A020756 -id:A020756 - cp's OEIS frontend

A051533 Numbers that are the sum of two positive triangular numbers.

2, 4, 6, 7, 9, 11, 12, 13, 16, 18, 20, 21, 22, 24, 25, 27, 29, 30, 31, 34, 36, 37, 38, 39, 42, 43, 46, 48, 49, 51, 55, 56, 57, 58, 60, 61, 64, 65, 66, 67, 69, 70, 72, 73, 76, 79, 81, 83, 84, 87, 88, 90, 91, 92, 93, 94, 97, 99, 100, 101, 102, 106, 108

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

Numbers n such that 8n+2 is in A085989. - Robert Israel, Mar 06 2017

			666 is in the sequence because we can write 666 = 435 + 231 = binomial(22,2) + binomial(30,2).

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Fermat's Polygonal Number Theorem

Cf. A000217, A020756 (sums of two triangular numbers), A001481 (sums of two squares), A007294, A051611 (complement).
Cf. A061336: minimal number of triangular numbers that sum up to n.
Cf. A085989.

a051533 n = a051533_list !! (n-1)
a051533_list = filter ((> 0) . a053603) [1..]
-- Reinhard Zumkeller, Jun 28 2013

isA051533 := proc(n)
    local a,ta;
    for a from 1 do
        ta := A000217(a) ;
        if 2*ta > n then
            return false;
        end if;
        if isA000217(n-ta) then
            return true;
        end if;
    end do:
end proc:
for n from 1 to 200 do
    if isA051533(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 16 2015

f[k_] := If[!
   Head[Reduce[m (m + 1) + n (n + 1) == 2 k && 0 < m && 0 < n, {m, n},
       Integers]] === Symbol, k, 0]; DeleteCases[Table[f[k], {k, 1, 108}], 0] (* Ant King, Nov 22 2010 *)
nn=50; tri=Table[n(n+1)/2, {n,nn}]; Select[Union[Flatten[Table[tri[[i]]+tri[[j]], {i,nn}, {j,i,nn}]]], #<=tri[[-1]] &]
With[{nn=70},Take[Union[Total/@Tuples[Accumulate[Range[nn]],2]],nn]] (* Harvey P. Dale, Jul 16 2015 *)

is(n)=for(k=ceil((sqrt(4*n+1)-1)/2),(sqrt(8*n-7)-1)\2, if(ispolygonal(n-k*(k+1)/2, 3), return(1))); 0 \\ Charles R Greathouse IV, Jun 09 2015

A053603(a(n)) > 0. - Reinhard Zumkeller, Jun 28 2013

A061336(a(n)) = 2. - M. F. Hasler, Mar 06 2017

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

Original entry on oeis.org

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

A020757 Numbers that are not the sum of two triangular numbers.

Original entry on oeis.org

5, 8, 14, 17, 19, 23, 26, 32, 33, 35, 40, 41, 44, 47, 50, 52, 53, 54, 59, 62, 63, 68, 71, 74, 75, 77, 80, 82, 85, 86, 89, 95, 96, 98, 103, 104, 107, 109, 113, 116, 117, 118, 122, 124, 125, 128, 129, 131, 134, 138, 140, 143, 145, 147, 149, 152, 155, 158, 161, 162, 166, 167

Offset: 1

David W. Wilson

nonn

A052343(a(n)) = 0. - Reinhard Zumkeller, May 15 2006
Numbers of the form (p^(2k+1)s-1)/4, where p is a prime number of the form 4n+3, and s is a number of the form 4m+3 and prime to p, are not expressible as the sum of two triangular numbers. See Satyanarayana (1961), Theorem 2. - Hans J. H. Tuenter, Oct 11 2009
An integer n is in this sequence if and only if at least one 4k+3 prime factor in the canonical form of 4n+1 occurs with an odd exponent. - Ant King, Dec 02 2010
A nonnegative integer n is in this sequence if and only if A000729(n) = 0. - Michael Somos, Feb 13 2011
4*a(n) + 1 are terms of A022544. - XU Pingya, Aug 05 2018 [Actually, k is here if and only if 4*k + 1 is in A022544. - Jianing Song, Feb 09 2021]
Integers m such that the smallest number of triangular numbers which sum to m is 3, hence A061336(a(n)) = 3. - Bernard Schott, Jul 21 2022

			3 = 0 + 3 and 7 = 1 + 6 are not terms, but 8 = 1 + 1 + 6 is a term.

