A333818 G.f.: Sum_{k>=1} x^(k*(3*k - 2)) / (1 - x^(6*k)).

1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1

Offset

1

Comments

Number of ways to write n as the difference of two octagonal numbers.

a(n) = 2 if n = 133, 175, 176, 217, 224, 259, 272, 280, 301, 320, 343, 368, 385, 400, ... a(n) = 3 if n = 560, 637, 896, 935, ... a(n) = 4 if n = 1729, 2240, 2275, ... - R. J. Mathar, Oct 08 2020 [modified by Ilya Gutkovskiy, Oct 09 2020]

Links

Table of n, a(n) for n=1..95.

Formula

G.f.: Sum_{i>=0} Sum_{j>=i} Product_{k=i..j} x^(6*k + 1).

Example

a(1729) = 4 with representations 1729 = 1825-96 = 2465-736 = 5985-4256 = 249985-248256. - R. J. Mathar, Oct 08 2020

Maple

A333818 := proc(n)
    local a, hi, hiO, lo, loO;
    a := 0 ;
    for hi from 1 do
        hiO := A000567(hi) ;
        for lo from hi-1 to 1 by -1 do
            loO := A000567(lo) ;
            if lo = hi-1 and hiO-loO > n then
                return a;
            end if;
            if hiO-loO = n then
                a := a+1 ;
            elif hiO-loO > n then
                break;
            end if ;
        end do:
    end do:
end proc:
seq( A333818(n), n=1..300) ; # R. J. Mathar, Oct 08 2020

Mathematica

nmax = 95; CoefficientList[Series[Sum[x^(k (3 k - 2))/(1 - x^(6 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Crossrefs

Cf. A000567, A001227, A034178, A333815, A333816, A333817, A334037.

Sequence in context: A266243 A266608 A066288 * A111412 A080111 A204545

Adjacent sequences: A333815 A333816 A333817 * A333819 A333820 A333821

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 06 2020

Status

approved