A053604 -id:A053604 - cp's OEIS frontend

A008443 Number of ordered ways of writing n as the sum of 3 triangular numbers.

1, 3, 3, 4, 6, 3, 6, 9, 3, 7, 9, 6, 9, 9, 6, 6, 15, 9, 7, 12, 3, 15, 15, 6, 12, 12, 9, 12, 15, 6, 13, 21, 12, 6, 15, 9, 12, 24, 9, 18, 12, 9, 18, 15, 12, 13, 24, 9, 15, 24, 6, 18, 27, 6, 12, 15, 18, 24, 21, 15, 12, 27, 9, 13, 18, 15, 27, 27, 9, 12, 27, 15, 24, 21, 12, 15, 30, 15, 12

Offset: 0

N. J. A. Sloane

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. See also Gauss, DA, art. 293.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Andrews (2016), Theorem 2, shows that A008443(n) = A290735(n) + A290737(n) + A290739(n). = N. J. A. Sloane, Aug 10 2017

			5 can be written as 3+1+1, 1+3+1, 1+1+3, so a(5) = 3.
G.f. = 1 + 3*x + 3*x^2 + 4*x^3 + 6*x^4 + 3*x^5 + 6*x^6 + 9*x^7 + 3*x^8 + ...
G.f. = q^3 + 3*q^11 + 3*q^19 + 4*q^27 + 6*q^35 + 3*q^43 + 6*q^51 + 9*q^59 + 3*q^67 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
C. F. Gauss, Disquisitiones Arithmeticae, Yale University Press, 1966, New Haven and London, p. 342, art. 293.
M. Nathanson, Additive Number Theory: The Classical Bases, Graduate Texts in Mathematics, Volume 165, Springer-Verlag, 1996. See Chapter 1.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000 (first 5050 terms from T. D. Noe)
George E. Andrews, EYPHKA! num = Delta + Delta + Delta, J. Number Theory 23 (1986), 285-293. [The Y in the title is really the Greek letter Upsilon and Delta is really the Greek letter of that name.]
George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016.
M. Doring, J. Haidenbauer, U.-G. Meissner, and A. Rusetsky, Dynamical coupled-channel approaches on a momentum lattice, arXiv preprint arXiv:1108.0676 [hep-lat], 2011.
M. D. Hirschhorn and J. A. Sellers, Partitions into three triangular numbers, Australasian Journal of Combinatorics, Volume 30 (2004), Pages 307-318; Submission.
M. D. Hirschhorn and J. A. Sellers, On Representations Of A Number As A Sum Of Three Triangles, Acta Arithmetica 77 (1996), 289-301.
K. Ono, S. Robins, and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440,A226252, A007331, A226253, A226254, A226255, A014787, A014809.
Cf. A053604, A002636.
Partial sums are in A038835.
Cf. A005869, A005875, A005878, A005886.
Cf. A290733-A290740.

Basis( ModularForms( Gamma0(16), 3/2), 630)[4]; /* Michael Somos, Aug 26 2015 */

s1 := sum(q^(n*(n+1)/2), n=0..30): s2 := series(s1^3, q, 250): for i from 0 to 200 do printf(`%d,`,coeff(s2, q, i)) od:

s1 = Sum[q^(n (n + 1)/2), {n, 0, 12}]; s2 = Series[s1^3, {q, 0, 80}]; CoefficientList[s2, q] (* Jean-François Alcover, Oct 04 2011, after Maple *)
a[ n_] := SeriesCoefficient[ (1/8) EllipticTheta[ 2, 0, q]^3, {q, 0, 2 n + 3/4}]; (* Michael Somos, May 29 2012 *)
QP = QPochhammer; CoefficientList[(QP[q^2]^2/QP[q])^3 + O[q]^80, q] (* Jean-François Alcover, Nov 24 2015 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(8*n + 1) - 1)\2, x^((k^2 + k)/2), x * O(x^n))^3, n))}; /* Michael Somos, Oct 25 2006 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A))^3, n))}; /* Michael Somos, Oct 25 2006 */

Expansion of Jacobi theta constant theta_2^3 /8. G.f. is cube of g.f. for A010054.
Expansion of psi(q)^3 in powers of q where psi() is a Ramanujan theta function (A010054). - Michael Somos, Oct 25 2006
Expansion of q^(-3/8) * (eta(q^2)^2 / eta(q))^3 in powers of q. - Michael Somos, May 29 2012
Euler transform of period 2 sequence [ 3, -3, ...]. - Michael Somos, Oct 25 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(-3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A213384. - Michael Somos, Jun 23 2012
a(3*n) = A213627(n). a(3*n + 1) = 3 * A213617(n). a(3*n + 2) = A181648(n). - Michael Somos, Jun 23 2012
G.f.: (Sum_{k>0} x^((k^2 - k)/2))^3 = (Product_{k>0} (1 + x^k) * (1 - x^(2*k)))^3. - Michael Somos, May 29 2012
a(n) = A005869(n)/2 = A005886(n)/4 = A005878(n)/8.
a(n) = A005875(8*n+3)/8. See, e.g., the Ono et al. link: The case k=3. - Wolfdieter Lang, Jan 12 2017
a(0) = 1, a(n) = (3/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017

More terms from James Sellers, Feb 07 2001

A002636 Number of ways of writing n as an unordered sum of at most 3 nonzero triangular numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 3, 2, 1, 2, 3, 2, 2, 2, 1, 4, 3, 2, 2, 2, 2, 3, 3, 1, 4, 4, 2, 2, 3, 2, 3, 4, 2, 3, 3, 2, 4, 3, 2, 4, 4, 2, 4, 4, 1, 4, 5, 1, 2, 3, 4, 6, 4, 3, 2, 5, 2, 3, 3, 3, 6, 5, 2, 2, 5, 3, 5, 4, 2, 4, 5, 3, 4, 5, 2, 4, 6, 2, 6, 3, 3, 6, 3, 2, 3, 7, 3, 6, 6, 2, 4, 6, 3, 2

Offset: 0

N. J. A. Sloane, Sep 18 2001

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: EYPHKA! num = DELTA + DELTA + DELTA.
a(n) <= A167618(n). - Reinhard Zumkeller, Nov 07 2009
Equivalently, number of ways of writing n as an unordered sum of exactly 3 triangular numbers. - Jon E. Schoenfield, Mar 28 2021

			0 : empty sum
1 : 1
2 : 1+1
3 : 3 = 1+1+1
4 : 3+1
5 : 3+1+1
6 : 6 = 3+3
7 : 6+1 = 3+3+1
...
13 : 10 + 3 = 6 + 6 + 1, so a(13) = 2.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102, eq. (8).
D. H. Lehmer, Review of Loria article, Math. Comp. 2 (1947), 301-302.
G. Loria, Sulla scomposizione di un intero nella somma di numeri poligonali. (Italian) Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 1, (1946). 7-15.
Mel Nathanson, Additive Number Theory: The Classical Bases, Graduate Texts in Mathematics, Volume 165, Springer-Verlag, 1996. See Chapter 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Gino Loria, Sulla scomposizione di un intero nella somma di numeri poligonali. (Italian). Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 1, (1946). 7-15. Also D. H. Lehmer, Review of Loria article, Math. Comp. 2 (1947), 301-302. [Annotated scanned copies]
Eric T. Mortenson, A Kronecker-type identity and the representations of a number as a sum of three squares, arXiv:1702.01627 [math.NT], 2017.
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.

Cf. A007294, A053604, A008443, A063993, A061262.

# reuses code in A000217
A002636 := proc(n)
    local a,i,Ti, j,Tj, Tk ;
    a := 0 ;
    for i from 0 do
        Ti := A000217(i) ;
        if Ti > n then
            break ;
        end if;
        for j from i do
            Tj := A000217(j) ;
            if Ti+Tj > n then
                break ;
            end if;
            Tk := n-Ti-Tj ;
            if Tk >= Tj and isA000217(Tk) then
                a := a+1 ;
            end if;
            if Tk < Tj then
                break ;
            end if;
        end do:
    end do:
    a ;
end proc:
seq(A002636(n),n=0..40) ; # R. J. Mathar, May 26 2025

a = Table[ n(n + 1)/2, {n, 0, 15} ]; b = {0}; c = Table[ 0, {100} ]; Do[ b = Append[ b, a[ [ i ] ] + a[ [ j ] ] + a[ [ k ] ] ], {k, 1, 15}, {j, 1, k}, {i, 1, j} ]; b = Delete[ b, 1 ]; b = Sort[ b ]; l = Length[ b ]; Do[ If[ b[ [ n ] ] < 100, c[ [ b[ [ n ] ] + 1 ] ]++ ], {n, 1, l} ]; c

first(n)=my(v=vector(n+1),A,B,C); for(a=0,n, A=a*(a+1)/2; if(A>n, break); for(b=0,a, B=A+b*(b+1)/2; if(B>n, break); for(c=0,b, C=B+c*(c+1)/2; if(C>n, break); v[C+1]++))); v \\ Charles R Greathouse IV, Jun 23 2017

