A275999 -id:A275999 - cp's OEIS frontend

A240088 The number of ways of writing n as an ordered sum of a triangular number (A000217), a square (A000290) and a pentagonal number (A000326).

1, 3, 3, 2, 3, 4, 4, 4, 3, 3, 5, 5, 5, 3, 3, 7, 7, 5, 2, 6, 5, 4, 8, 5, 6, 4, 8, 7, 5, 7, 4, 9, 6, 5, 4, 3, 9, 12, 9, 4, 7, 9, 8, 4, 6, 8, 7, 8, 4, 8, 9, 10, 9, 6, 10, 6, 7, 10, 9, 8, 7, 11, 7, 4, 10, 8, 10, 10, 7, 5, 10, 14, 11, 7, 6, 11, 10, 10, 4, 11, 10, 10, 13, 8, 7, 7, 13, 12, 8, 8, 6, 10, 17, 8, 10, 7, 16, 10, 3, 12, 9

Offset: 0

Robert G. Wilson v, Mar 31 2014

nonn

0 and 1 are triangular numbers, square numbers and pentagonal numbers.
It is conjectured that a(n) is always positive - this is one of the conjectures in Conjecture 1.1 of Sun (2009). - N. J. A. Sloane, Apr 01 2014
Note that both the conjecture in A160325 and the conjecture in A160324 imply that a(n) is always positive. - Zhi-Wei Sun, Apr 01 2014
a(n) > 0 for all n < 10^10. - Robert G. Wilson v, Aug 20 2016
Least number to be represented k ways, k >= 1: 0, 3, 1, 5, 10, 19, 15, 22, 31, 51, 61, 37, 82, 71, 126, 96, 92, 136, 162, 187, 206, 276, 191, 261, 236, 247, 317, 302, 401, 292, 422, 547, 456, 544, 551, 612, 591, 577, 521, 666, 742, 726, 682, 877, 796, 1052, 961, 1046, 1171, 1027, ..., . A275999.
Greatest number (conjectured) to be represented k ways, k >= 1: 0, 18, 168, 78, 243, 130, 553, 455, 515, 658, 865, 945, 633, 1918, 2258, 1385, 1583, 2828, 2135, 2335, 2785, 4533, 3168, 3478, 2790, 3868, 4193, 7328, 4953, 5278, 6390, 8148, 8015, 4585, 9160, 10485, 7613, 12333, 12025, 10178, 9923, 9720, 12558, 11340, 17420, 11753, 14893, 16155, 16415, 14343, ..., .
Conjectured lists of numbers that are represented in k >= 1 ways:
1: 0;
2: 3, 18;
3: 1, 2, 4, 8, 9, 13, 14, 35, 98, 168;
4: 5, 6, 7, 21, 25, 30, 34, 39, 43, 48, 63, 78;
5: 10, 11, 12, 17, 20, 23, 28, 33, 69, 193, 203, 230, 243;
6: 19, 24, 32, 44, 53, 55, 74, 90, 111, 130;
7: 15, 16, 27, 29, 40, 46, 56, 60, 62, 68, 73, 84, 85, 95, 108, 113, 123, 135, 139, 163, 165, 273, 553;
8: 22, 26, 42, 45, 47, 49, 59, 65, 83, 88, 89, 93, 112, 119, 125, 134, 140, 144, 186, 205, 233, 244, 320, 405, 455;
9: 31, 36, 38, 41, 50, 52, 58, 100, 109, 124, 160, 214, 249, 308, 358, 515; ..., .

Robert G. Wilson v and Robert Israel, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, Science China Mathematics, Vol. 58, No. 7 (2015), 1367-1396; arXiv:0905.0635 [math.NT], 2009-2015 [Edited, _Felix Fröhlich_, Aug 24 2016].
Zhi-Wei Sun, Various new conjectures involving polygonal numbers and primes, a message to the Number Theory List, May 8 2009.

Cf. A000925, A008443, A101428, A115171, A115172, A115173, A115174, A115175, A115176, A115177, A144642.

Cf. A160324, A160325, A160326. - Zhi-Wei Sun, Apr 01 2014

# requires Maple 17 and up
with(SignalProcessing):
N:= 10000;  # to get terms up to a(N)
A:= Array(0..N,datatype=float);
B:= Array(0..N,datatype=float);
C:= Array(0..N,datatype=float);
for i from 0 to floor(sqrt(N)) do A[i^2]:= 1 od:
for i from 0 to floor((1+sqrt(1+8*N))/2) do B[i*(i-1)/2]:= 1 od:
for i from 0 to floor((1+sqrt(1+24*N))/6) do C[i*(3*i-1)/2]:= 1 od:
R:= Convolution(Convolution(A,B),C);
R:= evalhf(map(round,R));
# Note that a(i) = R[i+1] for i from 0 to N
# Robert Israel, Apr 01 2014

p = Table[n (3n - 1)/2, {n, 0, 26}]; s = Table[n^2, {n, 0, 32}]; t = Table[n (n + 1)/2, {n, 0, 45}]; a = Sort@ Flatten@ Table[ p[[i]] + s[[j]] + t[[k]], {i, 26}, {j, 32}, {k, 45}]; Table[ Count[a, n], {n, 0, 105}]

A327792 a(n) is the greatest nonnegative number which has a partition into a triangular number (A000217), a square number (A000290), and a pentagonal number (A000326) in n different ways.

Original entry on oeis.org

0, 18, 168, 78, 243, 130, 553, 455, 515, 658, 865, 945, 633, 1918, 2258, 1385, 1583, 2828, 2135, 2335, 2785, 4533, 3168, 3478, 2790, 3868, 4193, 7328, 4953, 5278, 6390, 8148, 8015, 4585, 9160, 10485, 7613, 12333, 12025, 10178, 9923, 9720, 12558, 11340, 17420, 11753, 14893, 16155, 16415, 14343, 18053, 19803, 16608, 27283

Offset: 1

Robert G. Wilson v, Sep 25 2019

nonn

The largest nonnegative number k such that A240088(k) = n.

			a(0) does not exist since all numbers can be represented as the sum of a triangular, square & pentagonal number.

Robert G. Wilson v, Table of n, a(n) for n = 1..200

Cf. A000217, A000290, A000326, A240088, A275999, A327793.

f[n_] := Block[{j, k = 1, lenq, lenr, v = {}, t = PolygonalNumber[3, Range[0, 1 + Sqrt[2 n]]], s = PolygonalNumber[4, Range[0, 1 + Sqrt[n]]], p = PolygonalNumber[5, Range[0, 2 + Sqrt[2 n/3]]]}, u = Select[Union[Join[t, s, p]], # < n + 1 &]; q = IntegerPartitions[n, {3}, u]; lenq = 1 + Length@q; While[k < lenq, j = 1; r = q[[k]]; rr = Permutations@r; lenr = 1 + Length@rr; While[j < lenr, If[ MemberQ[t, rr[[j, 1]]] && MemberQ[s, rr[[j, 2]]] && MemberQ[p, rr[[j, 3]]], AppendTo[v, rr[[j]]]]; j++]; k++]; Length@v];

A327793 The number of nonnegative numbers that can be partitioned into a triangular number (A000217), a square number (A000290), and a pentagonal number (A000326) in n different ways.

Original entry on oeis.org

1, 2, 10, 12, 13, 10, 23, 25, 16, 36, 31, 34, 27, 45, 36, 50, 68, 61, 53, 68, 57, 72, 60, 59, 61, 87, 85, 88, 82, 97, 91, 106, 95, 98, 127, 93, 111, 125, 127, 124, 109, 127, 152, 122, 114, 146, 147, 132, 157, 169, 118, 180, 156, 158, 163, 168, 180, 178, 190, 184, 187, 196, 207, 191, 210, 204, 207, 206, 190, 227, 231, 203, 195, 219, 264

Offset: 1

Robert G. Wilson v, Sep 25 2019

nonn

The number of nonnegative numbers k such that A240088(k) = n.

			a(0) does not exist since all numbers can be represented as the sum of a triangular, square and pentagonal number;
a(1) = 1 because A240088({0}) = 1;
a(2) = 2 because A240088({3, 18}) = 2;
a(3) = 10 because A240088({1, 2, 4, 8, 9, 13, 14, 35, 98, 168}) = 3;
a(4) = 12 because A240088({5, 6, 7, 21, 25, 30, 34, 39, 43, 48, 63, 78}) = 4;
a(5) = 13 because A240088({10, 11, 12, 17, 20, 23, 28, 33, 69, 193, 203, 230, 243}) = 5;
a(6) = 10 because A240088({19, 24, 32, 44, 53, 55, 74, 90, 111, 130}) = 6;
a(7) = 23 because A240088({15, 16, 27, 29, 40, 46, 56, 60, 62, 68, 73, 84, 85, 95, 108, 113, 123, 135, 139, 163, 165, 273, 553}) = 7; etc.

Cf. A000217, A000290, A000326, A240088, A275999, A327792.

f[n_] := Block[{j, k = 1, lenq, lenr, v = {}, t = PolygonalNumber[3, Range[0, 1 + Sqrt[2 n]]], s = PolygonalNumber[4, Range[0, 1 + Sqrt[n]]], p = PolygonalNumber[5, Range[0, 2 + Sqrt[2 n/3]]]}, u = Select[Union[Join[t, s, p]], # < n + 1 &]; q = IntegerPartitions[n, {3}, u]; lenq = 1 + Length@q; While[k < lenq, j = 1; r = q[[k]]; rr = Permutations@r; lenr = 1 + Length@rr; While[j < lenr, If[ MemberQ[t, rr[[j, 1]]] && MemberQ[s, rr[[j, 2]]] && MemberQ[p, rr[[j, 3]]], AppendTo[v, rr[[j]]]]; j++]; k++]; Length@v];

cp's OEIS Frontend

A240088 The number of ways of writing n as an ordered sum of a triangular number (A000217), a square (A000290) and a pentagonal number (A000326).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A327792 a(n) is the greatest nonnegative number which has a partition into a triangular number (A000217), a square number (A000290), and a pentagonal number (A000326) in n different ways.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A327793 The number of nonnegative numbers that can be partitioned into a triangular number (A000217), a square number (A000290), and a pentagonal number (A000326) in n different ways.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica