms in {0,1,...,n}.
Original entry on oeis.org
0, 6, 14, 30, 46, 78, 94, 142, 174, 222, 254, 334, 366, 462, 510, 574, 638, 766, 814, 958, 1022, 1118, 1198, 1374, 1438, 1598, 1694, 1838, 1934, 2158, 2222, 2462, 2590, 2750, 2878, 3070, 3166, 3454, 3598, 3790, 3918, 4238, 4334, 4670, 4830
Offset: 0
a(2)=6 counts these matrices (using reduced matrix notation):
(1,0,0,1), determinant = 1, inverse = (1,0,0,1)
(1,0,1,1), determinant = 1, inverse = (1,0,-1,1)
(1,1,0,1), determinant = 1, inverse = (1,-1,0,1)
(0,1,1,0), determinant = -1, inverse = (0,1,1,0)
(0,1,1,1), determinant = -1, inverse = (-1,1,1,0)
(1,1,1,0), determinant = -1, inverse = (0,1,1,-1)
See also the very useful list of cross-references in the Comments section.
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a = 0; b = n; z1 = 50;
t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
Table[c[n, 0], {n, 0, z1}] (* A059306 *)
Table[c[n, 1], {n, 0, z1}] (* A171503 *)
2 % (* A210000 *)
Table[c[n, 2], {n, 0, z1}] (* A209973 *)
%/4 (* A209974 *)
Table[c[n, 3], {n, 0, z1}] (* A209975 *)
Table[c[n, 4], {n, 0, z1}] (* A209976 *)
Table[c[n, 5], {n, 0, z1}] (* A209977 *)
A106633
Number of ways to express n as k+l*m, with k, l, m all in the range [0..n].
Original entry on oeis.org
1, 4, 8, 12, 17, 21, 27, 31, 37, 42, 48, 52, 60, 64, 70, 76, 83, 87, 95, 99, 107, 113, 119, 123, 133, 138, 144, 150, 158, 162, 172, 176, 184, 190, 196, 202, 213, 217, 223, 229, 239, 243, 253, 257, 265, 273, 279, 283, 295, 300, 308, 314, 322, 326, 336, 342, 352
Offset: 0
0+1*2 = 0+2*1 = 1+1*1 = 2+0*0 = 2+0*1 = 2+0*2 = 2+1*0 = 2+2*0 = 2, so a(2)=8.
a(3)=12: 3+0*0, 3+0*m (6), 2+1*1, 1+2*1 (2), 0+3*1 (2).
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A106633 := proc(n)
local a, k, l, m ;
a := 0 ;
for k from 0 to n do
for l from 0 to n do
if l = 0 then
if k = n then
a := a+n+1 ;
end if;
else
m := (n-k)/l ;
if type(m,'integer') then
a := a+1 ;
end if;
end if;
end do:
end do:
a ;
end proc: # R. J. Mathar, Oct 17 2012
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A106633[n_] := Module[{a, m}, a = 0; Do[If[l == 0, If[k == n, a = a + n + 1], m = (n - k)/l; If[IntegerQ[m], a = a + 1]], {k, 0, n}, {l, 0, n}]; a];
Table[A106633[n], {n, 0, 56}] (* Jean-François Alcover, Jun 10 2023, after R. J. Mathar *)
-
list(n)={
my(v=vector(n),t);
for(i=2,n,for(j=1,min(n\i,i-1),v[i*j]+=2));
for(i=1,sqrtint(n),v[i^2]++);
concat(1,vector(n,k,2*k+1+t+=v[k]))
}; \\ Charles R Greathouse IV, Oct 17 2012
A106846
a(n) = Sum_{k + l*m <= n} (k + l*m), with 0 <= k,l,m <= n.
Original entry on oeis.org
0, 4, 22, 64, 144, 269, 461, 720, 1072, 1522, 2092, 2774, 3626, 4614, 5776, 7126, 8694, 10445, 12461, 14684, 17204, 19997, 23077, 26412, 30156, 34206, 38600, 43352, 48532, 54042, 60072, 66458, 73338, 80664, 88450, 96710, 105638, 114999
Offset: 0
-
A106846 := proc(n)
local a,k,l,m ;
a := 0 ;
for k from 0 to n do
for l from 0 to n do
if l = 0 then
m := n ;
else
m := floor((n-k)/l) ;
end if;
if m >=0 then
m := min(m,n) ;
a := a+(m+1)*k+l*m*(m+1)/2 ;
end if;
end do:
end do:
a ;
end proc: # R. J. Mathar, Oct 17 2012
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A106846[n_] := Module[{a, k, l, m }, a = 0; For[k = 0, k <= n, k++, For[l = 0, l <= n, l++, If[l == 0, m = n, m = Floor[(n - k)/l]]; If[m >= 0, m = Min[m, n]; a = a + (m + 1)*k + l*m*(m + 1)/2 ]]]; a];
Table[A106846[n], {n, 0, 40}] (* Jean-François Alcover, Apr 04 2024, after R. J. Mathar *)
A106847
a(n) = Sum {k + j*m <= n} (k + j*m), with 0 < k,j,m <= n.
Original entry on oeis.org
0, 0, 2, 11, 31, 71, 131, 229, 357, 537, 767, 1064, 1412, 1867, 2385, 3000, 3720, 4570, 5506, 6608, 7808, 9194, 10734, 12436, 14260, 16360, 18622, 21079, 23739, 26668, 29758, 33199, 36815, 40742, 44924, 49369, 54085, 59265, 64661, 70355
Offset: 0
We have 1+1*1=2<=3, 1+2*1=3, 1+1*2=3, 2+1*1=3, thus a(3)=2+3+3+3=11.
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A106847 := proc(n)
local a,k,l,m ;
a := 0 ;
for k from 1 to n do
for l from 1 to n-k do
m := floor((n-k)/l) ;
if m >=1 then
m := min(m,n) ;
a := a+m*k+l*m*(m+1)/2 ;
end if;
end do:
end do:
a ;
end proc: # R. J. Mathar, Oct 17 2012
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A106847[n_] := Module[{a, k, l, m}, a = 0; For[k = 1, k <= n, k++, For[l = 1, l <= n - k, l++, If[l == 0, m = n, m = Floor[(n - k)/l]]; If[m >= 1, m = Min[m, n]; a = a + m*k + l*m*(m + 1)/2]]]; a];
Table[A106847[n], {n, 0, 40}] (* Jean-François Alcover, Apr 04 2024, after R. J. Mathar *)
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A106847(n)=sum(m=1,n-1,sum(k=1,(n-1)\m,(n-m*k)*(n+m*k+1)))/2 \\ M. F. Hasler, Oct 17 2012
Showing 1-4 of 4 results.
Comments