

A254661


Number of ways to write n as the sum of a triangular number, an even square and a second pentagonal number.


3



1, 1, 1, 2, 1, 2, 2, 3, 2, 1, 3, 1, 3, 1, 2, 2, 3, 4, 2, 4, 1, 5, 3, 2, 2, 3, 4, 2, 3, 3, 3, 3, 4, 3, 3, 1, 5, 3, 3, 4, 4, 4, 3, 5, 5, 4, 5, 5, 2, 2, 2, 6, 5, 2, 4, 3, 2, 6, 3, 6, 2, 5, 5, 4, 5, 3, 7, 5, 4, 1, 4, 6, 8, 3, 5, 1, 6, 6, 5, 6, 4, 6, 6, 4, 4, 7, 3, 5, 2, 5, 2, 5, 5, 7, 6, 2, 7, 6, 4, 4, 5
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,4


COMMENTS

Conjecture: (i) a(n) > 0 for all n. Also, a(n) = 1 only for n = 0, 1, 2, 4, 9, 11, 13, 20, 35, 69, 75, 188.
(ii) For each a = 2,3, any nonnegative integer n can be written as x(x+1)/2 + a*y^2 + z*(3*z+1)/2 with x,y,z nonnegative integers.
Compare part (i) of this conjecture with the conjecture in A160325.


LINKS



EXAMPLE

a(20) = 1 since 20 = 1*2/2 + 2^2 + 3*(3*3+1)/2.
a(35) = 1 since 35 = 7*8/2 + 0^2 + 2*(3*2+1)/2.
a(69) = 1 since 69 = 2*3/2 + 8^2 + 1*(3*1+1)/2.
a(75) = 1 since 75 = 9*10/2 + 2^2 + 4*(3*4+1)/2.
a(188) = 1 since 188 = 1*2/2 + 0^2 + 11*(3*11+1)/2.


MATHEMATICA

TQ[n_]:=IntegerQ[Sqrt[8n+1]]
Do[r=0; Do[If[TQ[n4y^2z(3z+1)/2], r=r+1], {y, 0, Sqrt[n/4]}, {z, 0, (Sqrt[24(n4y^2)+1]1)/6}];
Print[n, " ", r]; Label[aa]; Continue, {n, 0, 100}]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



