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A308547 Number of ways to write n as a^2 + 2*b^2 + 2^c*3^d, where a,b,c,d are nonnegative integers. 3
1, 2, 3, 4, 4, 4, 4, 4, 5, 7, 5, 8, 6, 5, 6, 4, 6, 8, 7, 10, 8, 6, 4, 8, 8, 8, 10, 10, 6, 9, 6, 4, 10, 9, 11, 14, 8, 8, 9, 10, 8, 11, 8, 9, 13, 6, 5, 8, 9, 10, 11, 13, 7, 14, 8, 10, 13, 9, 11, 16, 7, 7, 13, 4, 12, 12, 10, 12, 10, 13, 5, 14, 13, 9, 17, 12, 7, 12, 6, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

As 3*(a^2 + 2*b^2 + 2^c*3^d) = (a+2*b)^2 + 2*(a-b)^2 + 2^c*3^(d+1), we have a(3*n) > 0 if a(n) > 0.

The first positive integer n with a(n) = 0 is 139571911. We also have a(142991573) = 0.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

EXAMPLE

a(1) = 1 with 1 = 0^2 + 2*0^2 + 2^0*3^0.

a(2) = 2 with 2 = 0^2 + 2*0^2 + 2^1*3^0 = 1^2 + 2*0^2 + 2^0*3^0.

a(1117) = 2 with 1117 = 10^2 + 2*12^2 + 2^0*3^6 = 19^2 + 2*18^2 + 2^2*3^3.

a(78373) = 1 with 78373 = 271^2 + 2*48^2 + 2^2*3^4.

a(448159) = 1 with 448159 = 610^2 + 2*195^2 + 2^0*3^2.

a(82816213) = 2 with 82816213 = 4353^2 + 2*5651^2 + 2^1*3^0 = 3681^2 + 2*5885^2 + 2^1*3^0.

a(90685253) = 2 with 90685253 = 7007^2 + 2*4560^2 + 2^2*3^0 = 607^2 + 2*6720^2 + 2^2*3^0.

MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

tab={}; Do[r=0; Do[If[SQ[n-3^k*2^m-2x^2], r=r+1], {k, 0, Log[3, n]}, {m, 0, Log[2, n/3^k]}, {x, 0, Sqrt[(n-3^k*2^m)/2]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]

CROSSREFS

Cf. A000079, A000244, A000290, A002479, A302984, A303637, A303656, A308411.

Sequence in context: A287422 A120510 A029117 * A195850 A253593 A174901

Adjacent sequences:  A308544 A308545 A308546 * A308548 A308549 A308550

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jun 06 2019

STATUS

approved

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Last modified September 20 08:13 EDT 2019. Contains 327214 sequences. (Running on oeis4.)