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A174685 Indices i such that 2*A000326(i)+1 is also a pentagonal number.

%I A174685 #43 Feb 16 2025 08:33:12
%S A174685 0,75,244,86359,281384,99658019,324716700,115005267375,374722790224,
%T A174685 132715978892539,432429775201604,153154124636722439,
%U A174685 499023585859860600,176739727114798801875,575872785652503930604,203957491936353180641119
%N A174685 Indices i such that 2*A000326(i)+1 is also a pentagonal number.
%H A174685 Ray Chandler, <a href="/A174685/b174685.txt">Table of n, a(n) for n = 1..653</a> (terms to 1000 digits)
%H A174685 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number</a>.
%H A174685 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SophieGermainPrime.html">Sophie Germain prime</a>.
%H A174685 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1154,-1154,-1,1).
%F A174685 G.f.: x^2*(-75-169*x+435*x^2+x^3) / ( (x-1)*(x^2-34*x+1)*(x^2+34*x+1) ) with a(n) = 1/6 -7*A029547(n) +239*A029547(n-1) +35*A029547(n)*(-1)^n/6 +1183*A029547(n-1)*(-1)^(n-1)/6. - _R. J. Mathar_, Oct 25 2011
%F A174685 a(n)= a(n-1)+ 1154*a(n-2)-1154*a(n-3)-a(n-4)+a(n-5). - _Harvey P. Dale_, Jul 17 2014
%e A174685 Using P(n) = A000326(n) we have:
%e A174685 a(1) = 0 because 2*P(0)+1 = 2*0+1 = 1 = P(1).
%e A174685 a(2) = 75 because 2*P(75)+1 = 2*8400+1 = 16801 = P(106).
%e A174685 a(3) = 244 because 2*P(244)+1 = 2*89182+1 = 178365 = P(345).
%t A174685 LinearRecurrence[{1,1154,-1154,-1,1},{0,75,244,86359,281384},30] (* _Harvey P. Dale_, Jul 17 2014 *)
%o A174685 (PARI) /* _Jack Brennen_, who extended this sequence, found that, other than the trivial pair (0,1), you need to solve: u^2 - 2*v^2 = 23. And take only those values where u and v are both congruent to 5 mod 6. Then you have P((u+1)/6) = 2*P((v+1)/6)+1. The following PARI/GP will give the nontrivial answers: */
%o A174685 {forstep(i=7,47,8,
%o A174685   print(Vec(lift(Mod((x+1),x^2-2)^i*(4*x-3)+(x+1))/6));
%o A174685   print(Vec(lift(Mod((x+1),x^2-2)^i*(4*x+3)+(x+1))/6)))}
%Y A174685 Cf. A000326, A005384, A124174, A260937.
%K A174685 nonn,easy
%O A174685 1,2
%A A174685 _Jonathan Vos Post_, Mar 27 2010