A051611 -id:A051611 - 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

A053614 Numbers that are not the sum of distinct triangular numbers.

Original entry on oeis.org

2, 5, 8, 12, 23, 33

Offset: 1

Jud McCranie, Mar 19 2000

The Mathematica program first computes A024940, the number of partitions of n into distinct triangular numbers. Then it finds those n having zero such partitions. It appears that A024940 grows exponentially, which would preclude additional terms in this sequence. - T. D. Noe, Jul 24 2006, Jan 05 2009

			a(2) = 5: the 7 partitions of 5 are 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1. Among those the distinct ones are 5, 4+1, 3+2. None contains all distinct triangular numbers.
12 is a term as it is not a sum of 1, 3, 6 or 10 taken at most once.

Joe Roberts, Lure of the Integers, The Mathematical Association of America, 1992, page 184, entry 33.
David Wells in "The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, page 94, states that "33 is the largest number that is not the sum of distinct triangular numbers".

Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 83-125.

Cf. A025524 (number of numbers not the sum of distinct n-th-order polygonal numbers)
Cf. A007419 (largest number not the sum of distinct n-th-order polygonal numbers)
Cf. A001422, A121405 (corresponding sequences for square and pentagonal numbers)
Cf. A000217, A002243, A002244, A014134, A014156, A014158, A020757, A050941, A050942, A051611, A007294, A051533, A060773.

nn=100; t=Rest[CoefficientList[Series[Product[(1+x^(k*(k+1)/2)), {k,nn}], {x,0,nn(nn+1)/2}], x]]; Flatten[Position[t,0]] (* T. D. Noe, Jul 24 2006 *)

Complement of A061208.

Entry revised by N. J. A. Sloane, Jul 23 2006

A053604 Number of ways to write n as an ordered sum of 3 nonzero triangular numbers.

Original entry on oeis.org

0, 0, 0, 1, 0, 3, 0, 3, 3, 1, 6, 0, 6, 3, 6, 3, 3, 9, 1, 12, 0, 6, 9, 6, 6, 6, 9, 6, 12, 0, 10, 9, 12, 6, 9, 9, 3, 18, 3, 12, 12, 9, 9, 9, 12, 10, 12, 9, 9, 18, 6, 6, 27, 6, 12, 6, 9, 18, 15, 15, 6, 21, 9, 13, 12, 9, 18, 21, 9, 6, 21, 15, 15, 15, 12, 15, 18, 15, 9

Offset: 0

N. J. A. Sloane, Jan 20 2000

nonn

Fermat asserted that every number is the sum of three triangular numbers. This was proved by Gauss, who recorded in his Tagebuch entry for Jul 10 1796 that: EYPHEKA! num = DELTA + DELTA + DELTA.

Mel Nathanson, Additive Number Theory: The Classical Bases, Graduate Texts in Mathematics, Volume 165, Springer-Verlag, 1996. See Chapter 1.

T. D. Noe, Table of n, a(n) for n=0..5050

Cf. A000217, A007294, A051611, A051533, A053603, A008443, A002636.

nmax = 100; m0 = 10; A053604 :=
Table[a[n], {n, 0, nmax}]; Clear[counts];
counts[m_] :=
counts[m] = (Clear[a]; a[_] = 0;
   Do[s = i*(i + 1)/2 + j*(j + 1)/2 + k*(k + 1)/2;
    a[s] = a[s] + 1, {i, 1, m}, {j, 1, m}, {k, 1, m}];
   A053603); counts[m = m0]; counts[m = 2*m]; While[
counts[m] != counts[m/2], m = 2*m]; A053604  (* G. C. Greubel, Dec 24 2016 *)

G.f.: ( Sum_{k>=1} x^(k*(k+1)/2) )^3. - Ilya Gutkovskiy, Dec 24 2016

A053603 Number of ways to write n as an ordered sum of two nonzero triangular numbers.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 1, 2, 0, 2, 0, 2, 1, 2, 0, 0, 4, 0, 2, 0, 1, 2, 2, 0, 2, 2, 0, 2, 0, 2, 1, 4, 0, 0, 2, 0, 2, 2, 2, 2, 0, 0, 3, 2, 0, 0, 4, 0, 2, 2, 0, 4, 0, 0, 0, 2, 3, 2, 2, 0, 2, 2, 0, 0, 2, 2, 2, 2, 0, 2, 2, 0, 3, 2, 0, 0, 4, 0, 0, 2, 0, 6, 0, 2, 2, 0, 0, 2, 2, 0, 1, 2, 2

Offset: 0

N. J. A. Sloane, Jan 20 2000

nonn

a(A051611(n)) = 0; A051533(a(n)) > 0. - Reinhard Zumkeller, Jun 27 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000217, A007294, A051611, A051533, A052343-A052348, A053604.

