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Search: a106847 -id:a106847
Displaying 1-3 of 3 results found.
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A106634
Number of ways to express n as i*j+k*l, with i,j,k,l in the range [0..n].
+10
5
1, 6, 21, 32, 62, 58, 124, 88, 173, 158, 226, 156, 380, 194, 340, 356, 466, 274, 613, 316, 690, 536, 596, 404, 1060, 552, 734, 728, 1032, 546, 1376, 596, 1213, 932, 1026, 976, 1858, 750, 1180, 1144, 1910, 854, 2048, 908, 1784, 1730, 1500, 1016, 2800
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Number of ordered 4-tuples [i,j,k,l] with n=i*j+k*l and i,j,k,l in the range [0..n].
a(n) is odd iff n is in A001105.
LINKS
R. J. Mathar and Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 (terms through 780 from Mathar)
FORMULA
From Ridouane Oudra, Jul 20 2024: (Start)
a(n) = (4*n + 2)*tau(n) + Sum_{i=1..n-1} tau(i)*tau(n-i), for n>0 ;
a(n) = (4*n + 2)*A000005(n) + A055507(n-1), for n>0 ;
a(n) = 4*A038040(n) + A062011(n) + A055507(n-1), for n>0. (End)
EXAMPLE
a(1)=6: the 4-tuples ijkl are 1100, 1101, 1110, 0011, 0111, 1011.
a(2)=21: 1111, 2100, 210x, 21x0, 1200, 120x, 12x0, where x = 1 or 2, and ten more with the two halves swapped.
MAPLE
A106634 := proc(n)
local a, i, j, k, l ;
a := 0 ;
for i from 0 to n do
for j from 0 to n do
if i*j > n then
break;
end if;
for k from 0 to n do
if i*j = n then
# treat l=0 separately
a := a+1 ;
end if;
# l=1..n
if k =0 then
if i*j=n then
a := a+n ;
end if;
else
l := (n-i*j)/k ;
if l >=1 and l <=n and type(l, 'integer') then
a := a+1 ;
end if;
end if;
end do:
end do:
end do:
a ;
end proc: # R. J. Mathar, Oct 17 2012
MATHEMATICA
list[n_] := Module[{v, i, j}, v[_] = 0;
For[i = 2, i <= n, i++, For[j = 1, j <= Min[Quotient[n, i], i-1], j++, v[i*j]+= 2]];
For[i = 1, i <= Floor@Sqrt[n], i++, v[i^2]++];
Join[{1}, Table[2 Sum[v[j] v[i-j], {j, Quotient[i, 2]+1, i-1}]+If[OddQ[i], 0, v[i/2]^2]+(4i+2) v[i], {i, 1, n}]]];
list[48] (* Jean-François Alcover, Jun 04 2023, after Charles R Greathouse IV *)
PROG
(PARI) list(n)={
my(v=vector(n));
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, i, 2*sum(j=i\2+1, i-1, v[j]*v[i-j])+if(i%2, , v[i/2]^2)+(4*i+2)*v[i]))
}; \\ Charles R Greathouse IV, Oct 17 2012
CROSSREFS
Cf. A001105, A055507, A106633, A106846, A106847.
Cf. A000005, A038040, A062011.
KEYWORD
nonn,easy
AUTHOR
Ralf Stephan, May 06 2005
EXTENSIONS
Definition clarified by N. J. A. Sloane, Jul 07 2012
STATUS
approved
A106633
Number of ways to express n as k+l*m, with k, l, m all in the range [0..n].
+10
4
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Number of ordered triples [k,l,m] with n = k+l*m and k, l, m all in the range [0..n].
From R. J. Mathar, Jun 30 2013: (Start)
A010766 is the following array A read by antidiagonals:
1, 1, 1, 1, 1, 1, ...
2, 1, 1, 1, 1, 1, ...
3, 2, 1, 1, 1, 1, ...
4, 2, 2, 1, 1, 1, ...
5, 3, 2, 2, 1, 1, ...
6, 3, 2, 2, 2, 1, ...
and apparently a(n) is the hook sum Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). (End)
LINKS
R. J. Mathar, Table of n, a(n) for n = 0..1000
FORMULA
From Ridouane Oudra, Apr 22 2024: (Start)
a(n) = 2*n + 1 + Sum_{k=1..n} floor(n/k);
a(n) = 2*n + 1 + Sum_{k=1..n} tau(k);
a(n) = A005408(n) + A006218(n). (End)
EXAMPLE
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).
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) 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
CROSSREFS
Cf. A006218, A106634, A106846, A106847.
Cf. A005408, A006218.
KEYWORD
nonn,easy
AUTHOR
Ralf Stephan, May 06 2005
EXTENSIONS
Definition clarified by N. J. A. Sloane, Jul 07 2012
STATUS
approved
A106846
a(n) = Sum_{k + l*m <= n} (k + l*m), with 0 <= k,l,m <= n.
+10
4
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
R. J. Mathar, Table of n, a(n) for n = 0..1000
FORMULA
From Ridouane Oudra, Jun 24 2024: (Start)
a(n) = (1/2) * (n*(n+1)*(2*n+1) + Sum_{k=1..n} (n^2 + n + k - k^2) * tau(k)).
a(n) = (1/2) * (A055112(n) + (n^2 + n) * A006218(n) + A143127(n) - A319085(n)).
a(n) = A059270(n) + A143127(n) + A106847(n). (End)
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
Cf. A106633, A106634, A106847.
Cf. A055112, A006218, A143127, A319085, A059270, A143127.
KEYWORD
nonn
AUTHOR
Ralf Stephan, May 06 2005
STATUS
approved
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