More terms from Robert G. Wilson v, Sep 20 2001

Entry revised by N. J. A. Sloane, Feb 25 2007

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

A230121 Number of ways to write n = x + y + z (0 < x <= y <= z) such that x(x+1)/2 + y(y+1)/2 + z*(z+1)/2 is a triangular number.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 2, 1, 1, 0, 2, 1, 2, 1, 2, 3, 2, 2, 6, 1, 3, 5, 1, 2, 3, 5, 2, 1, 3, 3, 3, 4, 3, 8, 2, 5, 11, 2, 5, 8, 4, 6, 4, 9, 4, 6, 5, 4, 6, 3, 8, 8, 5, 8, 10, 7, 7, 11, 8, 6, 7, 8, 5, 9, 7, 6, 8, 7, 7, 8, 13, 9, 11, 10, 7, 22, 9, 10, 13, 3, 6, 10, 8, 17, 12, 7, 9, 10, 16, 6, 18, 18, 10, 15, 9, 12, 20, 5

Offset: 1

Zhi-Wei Sun, Oct 10 2013

nonn

Conjecture: (i) a(n) > 0 except for n = 1, 2, 4, 5, 7, 12. Moreover, for each n = 20, 21, ... there are three distinct positive integers x, y and z with x + y + z = n such that x*(x+1)/2 + y*(y+1)/2 + z*(z+1)/2 is a triangular number.
(ii) A positive integer n cannot be written as x + y + z (x, y, z > 0) with x^2 + y^2 + z^2 a square if and only if n has the form 2^r*3^s or the form 2^r*7, where r and s are nonnegative integers.
(iii) Any integer n > 14 can be written as a + b + c + d, where a, b, c, d are positive integers with a^2 + b^2 + c^2 + d^2 a square. If n > 20 is not among 22, 28, 30, 38, 44, 60, then we may require additionally that a, b, c, d are pairwise distinct.
(iv) For each integer n > 50 not equal to 71, there are positive integers a, b, c, d with a + b + c + d = n such that both a^2 + b^2 and c^2 + d^2 are squares.
Part (ii) and the first assertion in part (iii) were confirmed by Chao Huang and Zhi-Wei Sun in 2021. - Zhi-Wei Sun, May 09 2021

			a(16) = 1 since 16 = 3 + 6 + 7 and 3*4/2 + 6*7/2 + 7*8/2 = 55 = 10*11/2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000
Chao Huang and Zhi-Wei Sun, On partitions of integers with restrictions involving squares, arXiv:2105.03416 [math.NT], 2021.
Zhi-Wei Sun, Diophantine problems involving triangular numbers and squares, a message to Number Theory List, Oct. 11, 2013.

Cf. A000217, A000290.

Cf. A008443, A053604, A053603, A052343, A052344, A008441.

SQ[n_]:=IntegerQ[Sqrt[n]]
T[n_]:=n(n+1)/2
a[n_]:=Sum[If[SQ[8(T[i]+T[j]+T[n-i-j])+1],1,0],{i,1,n/3},{j,i,(n-i)/2}]
Table[a[n],{n,1,100}]

a(n)=my(t=(n+1)*n/2,s);sum(x=1,n\3,s=t-n--*x;sum(y=x,n\2,is_A000217(s-(n-y)*y))) \\ - M. F. Hasler, Oct 11 2013

A307598 Number of partitions of n into 3 distinct positive triangular numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 1, 1, 0, 3, 0, 2, 1, 1, 1, 1, 2, 1, 2, 1, 0, 3, 1, 0, 4, 1, 2, 1, 1, 1, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 1, 3, 2, 2, 2, 2, 1, 3, 2, 0, 4, 1, 1, 5, 1, 3, 2, 2, 3, 2, 2, 1, 4, 1, 2, 4, 2, 2, 3, 2, 1, 3, 2, 4, 3, 3, 2, 2, 3, 1, 6

Offset: 0

Ilya Gutkovskiy, Apr 17 2019

nonn

The greedy inverse starts 0, 10, 19, 37, 52, 82, 109, 136, 241, 226, 217, 247, 364, 427, 457, 541, 532, 577, 637, 961, 721, 787, 1066, 1102, 1381, 1267, 1564, 1192, 1396, 1816, 1501, 1612, 1927, 1942, 2242, 1792, 2842, 2587, 2557, 2422, ... - R. J. Mathar, Apr 28 2020

			a(19) = 2 because we have [15, 3, 1] and [10, 6, 3].

