A008443 -id:A008443 - cp's OEIS frontend

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

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

A014455 Theta series of quadratic form with Gram matrix [ 1, 0, 0; 0, 1, 0; 0, 0, 2 ]. Number of integer solutions to x^2 + y^2 + 2*z^2 = n.

Original entry on oeis.org

1, 4, 6, 8, 12, 8, 8, 16, 6, 12, 24, 8, 24, 24, 0, 16, 12, 16, 30, 24, 24, 16, 24, 16, 8, 28, 24, 32, 48, 8, 0, 32, 6, 32, 48, 16, 36, 40, 24, 16, 24, 16, 48, 40, 24, 40, 0, 32, 24, 36, 30, 16, 72, 24, 32, 48, 0, 32, 72, 24, 48, 40, 0, 48, 12, 16, 48, 56, 48, 32, 48, 16, 30, 64

Offset: 0

N. J. A. Sloane

This is the tetragonal P lattice (the classical holotype) of dimension 3.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 4*q + 6*q^2 + 8*q^3 + 12*q^4 + 8*q^5 + 8*q^6 + 16*q^7 + 6*q^8 + 12*q^9 + ...

John Cannon, Table of n, a(n) for n = 0..10000
G. Nebe and N. J. A. Sloane, Home page for this lattice
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A004015, A005875, A005877, A008443, A033763, A045826, A045828, A156384, A213024, A246631.

A := Basis( ModularForms( Gamma1(8), 3/2), 40); A[1] + 4*A[2] + 6*A[3] + 8*A[4]; /* Michael Somos, Aug 31 2014 */

r[n_, z_] := Reduce[x^2 + y^2 + 2*z^2 == n, {x, y}, Integers]; a[n_] := Module[{rn0, rnz, k0, k}, rn0 = r[n, 0]; k0 = If[rn0 === False, 0, If[Head[rn0] === And, 1, Length[rn0]]]; For[k = 0; z = 1, z <= Ceiling[Sqrt[n/2]], z++, rnz = r[n, z]; If[rnz =!= False, k = If[Head[rnz] === And, k+1, k + Length[rnz]]]]; k0 + 2*k]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 07 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^2 EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* Michael Somos, Aug 31 2014 *)
QP = QPochhammer; s = QP[q^2]^8*(QP[q^4]/(QP[q]^4*QP[q^8]^2)) + O[q]^80; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015, after Michael Somos *)

{a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0, 0; 0, 1, 0; 0, 0, 2], n)[n])}; /* Michael Somos, Jul 05 2005 */

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^4 + A) / (eta(x + A)^4 * eta(x^8 + A)^2), n))}; /* Michael Somos, Jul 05 2005 */

Expansion of phi(q)^2 * phi(q^2) = psi(q)^4 / psi(q^4) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Apr 07 2012
Expansion of eta(q^2)^8 * eta(q^4) / (eta(q)^4 * eta(q^8)^2) in powers of q. - Michael Somos, Jul 05 2005
Euler transform of period 8 sequence [4, -4, 4, -5, 4, -4, 4, -3, ...]. - Michael Somos, Jul 07 2005
G.f.: theta_3(q)^2 * theta_3(q^2) = Product_{k>0} (1 - x^(2*k))^8 * (1 - x^(4*k)) / ((1 - x^k)^4 * (1 - x^(8*k))^2).
There is a classical formula (essentially due to Gauss): Write (uniquely) -2n=D(2^vf)^2, with D<0 fundamental discriminant, f odd, v>=-1. Then a(n)=12L((D/.),0)(1-(D/2))\sum_{d\mid f}\mu(d)(D/d)sigma(f/d) (the formula for A005875), except that the factor (1-(D/2)) has to be replaced by 1/3 if v=-1 and by 1 if v=0 (and kept if v>=1). Here mu() is the Moebius function, (D/2) and (D/d) are Kronecker-Legendre symbols, sigma() is the sum of divisors function, L((D/.),0)=h(D)/(w(D)/2) is the value at 0 of the L() function of the quadratic character (D/.), equal to the class number h(D) divided by 2 or 3 in the special cases D=-4 and -3. - Henri Cohen (Henri.Cohen(AT)math.u-bordeaux1.fr), May 12 2010
a(2*n) = a(8*n) = A005875(n). a(2*n + 1) = A005877(n) = 4 * A045828(n). a(4*n) = A004015(n). a(4*n + 2) = 2 * A045826(n). a(8*n + 4) = 12 * A045828(n). a(8*n + 7) = 16 * A033763(n). a(16*n + 6) = 8 * A008443(n). a(16*n + 14) = 0. - Michael Somos, Apr 07 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246631.

