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

0, 1, 2262, 11017977685, 24316671758562, 118442787685171571497, 261403178754290105125230, 1273254889025744028795358122877, 2810072963163120003620778537378426, 13687435462403616663579190345877254457425, 30208163863695025530402450846321663951473670

Offset: 1

Author

Kelvin Voskuijl, Jun 19 2025

Keywords

• nonn

Examples

			P = 2262 is a term because both P and 5*P are pentagonal numbers.
		

Crossrefs

Cf. A000326, A385146.

Programs

Maple

ispent:= x -> issqr(1+24*x) and sqrt(1+24*x) mod 6 = 5: ispent(0):= true:
A := (sqrt(5)/2 - 1/2)*(9 + 4*sqrt(5))^i + (-sqrt(5)/2 - 1/2)*(9 - 4*sqrt(5))^i:
B := (sqrt(5)/2 + 1/2)*(9 + 4*sqrt(5))^i + (-sqrt(5)/2 + 1/2)*(9 - 4*sqrt(5))^i:
M:= 10^60: # to get all terms <= M
R:= NULL:
for i from 0 do
  t:= simplify(A);
if t mod 6 = 5 then
     k:= (t+1)/6; x:= k*(3*k-1)/10;
   if x > M then break fi;
   if ispent(x) then R:= R, x fi;
fi
od:
for i from 0 do
  t:= simplify(B);
if t mod 6 = 5 then
     k:= (t+1)/6; x:= k*(3*k-1)/10;
   if x > M then break fi;
   if ispent(x) then R:= R, x fi;
fi
od:
sort([R]); # Robert Israel, Jun 30 2025

Formula

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

Extensions

a(6) from Michel Marcus, Jun 29 2025

More terms from Robert Israel, Jun 30 2025