A233474 Numbers m such that 5*T(m)-1 is a square, where T = A000217.

1, 4, 52, 169, 1993, 6436, 75700, 244417, 2874625, 9281428, 109160068, 352449865, 4145207977, 13383813460, 157408743076, 508232461633, 5977387028929, 19299449728612, 226983298356244, 732870857225641, 8619387950508361, 27829793124845764

Offset

1,2

Comments

Sum_{i=1..infinity} 1/a(i) = 1.2758228304947598524736181699610...

Links

Bruno Berselli, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (1,38,-38,-1,1)

Formula

G.f.: x*(1 + 3*x + 10*x^2 + 3*x^3 + x^4) / ((1 - x)*(1 - 38*x^2 + x^4)).

a(n) = a(n-1) + 38*a(n-2) - 38*a(n-3) - a(n-4) + a(n-5) for n>5, a(1)=1, a(2)=4, a(3)=52, a(4)=169, a(5)=1993.

a(n) = -1/2 + ( (15 + 4*sqrt(10)*(-1)^n)*(19 - 6*sqrt(10))^floor(n/2) + (15 - 4*sqrt(10)*(-1)^n)*(19 + 6*sqrt(10))^floor(n/2) )/20.

Example

169 is in the sequence because 5*169*170/2-1 = 268^2.

Mathematica

LinearRecurrence[{1, 38, -38, -1, 1}, {1, 4, 52, 169, 1993}, 30]

Crossrefs

Cf. A000217, A129556 (numbers m such that 5*A000217(m)+1 is a square), A132593 (numbers m such that 5*A000217(m) is a square).

Cf. numbers m such that k*A000217(m)-1 is a square: A072221 for k=1; m=1 for k=2; this sequence for k=5.

Sequence in context: A232507 A336428 A298073 * A297764 A101354 A229146

Adjacent sequences: A233471 A233472 A233473 * A233475 A233476 A233477

Keyword

nonn,easy

Author

Bruno Berselli, Dec 11 2013

Status

approved