OFFSET
1,5
COMMENTS
Conjecture: a(n) > 0 for all n > 0. Equivalently, each n = 1,2,3,... can be written as w^2 + x*(x+1) + 5^y*8^z with w,x,y,z nonnegative integers.
We have verified a(n) > 0 for all n = 1..4*10^8.
See also A308566 for a similar conjecture.
a(n) > 0 for all 0 < n < 10^10. - Giovanni Resta, Jun 10 2019
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.
EXAMPLE
a(13) = 1 with 13 = 3*4/2 + 3*4/2 + 5^0*8^0.
a(48) = 1 with 48 = 5*6/2 + 7*8/2 + 5^1*8^0.
a(87) = 1 with 87 = 1*2/2 + 12*13/2 + 5^0*8^1.
a(90) = 1 with 90 = 4*5/2 + 10*11/2 + 5^2*8^0.
a(423) = 1 with 423 = 9*10/2 + 22*23/2 + 5^3*8^0.
a(517) = 1 with 517 = 17*18/2 + 24*25/2 + 5^0*8^2.
a(985) = 1 with 985 = 19*20/2 + 34*35/2 + 5^2*8^1.
a(2694) = 1 with 2694 = 7*8/2 + 68*69/2 + 5^1*8^2.
a(42507) = 1 with 42507 = 178*179/2 + 223*224/2 + 5^2*8^2.
a(544729) = 1 with 544729 = 551*552/2 + 857*858/2 + 5^5*8^1.
a(913870) = 1 with 913870 = 559*560/2 + 700*701/2 + 5^3*8^4.
a(1843782) = 1 with 1843782 = 808*809/2 + 1668*1669/2 + 5^6*8^1.
MATHEMATICA
TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
tab={}; Do[r=0; Do[If[TQ[n-5^k*8^m-x(x+1)/2], r=r+1], {k, 0, Log[5, n]}, {m, 0, Log[8, n/5^k]}, {x, 0, (Sqrt[4(n-5^k*8^m)+1]-1)/2}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jun 08 2019
STATUS
approved