The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A180312 Number of solutions to n = x + 4*y + 4*z in triangular numbers. 2
 1, 1, 0, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 3, 3, 2, 2, 3, 4, 0, 1, 4, 1, 3, 5, 2, 5, 3, 3, 3, 4, 2, 2, 5, 0, 4, 4, 2, 5, 6, 2, 2, 4, 5, 6, 4, 2, 3, 5, 4, 3, 7, 3, 3, 5, 2, 4, 3, 4, 5, 6, 2, 4, 8, 6, 3, 8, 2, 4, 8, 2, 6, 6, 5, 4, 3, 0, 5, 7, 5, 5, 6, 3, 5, 10, 2, 6, 6, 4, 10, 5, 4, 3, 10, 5, 4, 4, 2, 9, 8, 3, 7, 7, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS From page 104 of the Sun reference: "(iii) A positive integer n is a sum of an odd square, an even square and a triangular number, unless it is a triangular number t_m (m>0) for which all prime divisors of 2m+1 are congruent to 1 mod 4 and hence t_m = x^2 + x^2 + t_z for some integers x > 0 and z = x == m/2 (mod 2)." Numbers of representations of n + 1 as a sum of an odd square, an even square and a triangular number. REFERENCES Z.-W. Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), no.2, 103--113, see page 104. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Z.-W. Sun, Mixed sums of squares and triangular numbers FORMULA Expansion of q^(-9/8) * eta(q^2)^2 * eta(q^8)^4 / (eta(q) * eta(q^4)^2) in powers of q Expansion of psi(q) * psi(q^8) * phi(q^4) = psi(q) * psi(q^4)^2 in powers of q where phi(), psi() are Ramanujan theta functions. Euler transform of period 8 sequence [ 1, -1, 1, 1, 1, -1, 1, -3, ...]. a(n) = 0 if and only if n+1 = A000217(2 * A094178(m)) for some integer m where A000217 is triangular numbers. G.f.: (Sum_{k>0} x^((n^2 - n)/2)) * (Sum_{k>0} x^(n^2 - n)). EXAMPLE a(10) = 3 since we have 10 = 6 + 4*1 + 4*0 = 6 + 4*0 + 4*1 = 10 + 4*0 + 4*0. a(10) = 3 since we have 10 + 1 = 1^2 + 0^2 + 10 = 1 + 2^2 + 6 = 1 + (-2)^2 + 6. 1 + x + x^3 + 2*x^4 + 2*x^5 + x^6 + 2*x^7 + x^8 + x^9 + 3*x^10 + x^11 + ... MATHEMATICA m=105; psi[q_] = Product[(1-q^(2n))/(1-q^(2n-1)), {n, 1, Floor[m/2]}]; Take[ CoefficientList[ Series[ psi[q]*psi[q^4]^2, {q, 0, m}], q], m] (* Jean-François Alcover, Sep 12 2011, after g.f. *) PROG (PARI) {a(n) = local(A) ; if( n<0, 0, A = x * O(x^n) ; polcoeff( eta(x^2 + A)^2 * eta(x^8 + A)^4 / (eta(x + A) * eta(x^4 + A)^2), n))} CROSSREFS Sequence in context: A087775 A089955 A352942 * A178819 A355855 A046816 Adjacent sequences:  A180309 A180310 A180311 * A180313 A180314 A180315 KEYWORD nonn AUTHOR Michael Somos, Aug 25 2010 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 13 22:39 EDT 2022. Contains 356109 sequences. (Running on oeis4.)