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A222393 Nonnegative integers m such that 18*m*(m+1)+1 is a square. 2
0, 4, 12, 152, 424, 5180, 14420, 175984, 489872, 5978292, 16641244, 203085960, 565312440, 6898944364, 19203981732, 234361022432, 652370066464, 7961375818340, 22161378278060, 270452416801144, 752834491387592, 9187420795420572, 25574211328900084 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n+2)/a(n) tends to A156164.

a(n) is congruent to {0,2,4} (mod 5, 6 and 10).

LINKS

Bruno Berselli, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (1,34,-34,-1,1).

FORMULA

G.f.: 4*x*(1+x)^2/((1-x)*(1-6*x+x^2)*(1+6*x+x^2)).

a(n) = a(-n+1) = a(n-1)+34*a(n-2)-34*a(n-3)-a(n-4)+a(n-5).

a(n) = -1/2+((3+t*(-1)^n)*(3-2*t)^(2*floor(n/2))+(3-t*(-1)^n)*(3+2*t)^(2*floor(n/2)))/12, where t=sqrt(2).

MATHEMATICA

LinearRecurrence[{1, 34, -34, -1, 1}, {0, 4, 12, 152, 424}, 23]

CoefficientList[Series[4 x (1 + x)^2 / ((1 - x) (1 - 6 x + x^2) (1 + 6 x + x^2)), {x, 0, 25}], x] (* Vincenzo Librandi, Aug 18 2013 *)

PROG

(MAGMA) m:=22; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(4*(1+x)^2/((1-x)*(1-6*x+x^2)*(1+6*x+x^2))));

(MAGMA) I:=[0, 4, 12, 152, 424]; [n le 5 select I[n] else Self(n-1)+34*Self(n-2)-34*Self(n-3)-Self(n-4)+Self(n-5): n in [1..25]]; // Vincenzo Librandi, Aug 18 2013

(Maxima) makelist(expand(-1/2+((3+sqrt(2)*(-1)^n)*(3-2*sqrt(2))^(2*floor(n/2))+(3-sqrt(2)*(-1)^n)*(3+2*sqrt(2))^(2*floor(n/2)))/12), n, 1, 23);

(PARI) x='x+O('x^30); concat([0], Vec(4*x*(1+x)^2/((1-x)*(1-6*x+x^2)*(1+6*x+x^2)))) \\ G. C. Greubel, Jul 15 2018

CROSSREFS

Cf. nonnegative integers n such that k*n*(n+1)+1 is a square: A001652 (k=2), A001921 (k=3), A001477 (k=4), A053606 (k=5), A105038 (k=6), A105040 (k=7), A053141 (k=8), A222390 (k=10), A105838 (k=11), A061278 (k=12), A104240 (k=13); A105063 (k=17), this sequence (k=18), A101180 (k=19), A077259 (k=20) [incomplete list].

Sequence in context: A052598 A230691 A032071 * A213143 A173603 A175718

Adjacent sequences:  A222390 A222391 A222392 * A222394 A222395 A222396

KEYWORD

nonn,easy

AUTHOR

Bruno Berselli, Feb 19 2013

STATUS

approved

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Last modified September 23 20:42 EDT 2021. Contains 347617 sequences. (Running on oeis4.)