Cf. A010054.

a053603 n = sum $ map (a010054 . (n -)) $
                  takeWhile (< n) $ tail a000217_list
-- Reinhard Zumkeller, Jun 27 2013

nmax = 100; m0 = 10; A053603 := Table[a[n], {n, 0, nmax}]; Clear[counts]; counts[m_] := counts[m] = (Clear[a]; a[A053603);%20counts%5Bm%20=%20m0%5D;%20counts%5Bm%20=%202*m%5D;%20While%5B%20counts%5Bm%5D%20!=%20counts%5Bm/2%5D,%20m%20=%202*m%5D;%20A053603%20(*%20_Jean-Fran%C3%A7ois%20Alcover">] = 0; Do[k = i*(i+1)/2 + j*(j+1)/2; a[k] = a[k]+1, {i, 1, m}, {j, 1, m}]; A053603); counts[m = m0]; counts[m = 2*m]; While[ counts[m] != counts[m/2], m = 2*m]; A053603 (* _Jean-François Alcover, Sep 05 2013 *)

istriang(n)={n>0 && issquare(8*n+1);}
a(n) = { my(t=1, ct=0, j=1); while (tJoerg Arndt, Sep 05 2013

G.f.: ( Sum_{k>=1} x^(k*(k+1)/2) )^2. - Ilya Gutkovskiy, Dec 24 2016

a(n) = Sum_{k=1..n-1} c(k) * c(n-k), where c(n) = A010054(n). - Wesley Ivan Hurt, Jan 06 2024

A061208 Numbers which can be expressed as sum of distinct triangular numbers (A000217).

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Amarnath Murthy, Apr 21 2001

nonn

These numbers were called "almost-triangular" numbers during the Peru's Selection Test for the XII IberoAmerican Olympiad (1998). All numbers >= 34 are almost-triangular: see link. [Bernard Schott, Feb 04 2013]

			25 = 1 + 3 + 6 + 15

R. E. Woodrow, The Olympiad Corner, No. 198, Crux Mathematicorum, v25-n4(2002), 207-208, exercise 2.

Cf. A000217, A007294, A051611, A051533. Complement of A053614.

gf := product(1+x^(j*(j+1)/2), j=1..100): s := series(gf, x, 200): for i from 1 to 200 do if coeff(s, x, i) > 0 then printf(`%d,`,i) fi:od:

Corrected and extended by James Sellers, Apr 24 2001

A152089 Numbers k such that k! is not the sum of two nonnegative triangular numbers.

Original entry on oeis.org

6, 11, 12, 14, 18, 19, 20, 22, 23, 25, 26, 30, 31, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 82, 83, 84, 87, 88, 89, 90, 91, 92, 93, 94, 97, 98, 100, 101, 103, 104, 105, 106, 107, 108

Offset: 1

N. J. A. Sloane, Sep 24 2010

nonn

Up to 300 these are also terms: 110, 111, 112, 113, 115, 116, 117, 119, 120, 121, 122, 123, 124, 127, 128, 130, 132, 136, 137, 138, 142, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 163, 166, 167, 168, 169, 170, 172, 174, 175, 176, 177, 179, 180, 181, 182, 183, 185, 186, 187, 188, 191, 193, 194, 195, 196, 197, 198, 200, 201, 202, 203, 204, 205, 206, 207, 210, 211, 213, 214, 215, 216, 218, 223, 225, 226, 227, 228, 229, 230, 231, 232, 236, 237, 238, 239, 240, 241, 243, 245, 246, 247, 249, 250, 251, 253, 255, 256, 257, 258, 260, 261, 262, 263, 265, 266, 267, 269, 270, 271, 272, 273, 274, 276, 278, 279, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 292, 293, 294, 295, 297, 298. - G. Guninski, Oct 12 2010

Tyler Busby, Table of n, a(n) for n = 1..85
Factor Database, Factors of the numbers 4z!+1
Factor Database, Factors [From G. Guninski, Oct 12 2010]
G. Guninski, Factors
R. K. Guy et al., Emails about A152089

Complement of A180590.

Cf. A051611.

69 added by N. J. A. Sloane, Sep 24 2010 using the data from the Factor Database web site.
Further terms from G. Guninski and D. S. McNeil, Sep 24 2010
Terms from 93 on from G. Guninski, Oct 12 2010

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A053614 Numbers that are not the sum of distinct triangular numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A053604 Number of ways to write n as an ordered sum of 3 nonzero triangular numbers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A053603 Number of ways to write n as an ordered sum of two nonzero triangular numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A061208 Numbers which can be expressed as sum of distinct triangular numbers (A000217).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Extensions

A152089 Numbers k such that k! is not the sum of two nonnegative triangular numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions