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.

A214772 Number of partitions of n into parts 6, 9 or 20.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 1, 0, 0, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 3, 0, 2, 2, 1, 1, 3, 0, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 4, 1, 3, 3, 2, 2, 5, 1, 3, 4, 2, 3, 5, 2, 3, 5, 2, 3, 6, 2, 4, 5, 3, 3, 7, 2, 5, 6, 3, 4, 7, 3
Offset: 0

Views

Author

Reinhard Zumkeller, Jul 28 2012

Keywords

Comments

a(A065003(n)) = 0; a(A214777(n)) > 0.

Examples

			a(10) = 0, cf. A065003(8) = 10;
a(20) = #{20} = 1;
a(30) = #{6+6+6+6+6, 6+6+9+9} = 2;
a(40) = #{20+20} = 1;
a(50) = #{5*6+20, 6+6+9+9+20} = 2;
a(60) = #{10*6, 7*6+9+9, 4*6+4*9, 6+6*9, 20+20+20} = 5;
a(70) = #{5*6+20+20, 6+6+9+9+20+20} = 2
a(80) = #{10*6+20, 7*6+9+9+20, 4*6+4*9+20, 6+6*99+20, 4*20} = 5;
a(90) = #{15*6, 12*6+9+9, 9*6+4*9, 6*6+6*99, 5*6+3*20, 3*6+8*9, 6+6+9+9+3*20, 10*9} = 8;
a(100) = #{10*6+2*20, 7*6+9+9+2*20, 4*6+4*9+2*20, 6+6*9+2*20, 5*20} = 5.

Links

Programs

  • Haskell
    a214772 = p [6, 9, 20] where
   p _      0 = 1
   p []     _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

Formula

G.f. 1/((1-x^6)*(1-x^9)*(1-x^20)). - R. J. Mathar, Jul 30 2012

A214777 McNugget numbers: numbers of the form 6*x + 9*y + 20*z for nonnegative integers x, y, z.

Original entry on oeis.org

0, 6, 9, 12, 15, 18, 20, 21, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 28 2012

Keywords

Comments

A214772(a(n)) > 0;
complement of A065003; all numbers greater than 43 are McNugget numbers: Frobenius(6,9,20) = 43.

Links

Crossrefs

Cf. A065003, A214772.

Programs

  • Haskell
    import Data.List (findIndices)
a214777 n = a214777_list !! (n-1)
a214777_list = findIndices (> 0) a214772_list
  • Mathematica
    CoefficientList[Series[- x (x^22 - x^21 + x^17 - x^16 + x^15 - x^14 + x^13 - x^12 + x^11 - x^10 + x^9 + x^8 - 2 x^7 + x^6 + x^5 + 3 x - 6)/(1 - x)^2, {x, 0, 70}], x] (* Vincenzo Librandi, Apr 27 2015 *)
  • Python
    def A214777(n): return (0, 6, 9, 12, 15, 18, 20, 21, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 40, 41, 42)[n-1] if n<23 else n+21 # Chai Wah Wu, Feb 24 2025

Formula

G.f.: -x^2*(x^22-x^21+x^17-x^16+x^15-x^14+x^13-x^12+x^11-x^10+x^9+x^8-2*x^7+x^6+x^5+3*x-6) / (x-1)^2. - Colin Barker, Dec 13 2012
a(n) = n + 21 for n >= 23. - Robert Israel, May 01 2015

A380697 Frobenius number of the set S = {e_i+2; 1 <= i <= m}, where the e_i are the exponents in the binary expansion n = Sum_{i=1..m} 2^e_i, or 0 if GCD(S) = A326674(2*n) > 1.

Original entry on oeis.org

0, 0, 1, 0, 0, 5, 1, 0, 3, 7, 1, 11, 3, 2, 1, 0, 0, 0, 1, 0, 0, 5, 1, 19, 3, 7, 1, 7, 3, 2, 1, 0, 5, 11, 1, 17, 5, 5, 1, 23, 3, 4, 1, 6, 3, 2, 1, 29, 5, 11, 1, 9, 5, 5, 1, 9, 3, 4, 1, 3, 3, 2, 1, 0, 0, 13, 1, 0, 0, 5, 1, 27, 3, 7, 1, 11, 3, 2, 1, 0, 0, 13, 1
Offset: 1

Views

Author

Pontus von Brömssen, Jan 30 2025

Keywords

Comments

The sequence gives the Frobenius numbers of all sets of integers greater than 1, encoded by the binary expansion of n.

Examples

			For n = 262288 = 2^4+2^7+2^18, a(n) is the Frobenius number of {6, 9, 20}, i.e., the last term of A065003, so a(262288) = 43.

Links

Crossrefs

Cf. A004767, A065003, A326674.

Formula

a(n) = 1 if and only if n == 3 (mod 4) (i.e., if and only if n is in A004767).
a(n) = 2 if and only if n == 14 (mod 16).
a(2^e+2^f) = (e+1)*(f+1)-1 for nonnegative integers e and f such that e+2 and f+2 are coprime.
Showing 1-3 of 3 results.

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