Cf. A000217, A002244, A008443, A024940, A025442, A053604, A063993, A224326, A307597.

a(n) = [x^n y^3] Product_{k>=1} (1 + y*x^(k*(k+1)/2)).

A063993 Number of ways of writing n as an unordered sum of exactly 3 nonzero triangular numbers.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 1, 2, 0, 2, 2, 2, 1, 1, 2, 2, 2, 0, 3, 2, 2, 2, 2, 2, 1, 3, 1, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 3, 1, 2, 5, 1, 2, 1, 2, 5, 3, 3, 1, 4, 2, 3, 2, 2, 4, 4, 2, 1, 4, 3, 3, 3, 2, 4, 3, 3, 3, 4, 2, 1, 6, 1, 5, 3, 3, 5, 2, 2, 2, 5, 2, 5, 4, 2, 4, 5, 3, 1

Offset: 0

N. J. A. Sloane, Sep 18 2001

a(A002097(n)) = 0; a(A111638(n)) = 1; a(A064825(n)) = 2. - Reinhard Zumkeller, Jul 20 2012

			5 = 3 + 1 + 1, so a(5) = 1.

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

Cf. A053604, A008443, A002636, A064181 (greedy inverse), A307598 (3 distinct positive).
Cf. A000217, A010054.
Column k=3 of A319797.

a063993 n = length [() | let ts = takeWhile (< n) $ tail a000217_list,
                    x <- ts, y <- takeWhile (<= x) ts,
                    let z = n - x - y, 0 < z, z <= y, a010054 z == 1]
-- Reinhard Zumkeller, Jul 20 2012

A063993 := proc(n)
    local a,t1idx,t2idx,t1,t2,t3;
    a := 0 ;
    for t1idx from 1 do
        t1 := A000217(t1idx) ;
        if 3*t1 > n then
            break;
        end if;
        for t2idx from t1idx do
            t2 := A000217(t2idx) ;
            if t1+t2 > n then
                break;
            end if;
            t3 :=  n-t1-t2 ;
            if t3 >= t2 then
                if isA000217(t3) then
                    a := a+1 ;
                end if;
            end if ;
        end do:
    end do:
    a ;
end proc: # R. J. Mathar, Apr 28 2020

a = Table[ n(n + 1)/2, {n, 1, 15} ]; b = {0}; c = Table[ 0, {100} ]; Do[ b = Append[ b, a[ [ i ] ] + a[ [ j ] ] + a[ [ k ] ] ], {k, 1, 15}, {j, 1, k}, {i, 1, j} ]; b = Delete[ b, 1 ]; b = Sort[ b ]; l = Length[ b ]; Do[ If[ b[ [ n ] ] < 100, c[ [ b[ [ n ] ] + 1 ] ]++ ], {n, 1, l} ]; c

trmx(n)=my(k=sqrtint(8*n+1)\2);if(k^2+k>2*n,k-1,k)
trmn(n)=trmx(ceil(n)-1)+1
a(n)=if(n<3, return(0)); sum(a=trmn(n/3),trmx(n-2),my(t=n-a*(a+1)/2);sum(b=trmn(t/2),min(trmx(t-1),a), ispolygonal(t-b*(b+1)/2,3))) \\ Charles R Greathouse IV, Jul 07 2022

More terms from Robert G. Wilson v, Sep 20 2001

A340949 Number of ways to write n as an ordered sum of 4 nonzero triangular numbers.

Original entry on oeis.org

1, 0, 4, 0, 6, 4, 4, 12, 1, 16, 6, 16, 12, 12, 22, 8, 36, 4, 30, 24, 21, 36, 18, 36, 28, 48, 16, 44, 36, 44, 48, 36, 46, 40, 72, 20, 73, 48, 54, 72, 42, 68, 56, 84, 50, 72, 78, 56, 84, 84, 62, 112, 60, 60, 110, 84, 97, 72, 120, 76, 116, 84, 72, 144, 102, 104, 96, 108, 102, 156, 102, 92

Offset: 4

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..10000

Cf. A000217, A008438, A010054, A053603, A053604, A319814, A340950, A340951, A340952, A340953, A340954, A340955.

b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
      if n=0 then `if`(k=0, 1, 0) else
      while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..75);  # Alois P. Heinz, Jan 31 2021

nmax = 75; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^4, {x, 0, nmax}], x] // Drop[#, 4] &

G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^4, where theta_2() is the Jacobi theta function.

