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

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, 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, 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

Offset: 0

Author

Christian G. Bower, Jan 23 2000

Keywords

• nonn
• changed

Comments

Number of ways of writing n as a sum of a square and twice a triangular number (zeros allowed). - Michael Somos, Aug 18 2003

a(A020757(n))=0; a(A020756(n))>0; a(A119345(n))=1; a(A118139(n))>1. - Reinhard Zumkeller, May 15 2006

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

The average value of a(n) for n <= x is Pi/4 + O(1/sqrt(x)). - Vladimir Shevelev, Feb 06 2009

Examples

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Vladimir Shevelev, Binary additive problems: recursions for numbers of representations, arXiv:0901.3102 [math.NT], 2009-2013.
• Vladimir Shevelev, Binary additive problems: theorems of Landau and Hardy-Littlewood type, arXiv:0902.1046 [math.NT], 2009-2012.

Crossrefs

Cf. A000217, A052344, A052345 (greedy inverse), A052346, A052347, A052348, A053587, A056303, A056304.

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

Cf. A005369, A010052.

Programs

Haskell

a052343 = (flip div 2) . (+ 1) . a008441
-- Reinhard Zumkeller, Jul 25 2014

Maple

A052343 := proc(n)
    local a,t1idx,t2idx,t1,t2;
    a := 0 ;
    for t1idx from 0 do
        t1 := A000217(t1idx) ;
        if t1 > n then
            break;
        end if;
        for t2idx from t1idx do
            t2 := A000217(t2idx) ;
            if t1+t2 > n then
                break;
            elif t1+t2 = n then
                a := a+1 ;
            end if;
        end do:
    end do:
    a ;
end proc: # R. J. Mathar, Apr 28 2020

Mathematica

Length[PowersRepresentations[4 # + 1, 2, 2]] & /@ Range[0, 101] (* Ant King, Dec 01 2010 *)
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 *)
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 *)
a[ n_] := Length @ FindInstance[ {j >= 0, k >= 0, j^2 + k^2 + k == n}, {k, j}, Integers, 10^9]; (* Michael Somos, Jul 28 2015 *)

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 */

Formula

a(n) = ceiling(A008441(n)/2). - Reinhard Zumkeller, Nov 03 2009

G.f.: (Sum_{k>=0} x^(k^2 + k)) * (Sum_{k>=0} x^(k^2)). - Michael Somos, Aug 18 2003

Recurrence: a(n) = Sum_{k=1..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

a(n) = A025426(8n+2). - Max Alekseyev, Mar 09 2009

a(n) = (A002654(4n+1) + A010052(4n+1)) / 2. - Ant King, Dec 01 2010

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

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

Convolution of A005369 and A010052. - Michael Somos, Jul 28 2015