A160324 -id:A160324 - cp's OEIS frontend

A160325 Number of ways to express n=0,1,2,... as the sum of a triangular number, an even square and a pentagonal number.

1, 2, 1, 1, 2, 3, 3, 2, 2, 1, 3, 3, 2, 1, 1, 5, 3, 3, 2, 4, 3, 2, 6, 2, 2, 2, 5, 4, 3, 3, 1, 4, 4, 3, 1, 1, 5, 7, 5, 3, 4, 6, 4, 3, 4, 5, 2, 3, 3, 5, 4, 5, 5, 2, 6, 2, 5, 5, 5, 3, 3, 6, 3, 2, 5, 4, 6, 6, 3, 3, 6, 9, 6, 5, 4, 5, 5, 6, 2, 7, 4, 3, 6, 6, 4, 2, 7, 7, 3, 3, 4, 5, 8, 5, 5, 5, 8, 4, 2, 4, 6, 6, 7, 6, 4

Offset: 0

Zhi-Wei Sun, May 08 2009

nonn

In April 2009, Zhi-Wei Sun conjectured that a(n)>0 for every n=0,1,2,3,... Note that pentagonal numbers are more sparse than squares. It is known that any positive integer can be written as the sum of a triangular number, a square and an even square (or an odd square).

			For n=15 the a(15)=5 solutions are 3+0+12, 6+4+5, 10+0+5, 10+4+1, 15+0+0.

Zhi-Wei Sun, Table of n, a(n) for n = 0..60000
B. K. Oh and Z. W. Sun, Mixed sums of squares and triangular numbers (III), J. Number Theory 129(2009), 964-969.
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.
Zhi-Wei Sun, Various new conjectures involving polygonal numbers and primes (a message to Number Theory List), 2009.
Zhi-Wei Sun, Mixed Sums of Primes and Other Terms (a webpage).
Z. W. Sun, On universal sums of polygonal numbers, preprint, arXiv:0905.0635 [math.NT], 2009-2015.

Cf. A000217, A000290, A000326, A160324.

SQ[x_]:=x>-1&&IntegerPart[Sqrt[x]]^2==x RN[n_]:=Sum[If[SQ[8(n-4y^2-(3z^2-z)/2)+1],1,0], {y,0,Sqrt[n/4]},{z,0,Sqrt[n-4y^2]}] Do[Print[n," ", RN[n]],{n,0,60000}]

a(n) = |{: x,y,z=0,1,2,... & x(x+1)/2+4y^2+(3z^2-z)/2}|.

More terms copied from author's b-file by Hagen von Eitzen, Jul 20 2009

A160326 Number of ways to express n=0,1,2,... as the sum of two squares and a pentagonal number.

Original entry on oeis.org

1, 3, 3, 1, 2, 5, 4, 1, 1, 5, 6, 2, 1, 5, 5, 2, 4, 6, 5, 1, 3, 6, 5, 3, 1, 8, 8, 4, 2, 4, 8, 4, 5, 1, 4, 5, 4, 10, 6, 6, 5, 8, 6, 1, 3, 6, 6, 4, 6, 4, 7, 8, 8, 8, 5, 7, 4, 4, 6, 5, 6, 8, 7, 4, 8, 8, 6, 5, 4, 7, 7, 8, 7, 7, 8, 8, 8, 7, 3, 4, 12, 4, 4, 7, 3, 13, 12, 12, 5, 2, 12, 4, 5, 6, 6, 8, 10, 8, 3, 5, 11

Offset: 0

Zhi-Wei Sun, May 08 2009

nonn

In April 2009, Zhi-Wei Sun conjectured that a(n)>0 for every n=0,1,2,3,... Note that pentagonal numbers are more sparse than squares. The Gauss-Legendre theorem asserts that n is the sum of three squares if and only if it is not of the form 4^a(8b+7) (a,b=0,1,2,...).

			For n=5 the a(5)=5 solutions are 0+0+5, 1+4+0, 4+1+0, 0+4+1, 4+0+1.

Zhi-Wei Sun, Table of n, a(n) for n = 0..50000
B. K. Oh and Z. W. Sun, Mixed sums of squares and triangular numbers (III), J. Number Theory 129(2009), 964-969.
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.
Zhi-Wei Sun, Various new conjectures involving polygonal numbers and primes (a message to Number Theory List), 2009.
Zhi-Wei Sun, Mixed Sums of Primes and Other Terms (a webpage).
Z. W. Sun, On universal sums of polygonal numbers, preprint, arXiv:0905.0635 [math.NT], 2009-2015.

Cf. A000290, A000326, A160324, A160325.

SQ[x_]:=x>-1&&IntegerPart[Sqrt[x]]^2==x RN[n_]:=Sum[If[SQ[n-y^2-(3z^2-z)/2],1,0], {y,0,Sqrt[n]},{z,0,Sqrt[n-y^2]}] Do[Print[n," ", RN[n]],{n,0,50000}]

a(n) = |{: x,y=0,1,2,... & x^2+y^2+(3z^2-z)/2=n}|.

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).

Original entry on oeis.org

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}]

A287616 Number of ways to write n as x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2 with x,y,z nonnegative integers.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 3, 1, 3, 1, 3, 3, 2, 4, 2, 3, 3, 3, 4, 3, 2, 5, 1, 2, 4, 3, 5, 4, 5, 4, 4, 3, 6, 3, 3, 2, 5, 2, 3, 7, 3, 7, 2, 6, 3, 5, 6, 7, 2, 4, 6, 3, 7, 2, 8, 4, 2, 6, 6, 3, 8, 3, 4, 6, 3, 7, 5, 6, 7, 4, 6, 9, 5, 6, 4, 4, 3, 4, 9, 5, 6

Offset: 0

Zhi-Wei Sun, May 27 2017

nonn

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 1, 2, 4, 7, 9, 22.
It was proved in arXiv:1502.03056 that each n = 0,1,2,... can be written as x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2 with x,y,z integers. The author would like to offer 135 US dollars as the prize for the first proof of the conjecture that a(n) is always positive.
See over 400 similar conjectures in the linked a-file.

			a(4) = 1 since 4 = 1*(1+1)/2 + 0*(3*0+1)/2 + 1*(5*1+1)/2.
a(7) = 1 since 7 = 0*(0+1)/2 + 2*(3*2+1)/2 + 0*(5*0+1)/2.
a(9) = 1 since 9 = 3*(3+1)/2 + 0*(3*0+1)/2 + 1*(5*1+1)/2.
a(22) = 1 since 22 = 5*(5+1)/2 + 2*(3*2+1)/2 + 0*(5*0+1)/2.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, List of conjectural tuples (a,b,c,d,e,f) with {x*(ax+b)/2 + y*(cy+d)/2 + z*(ez+f)/2: x,y,z = 0,1,2,...} = {0,1,2,...}
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127 (2007), 103-113.
Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58 (2015), No. 7, 1367-1396.
Zhi-Wei Sun, On universal sums x(ax+b)/2+y(cy+d)/2+z(ez+f)/2, arXiv:1502.03056 [math.NT], 2015-2017.

Cf. A000217, A000290, A005449, A160324, A160325, A160326, A254668, A286944.

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
Do[r=0;Do[If[TQ[n-x(3x+1)/2-y(5y+1)/2],r=r+1],{x,0,(Sqrt[24n+1]-1)/6},{y,0,(Sqrt[40(n-x(3x+1)/2)+1]-1)/10}];Print[n," ",r],{n,0,80}]

A085263 Number of ways to write n as the sum of a squarefree number (A005117) and a positive square (A000290).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 0, 3, 3, 2, 2, 3, 3, 2, 2, 3, 4, 2, 1, 4, 4, 2, 1, 5, 4, 3, 2, 2, 5, 2, 3, 6, 6, 3, 2, 6, 4, 3, 2, 5, 6, 3, 2, 5, 6, 3, 2, 4, 6, 4, 3, 4, 6, 4, 1, 7, 5, 3, 3, 7, 6, 4, 4, 6, 8, 3, 3, 6, 7, 2, 4, 8, 5, 4, 3, 7, 9, 4, 2, 8, 9, 4, 3, 6, 6, 5, 4, 7, 9, 5, 3, 8, 4, 3, 5, 9

