

A308656


Number of ways to write n as (2^a*9^b)^2 + c*(2c+1) + d*(3d+1), where a and b are nonnegative integers, and c and d are integers.


5



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

1,4


COMMENTS

Note that {x*(2x+1): x is an integer} = {n*(n+1)/2: n = 0,1,2,...}.
Conjecture 1: a(n) > 0 for all n > 0.
Conjecture 2: If f(x) is one of the polynomials x*(4x+1), x*(5x+2), x*(5x+4), x*(7x+3)/2 and x(7x+5)/2, then any positive integer n can be written as (2^a*9^b)^2 + f(c) + d*(3d+1)/2, where a and b are nonnegative integers, and c and d are integers.
Conjecture 3: Let r be 1 or 2. Then any positive integer n can be written as (2^a*7^b)^2 + c*(2c+1) + d*(3d+r), where a and b are nonnegative integers, and c and d are integers.
Conjecture 4: If g(x) is one of the polynomials x*(x+1), x*(4x+3), x*(7x+1)/2, x*(7x+3)/2 and x*(7x+5)/2, then any positive integer n can be written as (2^a*7^b)^2 + g(c) + d*(3d+1)/2, where a and b are nonnegative integers, and c and d are integers.
We have verified a(n) > 0 for all n = 1..10^8, and Conjectures 24 for all n = 1..10^6.
See also A308640, A308641, and A308644 for similar conjectures.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000


EXAMPLE

a(13) = 1 with 13 = (2^0*9^0)^2 + 2*(2*2+1) + (1)*(3*(1)+1).
a(3515) = 1 with 3515 = (2^0*9^1)^2 + 0*(2*0+1) + (34)*(3*(34)+1).
a(124076) = 1 with 124076 = (2^3*9^1)^2 + 206*(2*206+1) + 106*(3*106+1).
a(141518) = 1 with 141518 = (2^1*9^2)^2 + (188)*(2*(188)+1) + 122*(3*122+1).
a(345402) = 1 with 345402 = (2^7*9^0)^2 + 18*(2*18+1) + (331)*(3*(331)+1).


MATHEMATICA

PQ[n_]:=PQ[n]=IntegerQ[Sqrt[12n+1]];
tab={}; Do[r=0; Do[If[PQ[n81^a*4^bx(2x+1)], r=r+1], {a, 0, Log[81, n]}, {b, 0, Log[4, n/81^a]}, {x, Floor[(Sqrt[8(n81^a*4^b)+1]+1)/4], (Sqrt[8(n81^a*4^b)+1]1)/4}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]


CROSSREFS

A000079, A000217, A000420, A001019, A001318, A308566, A308584, A308621, A308623, A308640, A308641, A308644.
Sequence in context: A291674 A265157 A054263 * A178775 A124874 A230258
Adjacent sequences: A308653 A308654 A308655 * A308657 A308658 A308659


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jun 14 2019


STATUS

approved