T. D. Noe, Table of n, a(n) for n = 1..10000
John A. Ewell, On Sums of Triangular Numbers and Sums of Squares, The American Mathematical Monthly, Vol. 99, No. 8 (October 1992), pp. 752-757. [From _Ant King_, Dec 02 2010]
U. V. Satyanarayana, On the representation of numbers as sums of triangular numbers, The Mathematical Gazette, 45(351):40-43, February 1961. [From _Hans J. H. Tuenter_, Oct 11 2009]

Complement of A020756.
Cf. A000217, A052343, A022544, A061336.
Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m=1), A213250 (m=2), A014132 (m=3), A302056 (m=4), A302057 (m=5), this sequence (m=6), A322430 (m=8), A322431 (m=10), A322432 (m=14), A322043 (m=15), A322433 (m=26).

a020757 n = a020757_list !! (n-1)
a020757_list = filter ((== 0) . a052343) [0..]
-- Reinhard Zumkeller, Jul 25 2014

data = Reduce[m (m + 1) + n (n + 1) == 2 # && 0 <= m && 0 <= n, {m, n}, Integers] & /@ Range[167]; Position[data, False] // Flatten  (* Ant King, Dec 05 2010 *)
t = Array[PolygonalNumber, 18, 0]; Complement[Range@ 169, Flatten[ Outer[ Plus, t, t]]] (* Robert G. Wilson v, Aug 07 2024 *)

is(n)=my(m9=n%9,f); if(m9==5 || m9==8, return(1)); f=factor(4*n+1); for(i=1,#f~, if(f[i,1]%4==3 && f[i,2]%2, return(1))); 0 \\ Charles R Greathouse IV, Mar 17 2022

A117048 Prime numbers that are expressible as the sum of two positive triangular numbers.

Original entry on oeis.org

2, 7, 11, 13, 29, 31, 37, 43, 61, 67, 73, 79, 83, 97, 101, 127, 137, 139, 151, 157, 163, 181, 191, 193, 199, 211, 227, 241, 263, 277, 281, 307, 331, 353, 367, 373, 379, 389, 409, 421, 433, 443, 461, 463, 487, 499, 541, 571, 577, 587, 601, 619, 631, 659, 661

Offset: 1

Andrew S. Plewe, Apr 15 2006

If the triangular number 0 is allowed, only one additional prime occurs: 3. In that case, the sequence becomes A117112.

A subsequence of A051533. - Wolfdieter Lang, Jan 11 2017

			2 = 1 + 1
7 = 1 + 6
11 = 1 + 10
13 = 10 + 3, etc.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000040, A000217, A002243, A002244, A020756, A051533, A053614, A060773, A002636.

tri = Table[n (n + 1)/2, {n, 40}]; Select[Union[Flatten[Outer[Plus, tri, tri]]], # <= tri[[-1]]+1 && PrimeQ[#] &] (* T. D. Noe, Apr 07 2011 *)

is(n)=for(k=sqrtint(4*n+1)\2+1,(sqrtint(8*n+1)-1)\2, if(ispolygonal(n-k*(k+1)/2,3), return(n>3 && isprime(n)))); n==2 \\ Charles R Greathouse IV, Nov 07 2014

A117112 Primes expressible as the sum of two triangular numbers (including zero).

Original entry on oeis.org

2, 3, 7, 11, 13, 29, 31, 37, 43, 61, 67, 73, 79, 83, 97, 101, 127, 137, 139, 151, 157, 163, 181, 191, 193, 199, 211, 227, 241, 263, 277, 281, 307, 331, 353, 367, 373, 379, 389, 409, 421, 433, 443, 461, 463, 487, 499, 541, 571, 577, 587, 601, 619, 631, 659, 661

Offset: 1

Greg Huber, Apr 18 2006

See A117048 for the primes that are the sum of two positive triangular numbers. The only difference is that the prime 3 occurs here.

