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 A052343 Number of ways to write n as the unordered sum of two triangular numbers (zero allowed). 21

%I

%S 1,1,1,1,1,0,2,1,0,1,1,1,1,1,0,1,2,0,1,0,1,2,1,0,1,1,0,1,1,1,1,2,0,0,

%T 1,0,2,1,1,1,0,0,2,1,0,1,2,0,1,1,0,2,0,0,0,2,2,1,1,0,1,1,0,0,1,1,2,1,

%U 0,1,1,0,2,1,0,0,2,0,1,1,0,3,0,1,1,0,0,1,1,0,1,2,1,1,2,0,0,1,0,1,1,1

%N Number of ways to write n as the unordered sum of two triangular numbers (zero allowed).

%C Number of ways of writing n as a sum of a square and twice a triangular number (zeros allowed). - _Michael Somos_, Aug 18 2003

%C a(A020757(n))=0; a(A020756(n))>0; a(A119345(n))=1; a(A118139(n))>1. - _Reinhard Zumkeller_, May 15 2006

%C Also, number of ways to write 4n+1 as the unordered sum of two squares of nonnegative integers. - _Vladimir Shevelev_, Jan 21 2009

%C The average value of a(n) for n<=x is pi/4+O(1/sqrt(x)). - _Vladimir Shevelev_, Feb 06 2009

%H Reinhard Zumkeller, <a href="/A052343/b052343.txt">Table of n, a(n) for n = 0..10000</a>

%H V. Shevelev, <a href="http://www.arxiv.org/abs/0901.3102">Binary additive problems: recursions for numbers of representations</a>.

%H V. Shevelev, <a href="http://www.arxiv.org/abs/0902.1046">Binary additive problems: theorems of Landau and Hardy-Littlewood type</a>.

%F a(n) = ceiling(A008441(n)/2). - _Reinhard Zumkeller_, Nov 03 2009

%F G.f.: (Sum_{k>=0} x^(k^2 + k)) * (Sum_{k>=0} x^(k^2)). - _Michael Somos_, Aug 18 2003

%F Recurrence: a(n) = sum_{1<=k<=r(n)} r(2n-k^2+k) - C(r(n),2) - a(n-1) - a(n-2) - ... - a(0), n>=1,a (0)=1, where r(n)=A000194(n+1) is the nearest integer to square root of n+1. For example, since r(6)=3, a(6) = r(12) + r(10) + r(6) - C(3,2) - a(5) - ... - a(0) = 4 + 3 + 3 - 3 - 0 - 1 - 1 - 1 - 1 - 1 = 2. - _Vladimir Shevelev_, Feb 06 2009

%F a(n) = A025426(8n+2). - _Max Alekseyev_, Mar 09 2009

%F a(n) = (A002654(4n+1) + A010052(4n+1)) / 2. - _Ant King_, Dec 01 2010

%F a(2*n + 1) = A053692(n). a(4*n + 1) = A259287(n). a(4*n + 3) = A259285(n). a(6*n + 1) = A260415(n). a(6*n + 4) = A260516(n) - _Michael Somos_, Jul 28 2015

%F a(3*n) = A093518(n). a(3*n + 1) = A121444(n). a(9*n + 2) = a(n). a(9*n + 5) = a(9*n + 8) = 0. - _Michael Somos_, Jul 28 2015

%F Convolution of A005369 and A010052. - _Michael Somos_, Jul 28 2015

%e G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^6 + x^7 + x^9 + x^10 + x^11 + ...

%t Length[PowersRepresentations[4 # + 1, 2, 2]] & /@ Range[0, 101] (* _Ant King_, Dec 01 2010 *)

%t d1[k_]:=Length[Select[Divisors[k],Mod[#,4]==1&]];d3[k_]:=Length[Select[Divisors[k],Mod[#,4]==3&]];f[k_]:=d1[k]-d3[k];g[k_]:=If[IntegerQ[Sqrt[4k+1]],1/2 (f[4k+1]+1),1/2 f[4k+1]];g[#]&/@Range[0,101] (* _Ant King_, Dec 01 2010 *)

%t a[ n_] := Length @ Select[ Table[ Sqrt[n - i - i^2], {i, 0, Quotient[ Sqrt[4 n + 1] - 1, 2]}], IntegerQ]]; (* _Michael Somos_, Jul 28 2015 *)

%t a[ n_] := Length @ FindInstance[ {j >= 0, k >= 0, j^2 + k^2 + k == n}, {k, j}, Integers, 10^9]; (* _Michael Somos_, Jul 28 2015 *)

%o (PARI) {a(n) = if( n<0, 0, sum(i=0, (sqrtint(4*n + 1) - 1)\2, issquare(n - i - i^2)))}; /*_Michael Somos_, Aug 18 2003 */

%o a052343 = (flip div 2) . (+ 1) . a008441

%o -- _Reinhard Zumkeller_, Jul 25 2014

%Y Cf. A000217, A052344-A052348, A053587, A056303, A056304.

%Y Cf. A053692, A093518, A121444, A259285, A259287, A260415, A260516.

%Y Cf. A005369, A010052.

%K nonn

%O 0,7

%A _Christian G. Bower_, Jan 23 2000

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