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%I #27 Mar 20 2023 14:14:27
%S 1,3,4,5,9,10,11,14,15,16,17,19,21,23,24,25,26,29,30,31,35,36,37,39,
%T 41,43,44,45,49,50,51,54,55,56,57,59,61,63,64,65,66,69,70,71,75,76,77,
%U 79,81,83,84,85,89,90,91,94,95,96,97,99,101,103,104,105,106
%N Numbers that are congruent to {+-1, +-3, +-4, +-5, +-9, +-10, +-11, +-14, +-15, +-16, +-17, +-19} (mod 40).
%H G. E. Andrews, <a href="http://www.jstor.org/stable/2322727">Further Problems on Partitions</a>, Amer. Math. Monthly 94 (1987), no. 5, 437-439.
%H <a href="/index/Rec#order_24">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-2,2,-1).
%F From _Colin Barker_, Dec 01 2019: (Start)
%F G.f.: x*(1 + x)*(1 + x^3 + 2*x^4 - 4*x^5 + 7*x^6 - 7*x^7 + 8*x^8 - 8*x^9 + 9*x^10 - 8*x^11 + 9*x^12 - 8*x^13 + 8*x^14 - 7*x^15 + 7*x^16 - 4*x^17 + 2*x^18 + x^19 + x^22) / ((1 - x)^2*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 - x^2 + x^4)*(1 + x^4)*(1 - x^4 + x^8)).
%F a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - 2*a(n-6) + 2*a(n-7) - 2*a(n-8) + 2*a(n-9) - 2*a(n-10) + 2*a(n-11) - 2*a(n-12) + 2*a(n-13) - 2*a(n-14) + 2*a(n-15) - 2*a(n-16) + 2*a(n-17) - 2*a(n-18) + 2*a(n-19) - 2*a(n-20) + 2*a(n-21) - 2*a(n-22) + 2*a(n-23) - a(n-24) for n>24.
%F (End)
%p f:=proc(n,M,lis) local i; if member( n mod M, lis) or member( -n mod M, lis) then 1 else 0; fi; end;
%p lis3:=[1,3,4,5,9,10,11,14,15,16,17,19];
%p lis := lis3;
%p a:=[];
%p for n from 1 to 200 do
%p if f(n,40,lis) = 1 then a:=[op(a), n]; fi; od:
%p a;
%t okQ[n_] := AnyTrue[Join[lis = {1, 3, 4, 5, 9, 10, 11, 14, 15, 16, 17, 19}, 40 - lis], Mod[n, 40] == #&];
%t Select[Range[200], okQ] (* _Jean-François Alcover_, Mar 09 2023 *)
%Y Cf. A329779, A329780, A329782, A329783, A329784.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Nov 29 2019
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