A129263 Skylar (age 7) counts change by stacking all coins of the same type then arranging the stacks in a row. a(n) is the number of distinct Skylar stackings of n cents using any combination of pennies, nickels, dimes or quarters.

1, 1, 1, 1, 1, 2, 3, 3, 3, 3, 5, 7, 7, 7, 7, 10, 15, 15, 15, 15, 19, 25, 25, 25, 25, 31, 41, 41, 41, 41, 49, 63, 63, 63, 63, 74, 95, 95, 95, 95, 111, 147, 147, 147, 147, 166, 209, 209, 209, 209, 234, 293, 293, 293, 293, 322, 391, 391, 391, 391, 427, 515, 515, 515, 515

Offset

0,6

Comments

Sequence definition and Scratch program to compute the 100 terms due to Skylar Sutherland. Generating function contributed by Andrew V. Sutherland. Related to A001299, but distinguishes permutations of coin types.

References

Skylar Sutherland, student presentation at "The Undiscovered Country", a course for young mathematicians. Part of MIT's Educational Studies Program.

Links

Table of n, a(n) for n=0..64.

Formula

Let A_v(x,y) = 1-y+y/(1-x)^v and A(x,y) = A_1(x,y)A_5(x,y)A_10(x,y)A_25(x,y). Let A^(k)(x,y) denote the k-th partial derivative of A(x,y) w.r.t. y. The generating function of a(n) is A(x) = Sum A^(k)(x,0) for k from 0 to 4.

Example

a(16) = 15 = 1+2*4+6*1 since the distinct Skylar stackings of 16 cents are:
16p, 11p1n, 1n11p, 6p2n, 2n6p, 1p3n, 3n1p, 1p1d, 1d1p, 1p1n1d, 1p1d1n, 1n1p1d, 1n1d1p, 1d1p1n, 1d1n1p

Crossrefs

Cf. A001299.

Sequence in context: A369451 A029088 A253591 * A035367 A042959 A147815

Adjacent sequences: A129260 A129261 A129262 * A129264 A129265 A129266

Keyword

nonn

Author

Andrew V. Sutherland, Aug 20 2007

Status

approved