A053587 -id:A053587 - cp's OEIS frontend

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

Christian G. Bower, Jan 23 2000

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

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

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.

Cf. A000217, A052344, A052345 (greedy inverse), A052346, A052347, A052348, A053587, A056303, A056304.
Cf. A053692, A093518, A121444, A259285, A259287, A260415, A260516.
Cf. A005369, A010052.

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

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

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 *)

{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 */

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

A052344 Number of ways to write n as the unordered sum of two nonzero triangular numbers.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 2, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 2, 1, 0, 0, 2, 0, 1, 1, 0, 2, 0, 0, 0, 1, 2, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 2, 1, 0, 0, 2, 0, 0, 1, 0, 3, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 0, 0, 1, 0, 1, 1, 1

Offset: 0

Christian G. Bower, Jan 23 2000

nonn

Number of ways to write 8*n+2 as the unordered sum of two odd squares > 1. - Robert Israel, Feb 24 2016

Number of partitions of 2n into two promic numbers > 1. - Wesley Ivan Hurt, Jun 09 2021

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A000217, A052343-A052348, A053587, A056303, A056304.
Column k=2 of A319797.
Cf. A005369, A010054.

G:= (1/8)*(JacobiTheta2(0, sqrt(q))^2-4*JacobiTheta2(0, sqrt(q))*q^(1/8)+2*JacobiTheta2(0, q))/q^(1/4):
S:= series(G,q,1001):
seq(coeff(S,q,j),j=0..1000); # Robert Israel, Feb 24 2016

nn=150; tri=Accumulate[Range[nn]]; t=Table[0, {tri[[-1]]}]; Do[n=tri[[i]]+tri[[j]]; If[n <= tri[[-1]], t[[n]]++], {i,nn}, {j,i}]; t=Prepend[t,0]

G.f.: (Theta_2(sqrt(x))^2 - 4*x^(1/8)*Theta_2(sqrt(x)) + 2*Theta_2(x))/(8*x^(1/4)) where Theta_2 is a Jacobi theta function. - Robert Israel, Feb 24 2016
a(n) = Sum_{k=1..n} c(k) * c(2*n-k), where c(n) is the characteristic function of promic numbers (A005369). - Wesley Ivan Hurt, Jun 09 2021
a(n) = Sum_{k=1..floor(n/2)} c(k) * c(n-k), where c = A010054. - Wesley Ivan Hurt, Jan 06 2024

A052346 Smallest number which is the sum of two positive triangular numbers in exactly n different ways.

Original entry on oeis.org

1, 2, 16, 81, 471, 1056, 1381, 11781, 6906, 17956, 34531, 123256, 40056, 4462656, 305256, 448906, 200281, 1957231, 520731, 10563906, 1001406, 11222656, 539550781, 3454506, 1482081, 75865156, 7172606106, 8852431, 25035156, 334020781, 13018281, 38531031, 7410406, 7014160156

Offset: 0

Christian G. Bower, Jan 23 2000

nonn

From Chai Wah Wu, Oct 20 2023: (Start)
Other terms:
a(35) =  42980356
a(36) =  19267056
a(38) =  1289707656
a(39) =  2782318906
a(40) =  37052031
a(41) =  256720506
a(42) =  325457031
a(45) =  221310781
a(47) =  550240551
a(48) =  60765331
a(50) =  2200089531
a(54) =  327539956
a(56) =  926300781
a(59) =  7629645156
a(60) =  481676406
a(63) =  4598740656
a(64) =  303826656
a(68) =  6418012656
a(71) =  4579579956
a(72) =  789949306
a(80) =  1519133281
a(81) =  9498658731
a(84) =  12041910156
a(90) =  8188498906
a(96) =  3220562556
a(108) =  13429138206
(End)

			a(4) = 471 because 471 is the sum of two positive triangular numbers in exactly 4 different ways (as 300+171, 351+120, 435+36, and 465 + 6), and there is no smaller number that has this property.

Cf. A000217, A052343, A052344, A052345, A052347, A052348, A053587, A064816.

a(27), a(28) = 8852431, 25035156; a(26) not yet found
a(26) from Donovan Johnson, Nov 17 2008
Name edited (added the qualifier "positive"), example edited, and a(29)-a(32) added by Jon E. Schoenfield, Jul 16 2017
a(33) from Chai Wah Wu, Oct 20 2023

A052348 Indices of A052343 (ways to write n as sum of two triangular numbers) where record values are reached.

Original entry on oeis.org

0, 6, 81, 276, 1056, 1381, 6906, 17956, 34531, 40056, 200281, 520731, 1001406, 1482081, 7410406, 19267056, 37052031, 60765331, 303826656, 789949306, 1519133281, 3220562556, 13429138206, 16102812781, 41867313231, 80514063906, 196454315931, 711744324931

Offset: 1

Christian G. Bower, Jan 23 2000

nonn

			a(2) = 6 because 6 = 6 + 0 = 3 + 3. a(3) = 81 because 81 = 3 + 78 = 15 + 66 = 36 + 55.

Kenneth Korbin, Problem 665, The College Mathematics Journal Vol. 30 No. 5, Nov. 1999.

Probably differs from A053587 only at n=1, 2, 4.

Cf. A000217, A052343, A052344, A052345, A052346, A052347.

a(23)-a(26) from Donovan Johnson, May 24 2009

a(27)-a(28) from Donovan Johnson, Mar 20 2013

A334013 a(n) is the least integer that is the sum of two nonzero tetrahedral numbers in exactly n ways.

Original entry on oeis.org

2, 140, 102424, 43322844421220

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

			Let Te(k) denote the k-th tetrahedral number, then
a(1) = Te(1) + Te(1);
a(2) = Te(7) + Te(6) = Te(8) + Te(4);
a(3) = Te(82) + Te(34) = Te(83) + Te(27) = Te(84) + Te(7);
a(4) = Te(60500) + Te(33760) = Te(53311) + Te(47682) = Te(63383) + Te(17423) = Te(63495) + Te(15790);

Eric Weisstein's World of Mathematics, Tetrahedral Number
Index to sequences related to pyramidal numbers

Cf. A000292, A010329, A011541, A053587.

a(4) from Giovanni Resta, Apr 13 2020

A052347 Record values reached in A052343 and A052344.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 32, 36, 40, 48, 64, 72, 80, 96, 108, 128, 144, 160, 192, 216

Offset: 1

Christian G. Bower, Jan 23 2000

Cf. A052343-A052348, A053587.

a(25)-a(26) from Donovan Johnson, Jun 26 2010

a(27)-a(28) from Donovan Johnson, Mar 20 2013

A052345 Least k such that A052343(k)=n.

Original entry on oeis.org

5, 1, 6, 81, 276, 1056, 1381, 50781, 6906, 17956, 34531, 660156, 40056, 4462656, 305256, 448906, 200281, 412597656, 520731, 12397766113281, 1001406, 11222656, 539550781, 7631406, 1482081, 75865156, 422394133, 8852431, 25035156, 161170959472656

Offset: 0

Christian G. Bower, Jan 23 2000

nonn

Kenneth Korbin, Problem 665, The College Mathematics Journal Vol. 30 No. 5, Nov. 1999.

Cf. A052343, A052344, A052346, A052347, A052348, A053587.

a(19) and a(29) from Max Alekseyev, Mar 09 2009

a(19) corrected by Max Alekseyev, Mar 11 2009

A234004 Smallest number which is the sum of two positive triangular numbers (A000217) in at least n ways.

Original entry on oeis.org

2, 16, 81, 471, 1056, 1381, 6906, 6906, 17956, 34531, 40056, 40056, 200281, 200281, 200281, 200281, 520731, 520731, 1001406, 1001406, 1482081, 1482081, 1482081, 1482081, 7410406, 7410406, 7410406, 7410406, 7410406, 7410406, 7410406, 7410406

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Dec 18 2013

nonn

			2 = 1+1; 16 = 1+15 = 6+10; 81 = 3+78 = 15+66 = 36+45.

Donovan Johnson, Table of n, a(n) for n = 1..216

Cf. A000217, A048610, A199809, A052346, A053587.

poly[n_,k_]:=n (n-1)(k-2)/2+n;nn=10^5;order=3;t2=Table[0,{nn}];n2=Floor[(-4+#+Sqrt[(-4+#)^2+8 (-2+#) nn])/(2 (-2+#))&[order]];Do[r=Plus@@poly[{a,b},order];If[r<=nn,t2[[r]]++],{a,n2},{b,a,n2}];t={};n=1;While[a=Position[t2,?(#>=n&),1,1];a!={},AppendTo[t,a[[1,1]]];n++];t (* after _T. D. Noe (A048610) *)

A334077 a(n) is the smallest positive integer that can be expressed as the difference of two positive triangular numbers in at least n ways.

Original entry on oeis.org

2, 5, 9, 27, 45, 63, 105, 135, 225, 315, 315, 315, 945, 945, 945, 945, 1575, 1575, 2835, 2835, 3465, 3465, 3465, 3465, 10395, 10395, 10395, 10395, 10395, 10395, 10395, 10395, 17325, 17325, 17325, 17325, 31185, 31185, 31185, 31185, 45045, 45045, 45045, 45045

Offset: 1

Ilya Gutkovskiy, Apr 13 2020

nonn

Eric Weisstein's World of Mathematics, Triangular Number

Cf. A000217, A038547, A052346, A053587, A136107, A136108, A214771, A334078.

cp's OEIS Frontend

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

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A052344 Number of ways to write n as the unordered sum of two nonzero triangular numbers.

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A052346 Smallest number which is the sum of two positive triangular numbers in exactly n different ways.

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A052348 Indices of A052343 (ways to write n as sum of two triangular numbers) where record values are reached.

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A334013 a(n) is the least integer that is the sum of two nonzero tetrahedral numbers in exactly n ways.

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A052347 Record values reached in A052343 and A052344.

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A052345 Least k such that A052343(k)=n.

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A234004 Smallest number which is the sum of two positive triangular numbers (A000217) in at least n ways.

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A334077 a(n) is the smallest positive integer that can be expressed as the difference of two positive triangular numbers in at least n ways.

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