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.

A256321 Number of partitions of 5n into exactly 3 parts.

Original entry on oeis.org

0, 2, 8, 19, 33, 52, 75, 102, 133, 169, 208, 252, 300, 352, 408, 469, 533, 602, 675, 752, 833, 919, 1008, 1102, 1200, 1302, 1408, 1519, 1633, 1752, 1875, 2002, 2133, 2269, 2408, 2552, 2700, 2852, 3008, 3169, 3333, 3502, 3675, 3852, 4033, 4219, 4408, 4602
Offset: 0

Views

Author

Colin Barker, Mar 24 2015

Keywords

Examples

			For n=1 the 2 partitions of 5*1 = 5 are [1, 1, 3] and [1, 2, 2].

Links

Crossrefs

Cf. A033428 (6n), A256320 (4n), A256322 (7n).

Programs

  • Mathematica
    Length /@ (Total /@ IntegerPartitions[5 #, {3}] & /@ Range[0, 47]) (* Michael De Vlieger, Mar 24 2015 *)
LinearRecurrence[{1,1,0,-1,-1,1},{0,2,8,19,33,52},50] (* Harvey P. Dale, Oct 29 2017 *)
  • PARI
    concat(0, vector(40, n, k=0; forpart(p=5*n, k++, , [3,3]); k))
  • PARI
    concat(0, Vec(-x*(x^2+2*x+2)*(2*x^2+2*x+1)/((x-1)^3*(x+1)*(x^2+x+1)) + O(x^100)))

Formula

a(n) = a(n-1)+a(n-2)-a(n-4)-a(n-5)+a(n-6) for n>5.
G.f.: -x*(x^2+2*x+2)*(2*x^2+2*x+1) / ((x-1)^3*(x+1)*(x^2+x+1)).

A256322 Number of partitions of 7n into exactly 3 parts.

Original entry on oeis.org

0, 4, 16, 37, 65, 102, 147, 200, 261, 331, 408, 494, 588, 690, 800, 919, 1045, 1180, 1323, 1474, 1633, 1801, 1976, 2160, 2352, 2552, 2760, 2977, 3201, 3434, 3675, 3924, 4181, 4447, 4720, 5002, 5292, 5590, 5896, 6211, 6533, 6864, 7203, 7550, 7905, 8269, 8640
Offset: 0

Views

Author

Colin Barker, Mar 24 2015

Keywords

Examples

			For n=1 the 4 partitions of 7*1 = 7 are [1, 1, 5], [1, 2, 4], [1, 3, 3] and [2, 2, 3].

Links

Crossrefs

Cf. A033428 (6n), A256320 (4n), A256321 (5n).

Programs

  • Mathematica
    Length /@ (Total /@ IntegerPartitions[7 #, {3}] & /@ Range[0, 46]) (* Michael De Vlieger, Mar 24 2015 *)
LinearRecurrence[{1,1,0,-1,-1,1},{0,4,16,37,65,102},50] (* Harvey P. Dale, Aug 29 2024 *)
  • PARI
    concat(0, vector(40, n, k=0; forpart(p=7*n, k++, , [3,3]); k))
  • PARI
    concat(0, Vec(-x*(2*x^2+3*x+2)^2/((x-1)^3*(x+1)*(x^2+x+1)) + O(x^100)))

Formula

a(n) = a(n-1)+a(n-2)-a(n-4)-a(n-5)+a(n-6) for n>5.
G.f.: -x*(2*x^2+3*x+2)^2 / ((x-1)^3*(x+1)*(x^2+x+1)).

A264938 a(n) = n*(2*n-1) + floor(n/3).

Original entry on oeis.org

0, 1, 6, 16, 29, 46, 68, 93, 122, 156, 193, 234, 280, 329, 382, 440, 501, 566, 636, 709, 786, 868, 953, 1042, 1136, 1233, 1334, 1440, 1549, 1662, 1780, 1901, 2026, 2156, 2289, 2426, 2568, 2713, 2862, 3016, 3173, 3334, 3500, 3669, 3842, 4020, 4201, 4386, 4576, 4769
Offset: 0

Views

Author

Paul Curtz, Nov 29 2015

Keywords

Comments

Sequence extended to the left:
..., 133, 102, 76, 53, 34, 20, 9, 2, 0, 1, 6, 16, 29, 46, 68, 93, ...
Conjecture: after 0, a(n) provides the first bisection of A264041.
Conjecture: 2, 9, 20, 34, 53, 76, 102, 133, ... is A248121.

Links

Crossrefs

Second bisection of A260708.
Cf. A000217, A002264, A016777, A016969, A017197, A017641, A099837, A248121, A256320, A260699, A264041.
Cf. A171452: A000217(n-1) + A002264(n).

Programs

  • Magma
    [n*(2*n-1)+Floor(n/3): n in [0..60]]; // Vincenzo Librandi, Dec 02 2015
  • Maple
    seq(n*(2*n-1) + floor(n/3), n=0..100); # Robert Israel, Dec 02 2015
  • Mathematica
    Table[n (2 n - 1) + Floor[n/3], {n, 0, 50}] (* Vincenzo Librandi, Dec 02 2015 *)
LinearRecurrence[{2,-1,1,-2,1},{0,1,6,16,29},60] (* Harvey P. Dale, Oct 13 2020 *)
  • PARI
    concat(0, Vec(x*(1+x)^2*(1+2*x)/((1-x)^3*(1+x+x^2)) + O(x^100))) \\ Colin Barker, Dec 02 2015
  • PARI
    a(n) = n*(2*n-1) + n\3; \\ Altug Alkan, Dec 01 2015

Formula

a(n) = a(n-3) + 12*n - 20 for n>2.
From Colin Barker, Dec 02 2015: (Start)
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>4.
G.f.: x*(1+x)^2*(1+2*x) / ((1-x)^3*(1+x+x^2)).
(End)
a(n) = A000217(2n-1) + A002264(n).
a(n) + a(-n) = 3*A256320(n).
a(n +8) - a(n -7) = 20*A016777(n).
a(n+16) - a(n-14) = 20*A016969(n).
a(n+23) - a(n-22) = 20*A017197(n).
a(n+31) - a(n-29) = 20*A017641(n).
Generalization of the previous four formulas:
a(n+30*k +8) - a(n-30*k -7) = 20*(4*k+1)*(3*n+1).
a(n+30*k+16) - a(n-30*k-14) = 20*(2*k+1)*(6*n+5).
a(n+30*k+24) - a(n-30*k-21) = 20*(4*k+3)*(3*n+4).
a(n+30*k+32) - a(n-30*k-28) = 20*(2*k+2)*(6*n+11).
E.g.f.: (6*x^2+4*x-1)*exp(x)/3 + (cos(sqrt(3)*x/2)/3 +sqrt(3)*sin(sqrt(3)*x/2)/9)*exp(-x/2). - Robert Israel, Dec 02 2015

Extensions

Edited by Bruno Berselli, Dec 01 2015
Showing 1-3 of 3 results.

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