A174138 Numbers congruent to {5,6,7,8,9,15,16,17,18,19} mod 25.

5, 6, 7, 8, 9, 15, 16, 17, 18, 19, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44, 55, 56, 57, 58, 59, 65, 66, 67, 68, 69, 80, 81, 82, 83, 84, 90, 91, 92, 93, 94, 105, 106, 107, 108, 109, 115, 116, 117, 118, 119, 130, 131, 132, 133, 134, 140, 141, 142, 143, 144, 155, 156, 157

Offset

1,1

Comments

Numbers whose partition into parts of sizes 1, 5, 10, and 25 having a minimal number of parts includes a part of size 5.

For each number the partition is unique and exactly one part is of size 5.

Complement of A174139.

Amounts in cents requiring a nickel when the minimal number of coins is selected from pennies, nickels, dimes, and quarters (whether usage of bills for whole-dollar amounts is permitted or not).

For each n >= 0, floor(n/25) parts of size 25 (quarters) occur in the partition with minimal number of these parts (regardless of whether partition includes part of size 5).

Links

Table of n, a(n) for n=1..63.

Index entries for sequences related to making change.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Formula

a(10+n) = a(n) + 25 for n >= 1.

a(n) = a(n-1) + a(n-10) - a(n-11). G.f.: x*(5+x+x^2+x^3+x^4+6*x^5+x^6+x^7+x^8+x^9+6*x^10) / ( (1+x) *(x^4+x^3+x^2+x+1) *(x^4-x^3+x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

a(n) = n+9+5*floor((floor((n-1)/5)-1)/2)+10*floor(floor((n-1)/5)/2). - Wesley Ivan Hurt, Mar 22 2015

Example

As 15 = 10 + 5, 15 is a term since 5 is included and all other candidate partitions have more than two parts. Similarly, as 30 = 25 + 5, 30 is a term. However, 45 = 25 + 10 + 10 is not a term as it contains no part of size 5.

Mathematica

Table[n + 9 + 5 Floor[(Floor[(n - 1)/5] - 1)/2] + 10 Floor[Floor[(n - 1)/5]/2], {n, 100}] (* Wesley Ivan Hurt, Mar 22 2015 *)

Prog

(Magma) [n : n in [1..200] | n mod 25 in [5, 6, 7, 8, 9, 15, 16, 17, 18, 19]]; // Vincenzo Librandi, Mar 22 2015

Crossrefs

Cf. A174139, A174140, A174141, A047201 (requires at least one part of size 1 (penny)), A008587, A053344 (minimal number of parts), A001299 (number of all such partitions).

Sequence in context: A285219 A047577 A293481 * A108401 A125298 A037362

Adjacent sequences: A174135 A174136 A174137 * A174139 A174140 A174141

Keyword

easy,nonn

Author

Rick L. Shepherd, Mar 09 2010

Status

approved