OFFSET
0,2
COMMENTS
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; ..., .
LINKS
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.
MAPLE
# 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
MATHEMATICA
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Mar 31 2014
STATUS
approved