A195849 Column 5 of array A195825. Also column 1 of triangle A195839. Also 1 together with the row sums of triangle A195839.

1, 1, 1, 1, 1, 1, 2, 3, 4, 4, 4, 4, 5, 7, 10, 12, 13, 13, 14, 16, 21, 27, 32, 34, 36, 38, 44, 54, 67, 77, 84, 88, 95, 107, 128, 152, 174, 188, 200, 215, 242, 281, 329, 370, 402, 428, 462, 513, 589, 674, 754, 816, 873, 940, 1041, 1176, 1333, 1477, 1600, 1710, 1845

Offset: 0

Author

Omar E. Pol, Oct 07 2011

Keywords

• nonn

Comments

Note that this sequence contains three plateaus: [1, 1, 1, 1, 1, 1], [4, 4, 4, 4], [13, 13]. For more information see A210843. See also other columns of A195825. - Omar E. Pol, Jun 29 2012

Number of partitions of n into parts congruent to 0, 1 or 6 (mod 7). - Ludovic Schwob, Aug 05 2021

Links

• Ludovic Schwob, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000041, A001082, A006950, A036820, A057077, A118277, A195825, A195829, A195839, A195848, A195850, A195851, A195852, A196933, A210843, A210964, A211971.

Programs

Maple

A118277 := proc(n)
        7*n^2/8+7*n/8-3/16+3*(-1)^n*(1/16+n/8) ;
end proc:
A195839 := proc(n, k)
        option remember;
        local ks, a, j ;
        if A118277(k) > n then
                0 ;
        elif n <= 5 then
                return 1;
        elif k = 1 then
                a := 0 ;
                for j from 1 do
                        if A118277(j) <= n-1 then
                                a := a+procname(n-1, j) ;
                        else
                                break;
                        end if;
                end do;
                return a;
        else
                ks := A118277(k) ;
                (-1)^floor((k-1)/2)*procname(n-ks+1, 1) ;
        end if;
end proc:
A195849 := proc(n)
        A195839(n+1,1) ;
end proc:
seq(A195849(n), n=0..60) ; # R. J. Mathar, Oct 08 2011

Mathematica

m = 61;
Product[1/((1 - x^(7k))(1 - x^(7k - 1))(1 - x^(7k - 6))), {k, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Apr 13 2020, after Ilya Gutkovskiy *)

Formula

G.f.: Product_{k>=1} 1/((1 - x^(7*k))*(1 - x^(7*k-1))*(1 - x^(7*k-6))). - Ilya Gutkovskiy, Aug 13 2017

a(n) ~ exp(Pi*sqrt(2*n/7)) / (8*sin(Pi/7)*n). - Vaclav Kotesovec, Aug 14 2017