A213624 Expansion of psi(x)^2 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 4, 4, 2, 2, 5, 4, 2, 6, 3, 6, 7, 2, 5, 4, 5, 6, 6, 2, 5, 10, 3, 6, 10, 4, 6, 8, 3, 8, 7, 6, 7, 6, 4, 6, 11, 6, 9, 10, 3, 6, 14, 4, 8, 10, 8, 10, 5, 6, 4, 16, 7, 4, 10, 4, 13, 14, 7, 8, 8, 6, 10, 12, 7, 12, 15, 8, 8, 10, 4, 6, 17, 6, 10, 10

Offset: 0

Michael Somos, Jun 16 2012

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Dickson writes that Liouville proved several related results about sums of triangular number. In particular, that every nonnegative integer is the sum of t1 + t2 + 4*t3 where t1, t2, t3 are trianglular numbers. - Michael Somos, Feb 10 2020

			G.f. = 1 + 2*x + x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 2*x^8 + 2*x^9 + ...
G.f. = q^3 + 2*q^7 + q^11 + 2*q^15 + 3*q^19 + 2*q^23 + 4*q^27 + 4*q^31 + 2*q^35 + ...

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 23.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008443, A045818,

a[ n_] := SeriesCoefficient[ 1/8 EllipticTheta[ 2, 0, q]^2 EllipticTheta[ 2, 0, q^4], {q, 0, 2 n + 3/2}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)), n))};

Expansion of q^(-3/4) * eta(q^2)^4 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)) in powers of q.

Euler transform of period 8 sequence [2, -2, 2, -1, 2, -2, 2, -3, ...].

A240088 The number of ways of writing n as an ordered sum of a triangular number (A000217), a square (A000290) and a pentagonal number (A000326).

Original entry on oeis.org

1, 3, 3, 2, 3, 4, 4, 4, 3, 3, 5, 5, 5, 3, 3, 7, 7, 5, 2, 6, 5, 4, 8, 5, 6, 4, 8, 7, 5, 7, 4, 9, 6, 5, 4, 3, 9, 12, 9, 4, 7, 9, 8, 4, 6, 8, 7, 8, 4, 8, 9, 10, 9, 6, 10, 6, 7, 10, 9, 8, 7, 11, 7, 4, 10, 8, 10, 10, 7, 5, 10, 14, 11, 7, 6, 11, 10, 10, 4, 11, 10, 10, 13, 8, 7, 7, 13, 12, 8, 8, 6, 10, 17, 8, 10, 7, 16, 10, 3, 12, 9

Offset: 0

Robert G. Wilson v, Mar 31 2014

nonn

0 and 1 are triangular numbers, square numbers and pentagonal numbers.
It is conjectured that a(n) is always positive - this is one of the conjectures in Conjecture 1.1 of Sun (2009). - N. J. A. Sloane, Apr 01 2014
Note that both the conjecture in A160325 and the conjecture in A160324 imply that a(n) is always positive. - Zhi-Wei Sun, Apr 01 2014
a(n) > 0 for all n < 10^10. - Robert G. Wilson v, Aug 20 2016
Least number to be represented k ways, k >= 1: 0, 3, 1, 5, 10, 19, 15, 22, 31, 51, 61, 37, 82, 71, 126, 96, 92, 136, 162, 187, 206, 276, 191, 261, 236, 247, 317, 302, 401, 292, 422, 547, 456, 544, 551, 612, 591, 577, 521, 666, 742, 726, 682, 877, 796, 1052, 961, 1046, 1171, 1027, ..., . A275999.
Greatest number (conjectured) to be represented k ways, k >= 1: 0, 18, 168, 78, 243, 130, 553, 455, 515, 658, 865, 945, 633, 1918, 2258, 1385, 1583, 2828, 2135, 2335, 2785, 4533, 3168, 3478, 2790, 3868, 4193, 7328, 4953, 5278, 6390, 8148, 8015, 4585, 9160, 10485, 7613, 12333, 12025, 10178, 9923, 9720, 12558, 11340, 17420, 11753, 14893, 16155, 16415, 14343, ..., .
Conjectured lists of numbers that are represented in k >= 1 ways:
1: 0;
2: 3, 18;
3: 1, 2, 4, 8, 9, 13, 14, 35, 98, 168;
4: 5, 6, 7, 21, 25, 30, 34, 39, 43, 48, 63, 78;
5: 10, 11, 12, 17, 20, 23, 28, 33, 69, 193, 203, 230, 243;
6: 19, 24, 32, 44, 53, 55, 74, 90, 111, 130;
7: 15, 16, 27, 29, 40, 46, 56, 60, 62, 68, 73, 84, 85, 95, 108, 113, 123, 135, 139, 163, 165, 273, 553;
8: 22, 26, 42, 45, 47, 49, 59, 65, 83, 88, 89, 93, 112, 119, 125, 134, 140, 144, 186, 205, 233, 244, 320, 405, 455;
9: 31, 36, 38, 41, 50, 52, 58, 100, 109, 124, 160, 214, 249, 308, 358, 515; ..., .

Robert G. Wilson v and Robert Israel, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, Science China Mathematics, Vol. 58, No. 7 (2015), 1367-1396; arXiv:0905.0635 [math.NT], 2009-2015 [Edited, _Felix Fröhlich_, Aug 24 2016].
Zhi-Wei Sun, Various new conjectures involving polygonal numbers and primes, a message to the Number Theory List, May 8 2009.

Cf. A000925, A008443, A101428, A115171, A115172, A115173, A115174, A115175, A115176, A115177, A144642.

Cf. A160324, A160325, A160326. - Zhi-Wei Sun, Apr 01 2014

# requires Maple 17 and up
with(SignalProcessing):
N:= 10000;  # to get terms up to a(N)
A:= Array(0..N,datatype=float);
B:= Array(0..N,datatype=float);
C:= Array(0..N,datatype=float);
for i from 0 to floor(sqrt(N)) do A[i^2]:= 1 od:
for i from 0 to floor((1+sqrt(1+8*N))/2) do B[i*(i-1)/2]:= 1 od:
for i from 0 to floor((1+sqrt(1+24*N))/6) do C[i*(3*i-1)/2]:= 1 od:
R:= Convolution(Convolution(A,B),C);
R:= evalhf(map(round,R));
# Note that a(i) = R[i+1] for i from 0 to N
# Robert Israel, Apr 01 2014

p = Table[n (3n - 1)/2, {n, 0, 26}]; s = Table[n^2, {n, 0, 32}]; t = Table[n (n + 1)/2, {n, 0, 45}]; a = Sort@ Flatten@ Table[ p[[i]] + s[[j]] + t[[k]], {i, 26}, {j, 32}, {k, 45}]; Table[ Count[a, n], {n, 0, 105}]

A286180 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k in powers of x.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 1, 0, 1, 4, 3, 2, 0, 0, 1, 5, 6, 4, 2, 0, 0, 1, 6, 10, 8, 6, 0, 1, 0, 1, 7, 15, 15, 13, 3, 3, 0, 0, 1, 8, 21, 26, 25, 12, 6, 2, 0, 0, 1, 9, 28, 42, 45, 31, 14, 9, 0, 0, 0, 1, 10, 36, 64, 77, 66, 35, 24, 3, 2, 1, 0, 1, 11, 45

Offset: 0

Seiichi Manyama, May 07 2017

A(n, k) is the number of ways of writing n as the sum of k triangular numbers.

			Square array begins:
   1, 1, 1, 1,  1,  1, ...
   0, 1, 2, 3,  4,  5, ...
   0, 0, 1, 3,  6, 10, ...
   0, 1, 2, 4,  8, 15, ...
   0, 0, 2, 6, 13, 25, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-12 give A000007, A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787.

Main diagonal gives A106337.

Table[Function[k, SeriesCoefficient[Product[(1 + x^i) (1 - x^(2 i)), {i, Infinity}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten (* Michael De Vlieger, May 07 2017 *)

G.f. of column k: (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k.

A004013 Theta series of body-centered cubic (b.c.c.) lattice.

Original entry on oeis.org

1, 0, 0, 8, 6, 0, 0, 0, 12, 0, 0, 24, 8, 0, 0, 0, 6, 0, 0, 24, 24, 0, 0, 0, 24, 0, 0, 32, 0, 0, 0, 0, 12, 0, 0, 48, 30, 0, 0, 0, 24, 0, 0, 24, 24, 0, 0, 0, 8, 0, 0, 48, 24, 0, 0, 0, 48, 0, 0, 72, 0, 0, 0, 0, 6, 0, 0, 24, 48, 0, 0, 0, 36, 0, 0, 56, 24, 0, 0, 0, 24, 0, 0, 72, 48, 0, 0, 0, 24, 0, 0

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

The bcc lattice is also called the coweight lattice A^*3 (or D^*_3), dual to the root lattice A_3 (or D_3, or the fcc lattice), or the permutohedral lattice A^*_3. - _Andrey Zabolotskiy, Mar 08 2020

			G.f. = 1 + 8*x^3 + 6*x^4 + 12*x^8 + 24*x^11 + 8*x^12 + 6*x^16 + 24*x^19 + 24*x^20 + ...
G.f. = 1 + 8*q^(3/2) + 6*q^2 + 12*q^4 + 24*q^(11/2) + 8*q^6 + 6*q^8 + 24*q^(19/2) + 24*q^10 + 24*q^12 + 32*q^(27/2) + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 116.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Cannon, Table of n, a(n) for n = 0..10000
S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan's theta function, Proc. Amer. Math. Soc., 128 (1999), 1333-1338; F_4(q).
R. J. Mathar, Hierarchical Subdivision of the Simple Cubic Lattice, arXiv preprint arXiv:1309.3705 [math.MG], 2013.
G. Nebe and N. J. A. Sloane, Home page for this lattice
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Theta Series
Index entries for sequences related to b.c.c. lattice

Cf. A004015, A005875, A008443, A045826, A045828, A045834, A213056.
Indices of nonzero terms are A004014.
Cf. A023916-A023936.
Cf. A004011, A008422, A008425, A008423, A008427, A008424, A008426.

Basis( ModularForms( Gamma0(8), 3/2), 90) [1]; /* Michael Somos, Sep 04 2014 */

M:=100; M1:=M*(M+1)/2; ph:=series(add(q^(k^2),k=-M..M),q,M1): ps:=series(add(q^(k*(k+1)/2),k=0..M),q,M1): t1:=series(subs(q=q^2, ph)^3, q,M1): t2:=series((2*sqrt(q))^3*subs(q=q^4, ps)^3,q,M1): t3:=seriestolist(series(subs(q=q^2,t1+t2),q,M1)): for n from 0 to nops(t3)-1 do lprint(n,t3[n+1]); od:

m = 13; m1 = m*((m + 1)/2); ph[q_] = Series[ Sum[ q^k^2, {k, -m, m}], {q, 0, m1}]; ps[q_] = Series[ Sum[ q^(k*((k + 1)/2)), {k, 0, m}], {q, 0, m1}]; t1[q_] = Normal[ Series[ ph[q^2]^3, {q, 0, m1}]]; t2[q_] = Normal[ Series[ (2*Sqrt[q])^3*ps[q^4]^3, {q, 0, m1}]]; CoefficientList[ Series[ t1[q^2] + t2[q^2], {q, 0, m1}], q] (* Jean-François Alcover, Dec 20 2011, translated from Maple *)
(* From version 6 on *) terms=91; f[q_] = LatticeData["BodyCenteredCubic", "ThetaSeriesFunction"][-I Log[q]/Pi]; CoefficientList[Simplify[f[q] + O[q]^terms, q>0], q][[1 ;; terms]] (* Jean-François Alcover, May 15 2013, updated Jul 08 2017 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^4]^3 + EllipticTheta[ 2, 0, x^4]^3, {x, 0, n}]; (* Michael Somos, May 24 2013 *)