Offset: 1

Reinhard Zumkeller, Jun 23 2003

a(A085265(n))>0; a(A085266(n))=1; a(A085267(n))>1.
a(A085264(n))=n and a(i)<>n for i < A085264(n).
First occurrence of k: 2, 6, 11, 23, 30, 38, 62, 71, 83, 110, 138, 155, 182, 203, 227, 263, 302, 327, 383, 435, 447, 503, 542, 602, 635, ..., . Conjecture: For each k above, there is a finite number of terms; for example, only the two numbers 1 and 13 cannot be represented as the sum of a squarefree number and a square. The number of k terms beginning with 0: 2, 9, 19, 27, 38, 36, 57, 63, 62, 74, 94, ..., . - Robert G. Wilson v, May 16 2014

			a(11)=3:
11 = 1 + 10 = A000290(1) + A005117(7)
   = 4 + 7  = A000290(2) + A005117(6)
   = 9 + 2  = A000290(3) + A005117(2).

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square Numbers.
Eric Weisstein's World of Mathematics, Squarefree
Robert G. Wilson v, Plot of first 100000 terms

Cf. A000290, A002636, A005117, A115288, A160324, A160325, A160326, A240088, A242442, A242443.

f[n_] := Count[ SquareFreeQ@# & /@ (n - Range[1, Floor[ Sqrt[ n]]]^2), True]; Array[f, 105] (* Robert G. Wilson v, May 16 2014 *)

a(n) = sum(k=1, n-1, issquare(k) * issquarefree(n-k)); \\ Michel Marcus, Oct 30 2020

a(n+1) = Sum_{k=1..n} A008966(k)*A010052(n-k+1). - Reinhard Zumkeller, Nov 04 2009
a(n) < sqrt(n). - Robert G. Wilson v, May 17 2014
G.f.: (Sum_{i>=1} x^(i^2))*(Sum_{j>=1} mu(j)^2*x^j). - Ilya Gutkovskiy, Feb 06 2017

A242442 Number of ways of writing n, a positive integer, as an unordered sum of a triangular number (A000217), an odd square (A016754) and a pentagonal number (A000326).

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 2, 2, 2, 4, 2, 0, 2, 2, 2, 2, 3, 4, 2, 3, 3, 2, 4, 3, 5, 2, 2, 3, 2, 4, 5, 4, 1, 3, 3, 4, 1, 2, 3, 5, 5, 1, 3, 5, 5, 4, 4, 4, 4, 2, 5, 4, 5, 4, 5, 4, 2, 5, 4, 4, 4, 4, 2, 4, 5, 5, 2, 2, 6, 5, 4, 2, 4, 6, 7, 7, 2, 3, 5, 6, 5, 5, 5, 2, 5, 9, 3, 5, 2, 8, 6, 1, 8, 3

Offset: 1

Robert G. Wilson v, May 14 2014

It is conjectured that only 18 cannot be so represented. See Sun, p. 4, Remark 1.2 (b).

Zhi-Wei Sun, On Universal Sums Of Polygonal Numbers, arXiv:0905.0635v18 [math.NT] 26 Oct 2011

Cf. A002636, A115288, A160324, A160325, A160326, A240088, A242443.

planeFigurative[n_, r_] := (n - 2) Binomial[r, 2] + r; s = Sort@ Flatten@ Table[ planeFigurative[3, i] + planeFigurative[4, j] + planeFigurative[5, k], {i, 0, 20}, {j, 1, 11, 2}, {k, 0, 8}]; Table[ Count[s, n], {n, 0, 104}]

A242443 Number of ways of writing n, a positive integer, as an unordered sum of a triangular number (A000217), an even square (A016742) and a generalized pentagonal number (A001318).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, May 14 2014

It is conjectured (1.1) and then proved by theorem 1.2 that all positive integers can be so represented [Sun, pp. 4-5].

Zhi-Wei Sun, On Universal Sums Of Polygonal Numbers, arXiv:0905.0635v18 [math.NT] 26 Oct 2011

Cf. A002636, A115288, A160324, A160325, A160326, A240088, A242442.

planeFigurative[n_, r_] := pf[n, r] = (n - 2) Binomial[r, 2] + r; s = Sort@ Table[ planeFigurative[3, i] + planeFigurative[3, j] + planeFigurative[3, k], {i, 0, 14}, {j, 0, 10, 2}, {k, -8, 8}]; Table[ Count[s, n], {n, 0, 50}]

A254668 Number of ways to write n as the sum of a square, a second pentagonal number, and a hexagonal number.

Original entry on oeis.org

1, 2, 2, 2, 2, 1, 2, 3, 3, 3, 2, 2, 3, 1, 1, 3, 5, 6, 2, 3, 1, 2, 4, 2, 4, 3, 4, 3, 3, 2, 4, 7, 4, 4, 2, 2, 4, 3, 3, 4, 3, 5, 5, 3, 6, 3, 5, 4, 2, 4, 4, 6, 5, 3, 2, 6, 5, 7, 4, 3, 2, 4, 4, 4, 7, 3, 8, 4, 5, 3, 5, 6, 8, 3, 2, 3, 4, 9, 2, 8, 3, 7, 7, 4, 5, 5, 4, 4, 4, 6, 5, 4, 6, 7, 9, 2, 8, 4, 3, 4, 3

Offset: 0

Zhi-Wei Sun, Feb 04 2015

nonn

Conjecture: a(n) > 0 for all n. Also, a(n) = 1 only for n = 0, 5, 13, 14, 20, 112, 125.
Compare this conjecture with the conjecture in A160324.
The conjecture that a(n) > 0 for all n = 0,1,2,... appeared in Conjecture 1.2(ii) of the author's JNT paper in the links. - Zhi-Wei Sun, Oct 03 2016

			a(20) = 1 since 20 = 2^2 + 3*(3*3+1)/2 + 1*(2*1-1).
a(112) = 1 since 112 = 7^2 + 6*(3*6+1)/2 + 2*(2*2-1).
a(125) = 1 since 125 = 5^2 + 8*(3*8+1)/2 + 0*(2*0-1).

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.
Zhi-Wei Sun, On x(ax+1)+y(by+1)+z(cz+1) and x(ax+b)+y(ay+c)+z(az+d), J. Number Theory 171(2017), 275-283.

Cf. A000290, A000384, A005449, A160324, A160325, A254661.

SQ[n_]:=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-y(3y+1)/2-z(2z-1)],r=r+1],{y,0,(Sqrt[24n+1]-1)/6},{z,0,(Sqrt[8(n-y(3y+1)/2)+1]+1)/4}];
Print[n," ",r];Continue,{n,0,100}]

A165141 The least positive integer that can be written in exactly n ways as the sum of a square, a pentagonal number and a hexagonal number.

Original entry on oeis.org

3, 9, 1, 6, 16, 36, 50, 37, 66, 82, 167, 121, 162, 236, 226, 276, 302, 446, 478, 532, 457, 586, 677, 521, 666, 852, 976, 877, 1006, 1046, 1277, 1381, 1857, 1556, 1507, 1657, 1832, 1732, 2336, 2299, 2007, 2677, 2326, 2117, 2591, 2502, 2516, 2592, 3106, 3557

Offset: 1

Zhi-Wei Sun, Sep 05 2009

On Sep 04 2009, Zhi-Wei Sun conjectured that the sequence A160324 contains every positive integer, i.e., for each positive integer n there exists a positive integer s which can be written in exactly n ways as the sum of a square, a pentagonal number and a hexagonal number. Based on this conjecture we create the current sequence. It seems that 0.9 < a(n)/n^2 < 1.6 for n > 33. Zhi-Wei Sun conjectured that a(n)/n^2 has a limit c with 1.1 < c < 1.2. On Sun's request, his friend Qing-Hu Hou produced a list of a(n) for n = 1..913 (see the b-file).

			For n=5 the a(5)=16 solutions are 0^2+1+15 = 1^2+0+15 = 2^2+12+0 = 3^2+1+6 = 4^2+0+0 = 16.

