A290740 -id:A290740 - 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

A290735 a(n) = weighted sum over all the self-conjugate partitions of 4n + 1 into odd parts, with respect to a certain weight.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 5, 6, 4, 6, 7, 6, 7, 8, 6, 7, 11, 7, 8, 10, 6, 11, 12, 7, 10, 12, 8, 11, 13, 8, 11, 16, 10, 9, 15, 8, 13, 18, 9, 14, 14, 10, 15, 16, 10, 13, 20, 11, 13, 20, 8, 17, 22, 8, 14, 17, 15, 18, 20, 12, 14, 23, 12, 14, 20, 12, 21, 25, 9, 16, 22, 14, 21, 22, 12, 15, 26, 16, 14, 26

Offset: 0

N. J. A. Sloane, Aug 10 2017

nonn

See Andrews (2016) for the definition of the particular weight used here.
Andrews (2016), Theorem 2, shows that A008443(n) = a(n) + A290737(n) + A290739(n).
Andrews conjectures that a(n) > 0 for all n. The conjecture is known to be true for n <= 1000.
Andrews also conjectures that a(n) > |A290737(n) + A290739(n)| for n >= 2 (see A290740).

George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See Lemma 3.2.

Cf. A008443, A290733, A290734, A290736, A290737, A290738, A290739, A290740.

M:=101;
B:=proc(a,q,n) local j,t1; global M; t1:=1;
for j from 0 to M do t1:=t1*(1-a*q^j)/(1-a*q^(n+j)); od;
t1; end;
D1:=add( (-1)^m*q^(m*(m+1))/(B(q,q^2,m+1)*(1-q^(2*m+1))), m=0..M):
series(D1,q,M); d1seq:=seriestolist(%);

M = 101;
B[a_, q_, n_] := Module[{j, t1},  t1 = 1; For[j = 0, j <= M, j++, t1 = t1*(1-a*q^j)/(1-a*q^(n+j))]; t1];
D1 = Sum[(-1)^m*q^(m*(m+1))/(B[q, q^2, m+1]*(1-q^(2*m+1))), {m, 0, M}];
Series[D1, {q, 0, M}] // CoefficientList[#, q]& (* Jean-François Alcover, Mar 16 2023, after Maple code *)

See Maple code for g.f.

A290737 Weighted count of partitions of 2n+1 into odd parts in which the largest part appears an odd number of times and all other parts appear twice, with respect to a certain weight.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Aug 10 2017

sign

See Andrews (2016) for the definition of the particular weight used here.

Andrews (2016), Theorem 2, shows that A008443(n) = A290735(n) + a(n) + A290739(n).

George E. Andrews, The Bhargava-Adiga Summation and Partitions, 2016. See Lemma 3.3.

Cf. A008443, A290733-A290740.

M:=201;
B:=proc(a, q, n) local j, t1; global M; t1:=1;
for j from 0 to M do t1:=t1*(1-a*q^j)/(1-a*q^(n+j)); od;
t1; end;
D2:=add( q^(2*m+1)*B(q^2,q^4,m)/(1-q^(4*m+2)), m=0..M):
series(D2,q,M); d2seq:=seriestolist(%); BISECT(%,1);

See Maple code for g.f.

cp's OEIS Frontend

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

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A290735 a(n) = weighted sum over all the self-conjugate partitions of 4n + 1 into odd parts, with respect to a certain weight.

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A290737 Weighted count of partitions of 2n+1 into odd parts in which the largest part appears an odd number of times and all other parts appear twice, with respect to a certain weight.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula