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 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 (list; graph; refs; listen; history; text; internal format)
 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 2-4 for all n = 1..10^6. See also A308640, A308641, and A308644 for similar conjectures. LINKS Zhi-Wei 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[n-81^a*4^b-x(2x+1)], r=r+1], {a, 0, Log[81, n]}, {b, 0, Log[4, n/81^a]}, {x, -Floor[(Sqrt[8(n-81^a*4^b)+1]+1)/4], (Sqrt[8(n-81^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 Zhi-Wei Sun, Jun 14 2019 STATUS approved

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Last modified April 5 20:24 EDT 2020. Contains 333260 sequences. (Running on oeis4.)