cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A174141 Numbers congruent to k mod 25, where 0 <= k <= 9.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 150, 151, 152, 153, 154
Offset: 1

Views

Author

Rick L. Shepherd, Mar 09 2010

Keywords

Comments

Numbers whose partition into parts of sizes 1, 5, 10, and 25 having a minimal number of parts does not include a part of size 10.
For each number the partition is unique.
Complement of A174140.
Amounts in cents not including a dime 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).

Links

Crossrefs

Cf. A174138, A174139, A174140, A047201 (requires at least one part of size 1 (penny)), A008587, A053344 (minimal number of parts), A001299 (number of all such partitions).

Programs

  • Mathematica
    LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1},{0,1,2,3,4,5,6,7,8,9,25},70] (* Harvey P. Dale, May 30 2014 *)

Formula

a(n+10) = a(n) + 25 for n >= 1.
a(n)= +a(n-1) +a(n-10) -a(n-11). G.f. x^2*(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+16*x^9) / ( (1+x)*(1+x+x^2+x^3+x^4)*(x^4-x^3+x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

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

Original entry on oeis.org

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

Views

Author

Rick L. Shepherd, Mar 09 2010

Keywords

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).

Examples

			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.

Links

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).

Programs

  • 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
  • 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 *)

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

A174139 Numbers congruent to {0,1,2,3,4,10,11,12,13,14,20,21,22,23,24} mod 25.

Original entry on oeis.org

0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 85, 86, 87, 88, 89, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 110, 111, 112
Offset: 1

Views

Author

Rick L. Shepherd, Mar 09 2010

Keywords

Comments

Numbers whose partition into parts of sizes 1, 5, 10, and 25 having a minimal number of parts does not include a part of size 5.
For each number the partition is unique.
Complement of A174138.
Amounts in cents not including 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).
First differs from A032955 at n = 76. - Avi Mehra, Oct 08 2020

Links

Crossrefs

Cf. A174138, A174140, A174141, A047201 (requires at least one part of size 1 (penny)), A008587, A053344 (minimal number of parts), A001299 (number of all such partitions).

Programs

  • Mathematica
    Select[Range[0, 112], Mod[Mod[#, 25], 10] < 5 &] (* Amiram Eldar, Oct 08 2020 *)
  • PARI
    { my(table=[0,1,2,3,4, 10,11,12,13,14, 20,21,22,23,24]);
a(n) = my(r);[n,r]=divrem(n-1,15); 25*n + table[r+1]; } \\ Kevin Ryde, Oct 08 2020

Formula

a(15+n) = a(n) + 25 for n >= 1.
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = +a(n-1) +a(n-15) -a(n-16).
G.f.: x^2*(1 +x +x^2 +x^3 +6*x^4 +x^5 +x^6 +x^7 +x^8 +6*x^9 +x^10 +x^11 +x^12 +x^13+x^14) / ( (1+x+x^2) *(x^4+x^3+x^2+x+1) *(x^8-x^7+x^5-x^4+x^3-x+1) *(x-1)^2). (End)
Showing 1-3 of 3 results.

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