A197083 Number of solutions to a+b+c=d+e+f with 0<a<=n, 0<=b,c,d,e,f<=n.

0, 10, 96, 445, 1431, 3681, 8141, 16142, 29466, 50412, 81862, 127347, 191113, 278187, 394443, 546668, 742628, 991134, 1302108, 1686649, 2157099, 2727109, 3411705, 4227354, 5192030, 6325280, 7648290, 9183951, 10956925, 12993711, 15322711, 17974296, 20980872

Offset

0,2

Comments

When n<10, a(n) is the number of six digit numbers (with digits <= n) that have the property that the sum of the rightmost 3 digits equals the sum of the leftmost 3 digits. Some references call these balanced numbers. [Edited by M. F. Hasler, Mar 11 2013]

Links

Table of n, a(n) for n=0..32.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

G.f.: (x^4+19*x^3+36*x^2+10*x)/(x-1)^6.

a(n) = 6*a(n-1)-15*a(n-2)+20*a(n-3)-15*a(n-4)+6*a(n-5)-a(n-6) for n>5, a(0)=0, a(1)=10,a(2)=96, a(3)=445, a(4)=1431, a(5)=3681.

a(n) = (66*n^5+275*n^4+440*n^3+325*n^2+94*n)/120 = n*(n+1)*(66*n^3+209*n^2+231*n+94)/120.

Example

When n=1, a(n)=10 because there are 10 solutions when viewed as balanced numbers: 111111, 110110, 110101, 110011, 101110, 101101, 101011, 100100, 100010, 100001.

Mathematica

RecurrenceTable[{a[0] == 0, a[1] == 10, a[2] == 96, a[3] == 445, a[4] == 1431, a[5] == 3681, a[n] == 6 a[n - 1] - 15 a[n - 2] + 20 a[n - 3] - 15 a[n - 4] + 6 a[n - 5] - a[n - 6]}, a, {n, 0, 35}]

Crossrefs

Sequence in context: A233738 A277441 A307021 * A197086 A278359 A125945

Adjacent sequences: A197080 A197081 A197082 * A197084 A197085 A197086

Keyword

nonn,easy

Author

Bobby Milazzo, Mar 11 2013

Status

approved