|
%I #46 Jul 31 2021 22:58:26
%S 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,
%T 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,
%U 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
%N The number of ways of writing n as an ordered sum of a triangular number (A000217), a square (A000290) and a pentagonal number (A000326).
%C 0 and 1 are triangular numbers, square numbers and pentagonal numbers.
%C 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
%C 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
%C a(n) > 0 for all n < 10^10. - _Robert G. Wilson v_, Aug 20 2016
%C 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.
%C 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, ..., .
%C Conjectured lists of numbers that are represented in k >= 1 ways:
%C 1: 0;
%C 2: 3, 18;
%C 3: 1, 2, 4, 8, 9, 13, 14, 35, 98, 168;
%C 4: 5, 6, 7, 21, 25, 30, 34, 39, 43, 48, 63, 78;
%C 5: 10, 11, 12, 17, 20, 23, 28, 33, 69, 193, 203, 230, 243;
%C 6: 19, 24, 32, 44, 53, 55, 74, 90, 111, 130;
%C 7: 15, 16, 27, 29, 40, 46, 56, 60, 62, 68, 73, 84, 85, 95, 108, 113, 123, 135, 139, 163, 165, 273, 553;
%C 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;
%C 9: 31, 36, 38, 41, 50, 52, 58, 100, 109, 124, 160, 214, 249, 308, 358, 515; ..., .
%H Robert G. Wilson v and Robert Israel, <a href="/A240088/b240088.txt">Table of n, a(n) for n = 0..10000</a>
%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1007/s11425-015-4994-4">On universal sums of polygonal numbers</a>, Science China Mathematics, Vol. 58, No. 7 (2015), 1367-1396; arXiv:<a href="http://arxiv.org/abs/0905.0635">0905.0635</a> [math.NT], 2009-2015 [Edited, _Felix Fröhlich_, Aug 24 2016].
%H Zhi-Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;48c9be36.0905">Various new conjectures involving polygonal numbers and primes</a>, a message to the Number Theory List, May 8 2009.
%p # requires Maple 17 and up
%p with(SignalProcessing):
%p N:= 10000; # to get terms up to a(N)
%p A:= Array(0..N,datatype=float);
%p B:= Array(0..N,datatype=float);
%p C:= Array(0..N,datatype=float);
%p for i from 0 to floor(sqrt(N)) do A[i^2]:= 1 od:
%p for i from 0 to floor((1+sqrt(1+8*N))/2) do B[i*(i-1)/2]:= 1 od:
%p for i from 0 to floor((1+sqrt(1+24*N))/6) do C[i*(3*i-1)/2]:= 1 od:
%p R:= Convolution(Convolution(A,B),C);
%p R:= evalhf(map(round,R));
%p # Note that a(i) = R[i+1] for i from 0 to N
%p # _Robert Israel_, Apr 01 2014
%t 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}]
%Y Cf. A000925, A008443, A101428, A115171, A115172, A115173, A115174, A115175, A115176, A115177, A144642.
%Y Cf. A160324, A160325, A160326. - _Zhi-Wei Sun_, Apr 01 2014
%K nonn
%O 0,2
%A _Robert G. Wilson v_, Mar 31 2014
|
