cp's OEIS Frontend

A385152 Pentagonal numbers that are one-fifth of another pentagonal number.

%I A385152 #22 Jul 08 2025 19:02:31
%S A385152 0,1,2262,11017977685,24316671758562,118442787685171571497,
%T A385152 261403178754290105125230,1273254889025744028795358122877,
%U A385152 2810072963163120003620778537378426,13687435462403616663579190345877254457425,30208163863695025530402450846321663951473670
%N A385152 Pentagonal numbers that are one-fifth of another pentagonal number.
%F A385152 Empirical G.f.: (z + 2261*z^2 + 268018301*z^3 + 728035*z^4 + 330*z^5)/(1 - z - 10749957122*z^2 + 10749957122*z^3 + z^4 - z^5). - _Robert Israel_, Jun 30 2025
%e A385152 P = 2262 is a term because both P and 5*P are pentagonal numbers.
%p A385152 ispent:= x -> issqr(1+24*x) and sqrt(1+24*x) mod 6 = 5: ispent(0):= true:
%p A385152 A := (sqrt(5)/2 - 1/2)*(9 + 4*sqrt(5))^i + (-sqrt(5)/2 - 1/2)*(9 - 4*sqrt(5))^i:
%p A385152 B := (sqrt(5)/2 + 1/2)*(9 + 4*sqrt(5))^i + (-sqrt(5)/2 + 1/2)*(9 - 4*sqrt(5))^i:
%p A385152 M:= 10^60: # to get all terms <= M
%p A385152 R:= NULL:
%p A385152 for i from 0 do
%p A385152   t:= simplify(A);
%p A385152 if t mod 6 = 5 then
%p A385152      k:= (t+1)/6; x:= k*(3*k-1)/10;
%p A385152    if x > M then break fi;
%p A385152    if ispent(x) then R:= R, x fi;
%p A385152 fi
%p A385152 od:
%p A385152 for i from 0 do
%p A385152   t:= simplify(B);
%p A385152 if t mod 6 = 5 then
%p A385152      k:= (t+1)/6; x:= k*(3*k-1)/10;
%p A385152    if x > M then break fi;
%p A385152    if ispent(x) then R:= R, x fi;
%p A385152 fi
%p A385152 od:
%p A385152 sort([R]); # _Robert Israel_, Jun 30 2025
%Y A385152 Cf. A000326, A385146.
%K A385152 nonn
%O A385152 1,3
%A A385152 _Kelvin Voskuijl_, Jun 19 2025
%E A385152 a(6) from _Michel Marcus_, Jun 29 2025
%E A385152 More terms from _Robert Israel_, Jun 30 2025