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 A251990 Numbers n such that the sum of the hexagonal numbers H(n) and H(n+1) is equal to the sum of the pentagonal numbers P(m) and P(m+1) for some m. 2
 52, 10136, 1966380, 381467632, 74002754276, 14356152861960, 2785019652466012, 540279456425544416, 104811429526903150740, 20332877048762785699192, 3944473336030453522492556, 765207494312859220577856720, 148446309423358658338581711172 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Also nonnegative integers x in the solutions to 4*x^2-3*y^2+2*x-2*y = 0, the corresponding values of y being A251991. LINKS Colin Barker, Table of n, a(n) for n = 1..437 Index entries for linear recurrences with constant coefficients, signature (195,-195,1). FORMULA a(n) = 195*a(n-1)-195*a(n-2)+a(n-3). G.f.: 4*x*(x-13) / ((x-1)*(x^2-194*x+1)). a(n) = (-6+(3-2*sqrt(3))*(97+56*sqrt(3))^(-n)+(3+2*sqrt(3))*(97+56*sqrt(3))^n)/24. - Colin Barker, Mar 02 2016 a(n) = 194*a(n-1)-a(n-2)+48. - Vincenzo Librandi, Mar 03 2016 EXAMPLE 52 is in the sequence because H(52)+H(53) = 5356+5565 = 10921 = 5370+5551 = P(60)+P(61). MATHEMATICA LinearRecurrence[{195, -195, 1}, {52, 10136, 1966380}, 30] (* Vincenzo Librandi, Mar 03 2016 *) PROG (PARI) Vec(4*x*(x-13)/((x-1)*(x^2-194*x+1)) + O(x^100)) (Magma) I:=[52, 10136]; [n le 2 select I[n] else 194*Self(n-1)- Self(n-2)+48: n in [1..20]]; // Vincenzo Librandi, Mar 03 2016 CROSSREFS Cf. A000326, A000384, A251991. Sequence in context: A294507 A209500 A093252 * A209741 A209462 A209524 Adjacent sequences: A251987 A251988 A251989 * A251991 A251992 A251993 KEYWORD nonn,easy AUTHOR Colin Barker, Dec 12 2014 STATUS approved

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Last modified December 2 23:17 EST 2023. Contains 367526 sequences. (Running on oeis4.)