A340950 Number of ways to write n as an ordered sum of 5 nonzero triangular numbers.

Original entry on oeis.org

1, 0, 5, 0, 10, 5, 10, 20, 5, 35, 11, 40, 30, 35, 55, 30, 90, 25, 100, 60, 80, 120, 60, 140, 90, 161, 100, 165, 135, 165, 210, 140, 220, 180, 265, 170, 295, 200, 285, 330, 205, 365, 260, 395, 295, 391, 350, 355, 480, 340, 455, 490, 415, 480, 515, 445, 600, 510, 565, 550, 680, 545, 555

Offset: 5

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..10000

Cf. A000217, A008439, A010054, A053603, A053604, A319815, A340949, A340951, A340952, A340953, A340954, A340955.

b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
      if n=0 then `if`(k=0, 1, 0) else
      while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..67);  # Alois P. Heinz, Jan 31 2021

nmax = 67; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^5, {x, 0, nmax}], x] // Drop[#, 5] &

G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^5, where theta_2() is the Jacobi theta function.

A340951 Number of ways to write n as an ordered sum of 6 nonzero triangular numbers.

Original entry on oeis.org

1, 0, 6, 0, 15, 6, 20, 30, 15, 66, 21, 90, 61, 90, 126, 86, 210, 90, 270, 156, 261, 320, 210, 450, 261, 516, 375, 542, 495, 570, 727, 540, 870, 650, 966, 816, 1050, 906, 1155, 1266, 1020, 1560, 1090, 1710, 1416, 1698, 1635, 1746, 2120, 1650, 2376, 1980, 2316, 2490, 2368, 2520, 2835

Offset: 6

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..10000

Cf. A000217, A008440, A010054, A053603, A053604, A319816, A340949, A340950, A340952, A340953, A340954, A340955.

b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
      if n=0 then `if`(k=0, 1, 0) else
      while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
    end:
a:= n-> b(n, 6):
seq(a(n), n=6..62);  # Alois P. Heinz, Jan 31 2021

nmax = 62; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^6, {x, 0, nmax}], x] // Drop[#, 6] &

G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^6, where theta_2() is the Jacobi theta function.

A340952 Number of ways to write n as an ordered sum of 7 nonzero triangular numbers.

Original entry on oeis.org

1, 0, 7, 0, 21, 7, 35, 42, 35, 112, 42, 182, 112, 210, 260, 217, 462, 252, 651, 399, 728, 777, 672, 1232, 749, 1533, 1127, 1659, 1617, 1792, 2289, 1890, 2926, 2212, 3339, 2990, 3584, 3654, 4046, 4613, 4263, 5754, 4487, 6636, 5733, 6825, 7014, 7203, 8617, 7560, 10087, 8302

Offset: 7

Ilya Gutkovskiy, Jan 31 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..10000

Cf. A000217, A010054, A053603, A053604, A226252, A319817, A340949, A340950, A340951, A340953, A340954, A340955.

b:= proc(n, k) option remember; local r, t, d; r, t, d:= $0..2;
      if n=0 then `if`(k=0, 1, 0) else
      while t<=n do r:= r+b(n-t, k-1); t, d:= t+d, d+1 od; r fi
    end:
a:= n-> b(n, 7):
seq(a(n), n=7..58);  # Alois P. Heinz, Jan 31 2021

nmax = 58; CoefficientList[Series[(EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)) - 1)^7, {x, 0, nmax}], x] // Drop[#, 7] &

G.f.: (theta_2(sqrt(x)) / (2 * x^(1/8)) - 1)^7, where theta_2() is the Jacobi theta function.

cp's OEIS Frontend

A008443 Number of ordered ways of writing n as the sum of 3 triangular numbers.

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A002636 Number of ways of writing n as an unordered sum of at most 3 nonzero triangular numbers.

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

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A230121 Number of ways to write n = x + y + z (0 < x <= y <= z) such that x*(x+1)/2 + y*(y+1)/2 + z*(z+1)/2 is a triangular number.

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A307598 Number of partitions of n into 3 distinct positive triangular numbers.

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A063993 Number of ways of writing n as an unordered sum of exactly 3 nonzero triangular numbers.

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A340949 Number of ways to write n as an ordered sum of 4 nonzero triangular numbers.

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A230121 Number of ways to write n = x + y + z (0 < x <= y <= z) such that x(x+1)/2 + y(y+1)/2 + z*(z+1)/2 is a triangular number.