{a(n) = if( n<0, 0, if( n%4==0, n/=4; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1 + x * O(x^n))^3, n), n%8==3, n\=8; 8*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^8 + A)^5 / eta(x^4 + A)^2 / eta(x^16 + A)^2)^3 + (2 * x * eta(x^16 + A)^2 / eta(x^8 + A))^3, n))}; /* Michael Somos, May 17 2008 */

subs(q=q^2, ph)^3+(2*sqrt(q))^3*subs(q=q^4, ps)^3, where ps = A010054 = Sum_{k=0..infinity} q^(k*(k+1)/2), ph = A000122 = Sum_{k=-infinity, infinity} q^(k^2).
Expansion of phi(q^4)^3 + 8 * q^3 * psi(q^8)^3 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Oct 25 2006
a(4*n + 1) = a(4*n + 2) = a(8*n + 7) = 0. a(4*n) = A005875(n).
Expansion of theta_3(q)^3 + theta_2(q)^3 in powers of q^(1/4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A004015.
a(8*n) = A004015(n). a(8*n + 3) = 8 * A008443(n). a(8*n + 4) = 2 * A045826(n). - Michael Somos, Jul 19 2015
a(12*n + 4) = 6 * A213056(n). a(16*n + 4) = 6 * A045834(n). a(16*n + 8) = 12 * A045828(n).

A213384 Expansion of phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 12, -8, 6, -24, 24, 0, 12, -30, 24, -24, 8, -24, 48, 0, 6, -48, 36, -24, 24, -48, 24, 0, 24, -30, 72, -32, 0, -72, 48, 0, 12, -48, 48, -48, 30, -24, 72, 0, 24, -96, 48, -24, 24, -72, 48, 0, 8, -54, 84, -48, 24, -72, 96, 0, 48, -48, 24, -72, 0, -72, 96

Offset: 0

Michael Somos, Jun 10 2012

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - 6*q + 12*q^2 - 8*q^3 + 6*q^4 - 24*q^5 + 24*q^6 + 12*q^8 - 30*q^9 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from G. C. Greubel)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A004015, A005875, A008443, A045826, A045834.

# JacobiTheta4 is defined in A002448.
A213384List(len) = JacobiTheta4(len, 3)
A213384List(63) |> println # Peter Luschny, Mar 12 2018

A := Basis( ModularForms( Gamma0(16), 3/2), 63); A[1] - 6*A[2] + 12*A[3] - 8*A[4]; /* Michael Somos, May 21 2015 */

a[ n_] := (-1)^n SquaresR[ 3, n]; (* Michael Somos, May 21 2015 *)
a[ n_] := (-1)^n Length @ FindInstance[ n == x^2 + y^2 + z^2, {x, y, z}, Integers, 10^9]; (* Michael Somos, May 21 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^3, {q, 0, n}]; (* Michael Somos, May 21 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^2 / QPochhammer[ q^2])^3, {q, 0, n}]; (* Michael Somos, May 21 2015 *)

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

{a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), 2 * (-x)^k^2, 1 + x * O(x^n))^3, n))}; /* Michael Somos, May 21 2015 */

{a(n) = my(G); if( n<0, 0, G = [ 1, 0, 0; 0, 1, 0; 0, 0, 1]; (-1)^n * polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* Michael Somos, May 21 2015 */

Expansion of (eta(q)^2 / eta(q^2))^3 in powers of q.
Euler transform of period 2 sequence [ -6, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(15/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008443.
G.f.: (Sum_{k in Z} (-1)^k * x^k^2)^3.
a(n) = (-1)^n * A005875(n). a(2*n) = A004015(n). a(2*n + 1) = -2 * A045826(n). a(4*n) = A005875(n). a(4*n + 1) = -6 * A045834(n). a(4*n + 2) = 12 * A045828(n). a(8*n + 3) = -8 * A008443(n). a(8*n + 7) = 0.

A212885 Expansion of phi(q) * phi(-q)^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, -4, 8, 6, -8, -8, 0, 12, -10, -8, 24, 8, -8, -16, 0, 6, -16, -12, 24, 24, -16, -8, 0, 24, -10, -24, 32, 0, -24, -16, 0, 12, -16, -16, 48, 30, -8, -24, 0, 24, -32, -16, 24, 24, -24, -16, 0, 8, -18, -28, 48, 24, -24, -32, 0, 48, -16, -8, 72, 0, -24, -32

Offset: 0

Michael Somos, May 29 2012

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 - 2*q - 4*q^2 + 8*q^3 + 6*q^4 - 8*q^5 - 8*q^6 + 12*q^8 - 10*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A004015, A004024, A005875, A005877, A005878, A005887, A008443, A045828, A045834, A213624.

a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]* EllipticTheta[3, 0, -q]^2, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Nov 30 2017 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x + A)^2 / eta(x^4 + A)^2, n))};

Expansion of phi(-x) * phi(-x^2)^2 = phi(-x^2)^4 / phi(x) in powers of x where phi() is a Ramanujan theta function.
Expansion of eta(q^2)^3 * eta(q)^2 / eta(q^4)^2 in powers of q.
Euler transform of period 4 sequence [-2, -5, -2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A045828.
G.f.: Product_{k>0} (1 - x^(2*k))^3 * (1 - x^k)^2 / (1 - x^(4*k))^2.
a(4*n) = A005875(n). a(4*n + 1) = -2 * A045834(n). a(4*n + 2) = - A005877(n) = -4 * A045828(n).
a(8*n) = A004015(n). a(8*n + 3) = A005878(n) = 8 * A008443(n). a(8*n + 4)= A005887(n). a(8*n + 5) = -2 * A004024(n). a(8*n + 6) = -8 * A213624(n). a(8*n + 7) = 0.

A005878 Theta series of cubic lattice with respect to deep hole.

Original entry on oeis.org

8, 24, 24, 32, 48, 24, 48, 72, 24, 56, 72, 48, 72, 72, 48, 48, 120, 72, 56, 96, 24, 120, 120, 48, 96, 96, 72, 96, 120, 48, 104, 168, 96, 48, 120, 72, 96, 192, 72, 144, 96, 72, 144, 120, 96, 104, 192, 72, 120, 192, 48, 144, 216, 48, 96, 120, 144, 192, 168, 120, 96, 216, 72

Offset: 0

N. J. A. Sloane

Number of ways of writing 8*n+3 as the sum of three odd squares. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

Expansion of Jacobi theta constant theta_2^3. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..99 from Herman Jamke (hermanjamke(AT)fastmail.fm))
G. Nebe and N. J. A. Sloane, Home page for this lattice

Equals 8 times A008443. Cf. A085121.

QP = QPochhammer; CoefficientList[(2 QP[q^2]^2/QP[q])^3 + O[q]^63, q] (* Jean-François Alcover, Jul 04 2017 *)

{a(n)=if(n<0, 0, 8*polcoeff( sum(k=0, (sqrtint(8*n+1)-1)\2, x^((k^2+k)/2), x*O(x^n))^3, n))} {a(n)=local(A); if(n<0, 0, A=x*O(x^n); 8*polcoeff( (eta(x^2+A)^2/eta(x+A))^3, n))} \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

G.f.: Form (Sum_{n=-inf..inf} q^((2n+1)^2))^3, then divide by q^3 and set q = x^(1/8).

a(n) = 8 * A008443(n).

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 23 2008

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A213624 Expansion of psi(x)^2 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.

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A240088 The number of ways of writing n as an ordered sum of a triangular number (A000217), a square (A000290) and a pentagonal number (A000326).

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A286180 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k in powers of x.

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A004013 Theta series of body-centered cubic (b.c.c.) lattice.

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