			2 = 1 + 1
3 = 0 + 3
7 = 1 + 6
and so on.

Cf. A000217, A117048.

tri = Table[n (n + 1)/2, {n, 0, 40}]; Select[Union[Flatten[Outer[Plus, tri, tri]]], # <= tri[[-1]]+1 && PrimeQ[#] &] (* T. D. Noe, Apr 07 2011 *)
Select[Total/@Tuples[Accumulate[Range[0,40]],2],PrimeQ]//Union (* Harvey P. Dale, Apr 21 2019 *)

A000040 intersect A020756. - Jonathan Vos Post, Apr 17 2006

A119345 Numbers having exactly one representation as sum of two triangular numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 9, 10, 11, 12, 13, 15, 18, 20, 22, 24, 25, 27, 28, 29, 30, 34, 37, 38, 39, 43, 45, 48, 49, 57, 58, 60, 61, 64, 65, 67, 69, 70, 73, 78, 79, 83, 84, 87, 88, 90, 92, 93, 97, 99, 100, 101, 102, 105, 108, 110, 112, 114, 115, 119, 127, 130, 132, 135, 137, 139, 142

Offset: 1

Reinhard Zumkeller, May 15 2006

nonn

A052343(a(n)) = 1; gives A020756 together with A118139.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A020757, A000217.

Cf. A020756, A052343, A118139.

a119345 n = a119345_list !! (n-1)
a119345_list = filter ((== 1) . a052343) [0..]
-- Reinhard Zumkeller, Jul 25 2014

trn=SortBy[{First[#],Last[#],Total[#]}& /@ (Union[Sort/@Tuples[Accumulate[Range[0,70]],{2}]]),Last]; Take[With[{x=Transpose[trn][[3]]}, Complement[Union[x], Union[Flatten[Select[Split[x], Length[#]>1&]]]]],70]  (* Harvey P. Dale, Feb 14 2011 *)
nn=100; tri=Table[n(n+1)/2,{n,0,nn}]; sums=Select[Flatten[Table[tri[[i]]+tri[[j]], {i,nn}, {j,i}]], #

A118139 Numbers expressible as the sum of two triangular numbers in at least two different ways.

Original entry on oeis.org

6, 16, 21, 31, 36, 42, 46, 51, 55, 56, 66, 72, 76, 81, 91, 94, 106, 111, 120, 121, 123, 126, 133, 136, 141, 146, 156, 157, 171, 172, 174, 181, 186, 191, 196, 198, 210, 211, 216, 225, 226, 231, 237, 241, 246, 256, 259, 268, 276, 281, 286, 289, 291, 297, 301, 306

Offset: 1

Greg Huber, May 13 2006

A052343(a(n)) > 1; gives A020756 together with A119345. - Reinhard Zumkeller, May 15 2006

			a(1) = 6 = 0 + 6 = 3 +3.
a(2) = 16 = 1 + 15 = 6 + 10.
a(3) = 21 = 0 + 21 = 6 + 15.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A020757, A000217.

Cf. A052343.

a118139 n = a118139_list !! (n-1)
a118139_list = filter ((> 1) . a052343) [0..]
-- Reinhard Zumkeller, Jul 25 2014

Sort[Transpose[Select[Tally[Total/@(Union[Sort/@Tuples[Accumulate[ Range[ 0,30]],2]])],#[[2]]>1&]][[1]]] (* Harvey P. Dale, Jul 21 2015 *)

More terms from Reinhard Zumkeller, May 15 2006

A185978 Nontriangular numbers which are the sum of two (positive) triangular numbers.

Original entry on oeis.org

2, 4, 7, 9, 11, 12, 13, 16, 18, 20, 22, 24, 25, 27, 29, 30, 31, 34, 37, 38, 39, 42, 43, 46, 48, 49, 51, 56, 57, 58, 60, 61, 64, 65, 67, 69, 70, 72, 73, 76, 79, 81, 83, 84, 87, 88, 90, 92, 93, 94, 97, 99, 100, 101, 102, 106, 108

Offset: 1

Wolfdieter Lang, Feb 14 2011

This is A051533 (sums of two positive triangular numbers) excluding triangular numbers.
This is also A020756 (sums of two triangular numbers) excluding triangular numbers.
There may be multiple representations, e.g., 16 = 1 + 15 = 6 + 10.

			a(6) = 12 = 6 + 6,
a(17) = 31 = 3 + 28 = 10 + 21.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000217, A020756 (sums of two triangular numbers), A051533 (sums of two positive triangular numbers).

(* First run the program for A051533 *) Complement[%, tri] (* Alonso del Arte, Feb 15 2011 *)

This is the sorted, made unique set {binomial(k+1,2) + binomial(L+1,2), 1 <= k <= L sufficiently large}, excluding members from A000217 (triangular numbers).

A288631 Numbers that are the sum of two nonzero square pyramidal numbers (A000330).

Original entry on oeis.org

2, 6, 10, 15, 19, 28, 31, 35, 44, 56, 60, 69, 85, 92, 96, 105, 110, 121, 141, 145, 146, 154, 170, 182, 195, 205, 209, 218, 231, 234, 259, 280, 286, 290, 295, 299, 315, 340, 344, 376, 386, 390, 399, 408, 415, 425, 440, 476, 489, 507, 511, 520, 525, 536, 561, 570, 589, 597, 646, 651, 655, 664, 670, 680

Offset: 1

Ilya Gutkovskiy, Jun 12 2017

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000330, A000404, A020756, A051533, A102795.

M:= 20: # to get all terms <= A000330(M)
sqp:= [seq(k*(k+1)*(2*k+1)/6, k=1..M)]:
sort(convert(select(`<=`, {seq(seq(sqp[i]+sqp[j], j=1..i),i=1..M-1)},sqp[M]),list)); # Robert Israel, Jun 12 2017

nmax = 700; f[x_] := Sum[x^(k (k + 1) (2 k + 1)/6), {k, 1, 20}]^2; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

A185979 Numbers which are the sum of two positive triangular numbers in more than one way.

Original entry on oeis.org

16, 31, 42, 46, 51, 56, 72, 76, 81, 94, 106, 111, 121, 123, 126, 133, 141, 146, 156, 157, 172, 174, 181, 186, 191, 196, 198, 211, 216, 225, 226, 231, 237, 241, 246, 256, 259, 268, 276, 281, 286, 289, 291, 297, 301, 306, 310, 315, 321, 326, 328, 331, 336, 342, 346, 354, 361, 366, 367

Offset: 1

Wolfdieter Lang, Feb 15 2011

This is a subsequence of A020756 (sums of two triangular numbers).
This is also a subsequence of A051533 (sums of two positive triangular numbers). This is not a subsequence of A185978 (nontriangular numbers as sums of (positive) triangular numbers). E.g., a(32)=231 is missing there because 231=A000217(21). See A185980.
For the numbers which are sums of two positive triangular numbers in exactly two ways see A064816.
The first number which can be written in exactly three ways as sums of positive triangular numbers is 81.
a(n) gives the positions where A052344 entries are >= 2: A052344(a(n)) >= 2.

			16 = 15 + 1 = 10 + 6.
81 = 45 + 36 = 66 + 15 = 78 + 3.
231= 210 + 21 = 153 + 78

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000217, A020756, A051533, A052344, A064816, A185980.

cp's OEIS Frontend

A051533 Numbers that are the sum of two positive triangular numbers.

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

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A020757 Numbers that are not the sum of two triangular numbers.

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A117048 Prime numbers that are expressible as the sum of two positive triangular numbers.

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A117112 Primes expressible as the sum of two triangular numbers (including zero).

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A119345 Numbers having exactly one representation as sum of two triangular numbers.

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A118139 Numbers expressible as the sum of two triangular numbers in at least two different ways.

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