Robert G. Wilson v, Table of n, a(n) for n = 1..2658(terms 1..913 from Quig-Hu Hou)
F. Ge and Z. W. Sun, On some universal sums of generalized polygonal numbers, preprint, arXiv:0906.2450 [math.NT], 2009-2016.
M. B. Nathanson, A short proof of Cauchy's polygonal number theorem, Proc. Amer. Math. Soc. 99(1987), 22-24.
Zhi-Wei Sun, A challenging conjecture on sums of polygonal number (a message to Number Theory List), 2009.
Zhi-Wei Sun, Polygonal numbers, primes and ternary quadratic forms (a talk given at a number theory conference), 2009.
Zhi-Wei Sun, Mixed Sums of Primes and Other Terms (a webpage).
Zhi-Wei Sun, On universal sums of polygonal numbers, preprint, arXiv:0905.0635 [math.NT], 2009-2015.

Cf. A160324, A000290, A000326, A000384, A160325, A160326.

SQ[x_] := x>-1 && IntegerPart[Sqrt[x]]^2==x;
RN[n_] := RN[n] = Sum[If[SQ[n - (3 y^2 - y)/2 - (2 z^2 - z)], 1, 0], {y, 0, (1 + Sqrt[1 + 24 n])/6}, {z, 0, (1 + Sqrt[8 (n - (3 y^2 - y)/2) + 1])/4}] (* iterators modified by Robert G. Wilson v, Apr 05 2025 *);
Do[Do[If[RN[m]==n,Print[n," ", m];Goto[aa]],{m,1,1000000}];Label[aa];Continue,{n,1,100}]
f = Compile[{{n, Integer}}, Block[{cnt = 0, pentind, hex, hexind = Floor[(1 + Sqrt[8n +1])/4]}, While[ hexind > -1, hex = hexind (2 hexind - 1); pentind = Floor[(1 + Sqrt[1 + 24 (n - hex)])/6]; While[pentind > -1, If[ Floor[ Sqrt[ n - hex - pentind (3 pentind - 1)/2]]^2 == n - hex - pentind (3 pentind - 1)/2, cnt++]; pentind--]; hexind--]; cnt]]; t[] := 0; k = 1; While[k < 100001, If[ t[f[k]] == 0, t[f[k]] = k]; k++]; t /@ Range@ 250 (* Robert G. Wilson v, Apr 04 2025 *)

a(n) = min{m>0: m=x^2+(3y^2-y)/2+(2z^2-z) has exactly n solutions with x,y,z=0,1,2,...}.

A374406 Number of ways to write n as an ordered sum of a triangular number, a square and a hexagonal number.

Original entry on oeis.org

1, 3, 3, 2, 3, 3, 3, 5, 3, 2, 6, 5, 3, 3, 1, 5, 9, 5, 3, 5, 5, 4, 7, 3, 2, 9, 5, 4, 5, 6, 6, 8, 8, 2, 6, 4, 5, 11, 8, 3, 6, 5, 4, 9, 4, 7, 11, 8, 2, 5, 8, 7, 13, 5, 6, 10, 7, 6, 4, 7, 8, 9, 4, 2, 11, 12, 5, 12, 6, 3, 15, 10, 6, 9, 7, 4, 12, 6, 5, 8, 9, 10, 14, 7, 4, 15, 4, 9, 7, 4, 5, 12, 15, 7, 10, 10, 7, 13, 10, 3

Offset: 0

Seiichi Manyama, Jul 08 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000217, A000290, A000384, A160324, A240088.

G.f.: (Sum_{k>=0} x^(k*(k+1)/2)) * (Sum_{k>=0} x^(k^2)) * (Sum_{k>=0} x^(k*(2*k-1))).

cp's OEIS Frontend

A160325 Number of ways to express n=0,1,2,... as the sum of a triangular number, an even square and a pentagonal number.

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A160326 Number of ways to express n=0,1,2,... as the sum of two squares and a pentagonal number.

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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).

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Maple

Mathematica

A287616 Number of ways to write n as x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2 with x,y,z nonnegative integers.

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Mathematica

A085263 Number of ways to write n as the sum of a squarefree number (A005117) and a positive square (A000290).

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Mathematica

PARI

Formula

A242442 Number of ways of writing n, a positive integer, as an unordered sum of a triangular number (A000217), an odd square (A016754) and a pentagonal number (A000326).

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Mathematica

A242443 Number of ways of writing n, a positive integer, as an unordered sum of a triangular number (A000217), an even square (A016742) and a generalized pentagonal number (A001318).

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Mathematica

A254668 Number of ways to write n as the sum of a square, a second pentagonal number, and a hexagonal number.

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