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 A251963 Numbers n such that the sum of the triangular numbers T(n) and T(n+1) is equal to an octagonal number N(m) for some m. 2
 0, 14, 208, 2910, 40544, 564718, 7865520, 109552574, 1525870528, 21252634830, 296011017104, 4122901604638, 57424611447840, 799821658665134, 11140078609864048, 155161278879431550, 2161117825702177664, 30100488280951055758, 419245718107612602960 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also nonnegative integers x in the solutions to 2*x^2-6*y^2+4*x+4*y+2+2 = 0, the corresponding values of y being A046184. LINKS Colin Barker, Table of n, a(n) for n = 1..874 Index entries for linear recurrences with constant coefficients, signature (15,-15,1). FORMULA a(n) = 15*a(n-1)-15*a(n-2)+a(n-3). G.f.: 2*x^2*(x-7) / ((x-1)*(x^2-14*x+1)). a(n) = (-6-(7-4*sqrt(3))^n*(3+2*sqrt(3))+(-3+2*sqrt(3))*(7+4*sqrt(3))^n)/6. - Colin Barker, Mar 05 2016 a(n) = 14*a(n-1) - a(n-2) + 12. - Vincenzo Librandi, Mar 05 2016 EXAMPLE 14 is in the sequence because T(14)+T(15) = 105+120 = 225 = N(9). MATHEMATICA RecurrenceTable[{a[1] == 0, a[2] == 14, a[n] == 14 a[n-1]- a[n-2] + 12}, a, {n, 20}] (* Vincenzo Librandi, Mar 05 2016 *) PROG (PARI) concat(0, Vec(2*x^2*(x-7) / ((x-1)*(x^2-14*x+1)) + O(x^100))) (Magma) I:=[0, 14]; [n le 2 select I[n] else 14*Self(n-1)-Self(n-2)+12: n in [1..20]]; // Vincenzo Librandi, Mar 05 2016 CROSSREFS Cf. A000217, A000567, A046184. Sequence in context: A333949 A002961 A063071 * A192007 A160682 A097261 Adjacent sequences: A251960 A251961 A251962 * A251964 A251965 A251966 KEYWORD nonn,easy AUTHOR Colin Barker, Dec 11 2014 STATUS approved

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Last modified June 14 10:31 EDT 2024. Contains 373395 sequences. (Running on oeis4.)