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 A052343 Number of ways to write n as the unordered sum of two triangular numbers (zero allowed). 21
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 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 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 V. Shevelev, Binary additive problems: recursions for numbers of representations, arXiv:0901.3102 [math.NT], 2009-2013. V. Shevelev, Binary additive problems: theorems of Landau and Hardy-Littlewood type, arXiv:0902.1046 [math.NT], 2009-2012. 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 EXAMPLE G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^6 + x^7 + x^9 + x^10 + x^11 + ... 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 *) PROG (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 */ (Haskell) a052343 = (flip div 2) . (+ 1) . a008441 -- Reinhard Zumkeller, Jul 25 2014 CROSSREFS Cf. A000217, A052344, A052345 (greedy inverse), A052346, A052347, A052348, A053587, A056303, A056304. Cf. A053692, A093518, A121444, A259285, A259287, A260415, A260516. Cf. A005369, A010052. Sequence in context: A198068 A121361 A191907 * A073484 A203947 A081396 Adjacent sequences:  A052340 A052341 A052342 * A052344 A052345 A052346 KEYWORD nonn AUTHOR Christian G. Bower, Jan 23 2000 STATUS approved

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Last modified April 10 16:05 EDT 2021. Contains 342845 sequences. (Running on oeis4.)