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 A222390 Nonnegative integers m such that 10*m*(m+1)+1 is a square. 4
 0, 3, 15, 132, 588, 5031, 22347, 191064, 848616, 7255419, 32225079, 275514876, 1223704404, 10462309887, 46468542291, 397292260848, 1764580902672, 15086643602355, 67007605759263, 572895164628660, 2544524437949340, 21754929612286743, 96624921036315675 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n+1)/a(n) tends alternately to (7+2*sqrt(10))/3 and (13+4*sqrt(10))/3; a(n+2)/a(n) tends to A176398^2. Subsequence of A014601. LINKS Bruno Berselli, Table of n, a(n) for n = 1..500 Index entries for linear recurrences with constant coefficients, signature (1,38,-38,-1,1). FORMULA G.f.: 3*x*(1+4*x+x^2)/((1-x)*(1-6*x-x^2)*(1+6*x-x^2)). a(n) = a(-n+1) = a(n-1)+38*a(n-2)-38*a(n-3)-a(n-4)+a(n-5). a(n) = -1/2+((5+t*(-1)^n)*(3-t)^(2*floor(n/2))+(5-t*(-1)^n)*(3+t)^(2*floor(n/2)))/20, where t=sqrt(10). 2*a(n)+1 = A221875(n). MATHEMATICA LinearRecurrence[{1, 38, -38, -1, 1}, {0, 3, 15, 132, 588}, 23] CoefficientList[Series[3 x (1 + 4 x + 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:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(3*(1+4*x+x^2)/((1-x)*(1-6*x-x^2)*(1+6*x-x^2)))); (Magma) I:=[0, 3, 15, 132, 588]; [n le 5 select I[n] else Self(n-1) +38*Self(n-2)-38*Self(n-3)-Self(n-4)+Self(n-5): n in [1..25]]; // Vincenzo Librandi, Aug 18 2013 (Maxima) makelist(expand(-1/2+((5+(-1)^n*sqrt(10))*(3-sqrt(10))^(2*floor(n/2))+(5-(-1)^n*sqrt(10))*(3+sqrt(10))^(2*floor(n/2)))/20), n, 1, 23); (PARI) x='x+O('x^30); concat([0], Vec(3*x*(1+4*x+x^2)/((1-x)*(1-6*x-x^2)*(1+6*x-x^2)))) \\ G. C. Greubel, Jul 15 2018 CROSSREFS Cf. nonnegative integers m such that k*m*(m+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), this sequence (k=10), A105838 (k=11), A061278 (k=12), A104240 (k=13); A105063 (k=17), A222393 (k=18), A101180 (k=19), A077259 (k=20) [incomplete list]. Cf. A221875. Sequence in context: A228365 A117694 A264795 * A281186 A349590 A108210 Adjacent sequences: A222387 A222388 A222389 * A222391 A222392 A222393 KEYWORD nonn,easy AUTHOR Bruno Berselli, Feb 18 2013 STATUS approved

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Last modified June 7 00:08 EDT 2023. Contains 363151 sequences. (Running on oeis4.)