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 A077288 First member of the Diophantine pair (m,k) that satisfies 6(m^2 + m) = k^2 + k: a(n) = m. 7
 0, 1, 3, 14, 34, 143, 341, 1420, 3380, 14061, 33463, 139194, 331254, 1377883, 3279081, 13639640, 32459560, 135018521, 321316523, 1336545574, 3180705674, 13230437223, 31485740221, 130967826660, 311676696540, 1296447829381, 3085281225183, 12833510467154 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Also nonnegative m such that 24*m^2 + 24*m + 1 is a square. - Gerald McGarvey, Apr 02 2005 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Mohammad K. Azarian, Diophantine Pair, Problem B-881, Fibonacci Quarterly, Vol. 37, No. 3, August 1999, pp. 277-278; Solution to Problem B-881, Fibonacci Quarterly, Vol. 38, No. 2, May 2000, pp. 183-184. Index entries for linear recurrences with constant coefficients, signature (1,10,-10,-1,1). FORMULA Let b(n) be A072256. Then a(2n+2) = 2*a(2*n+1) - a(2*n) + b(n+1), a(2*n+3) = 2*a(2*n+2) - a(2*n+1) + b(n+2), with a(0)=0, a(1)=1. G.f.: x*(1+x)^2/((1-x)*(1-10*x^2+x^4)). a(n) = a(-1-n) for all n in Z. - Michael Somos, Jul 15 2018 EXAMPLE a(3) = 2*3 - 1 + 9 = 14, a(4) = 2*14 - 3 + 9 = 34, etc. G.f. = x + 3*x^2 + 14*x^3 + 34*x^4 + 143*x^5 + 341*x^6 + 1420*x^7 + 3380*x^8 + ... - Michael Somos, Jul 15 2018 MATHEMATICA CoefficientList[Series[x*(1 + x)^2/((1 - x)*(1 - 10 x^2 + x^4)), {x, 0, 40}], x] (* T. D. Noe, Jun 04 2012 *) LinearRecurrence[{1, 10, -10, -1, 1}, {0, 1, 3, 14, 34}, 50] (* G. C. Greubel, Jul 15 2018 *) a[ n_] := With[{m = Max[n, -1 - n]}, SeriesCoefficient[ x (1 + x)^2 / ((1 - x) (1 - 10 x^2 + x^4)), {x, 0, m}]]; (* Michael Somos, Jul 15 2018 *) PROG (PARI) x='x+O('x^30); concat([0], Vec(x*(1+x)^2/((1-x)*(1-10*x^2+x^4)))) \\ G. C. Greubel, Jul 15 2018 (MAGMA) m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x)^2/((1-x)*(1-10*x^2+x^4)))); // G. C. Greubel, Jul 15 2018 CROSSREFS The k values are in A077291 Cf. A077289, A077290, A077291. Cf. A053141. Sequence in context: A081269 A140064 A064226 * A094627 A009394 A076533 Adjacent sequences:  A077285 A077286 A077287 * A077289 A077290 A077291 KEYWORD easy,nonn AUTHOR Bruce Corrigan (scentman(AT)myfamily.com), Nov 03 2002 STATUS approved

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Last modified October 15 12:31 EDT 2019. Contains 328026 sequences. (Running on oeis4.)