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 A279275 Numbers k such that 2*k+1 and 5*k+1 are both pentagonal numbers (A000326). 3
 500, 721525, 1040439075, 1500312425150, 2163449476627750, 3119692644984790875, 4498594630618591814525, 6486970337659364411754700, 9354206728310172863158463400, 13488759615252931609310092468625, 19450782010987999070452290181294375 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS Colin Barker, Table of n, a(n) for n = 1..300 Index entries for linear recurrences with constant coefficients, signature (1443,-1443,1). FORMULA a(n) = 5*((7-2*sqrt(10))*(721+228*sqrt(10))^(-n) + (7+2*sqrt(10))*(721+228*sqrt(10))^n - 14)/192. a(n) = 1443*a(n-1) - 1443*a(n-2) + a(n-3) for n>3. G.f.: 25*x*(20 + x) / ((1 - x)*(1 - 1442*x + x^2)). EXAMPLE 500 is in the sequence because 2*500+1 = 1001 and 5*500+1 = 2501 are both pentagonal numbers. MATHEMATICA Table[Simplify[5 ((7 - 2 #) (721 + 228 #)^(-n) + (7 + 2 #) (721 + 228 #)^n - 14)/192 &@ Sqrt@ 10], {n, 11}] (* or *) Rest@ CoefficientList[Series[25 x (20 + x)/((1 - x) (1 - 1442 x + x^2)), {x, 0, 11}], x] (* Michael De Vlieger, Dec 09 2016 *) LinearRecurrence[{1443, -1443, 1}, {500, 721525, 1040439075}, 20] (* Harvey P. Dale, Apr 21 2020 *) PROG (PARI) isok(k) = ispolygonal(2*k+1, 5) & ispolygonal(5*k+1, 5) (PARI) Vec(25*x*(20 + x) / ((1 - x)*(1 - 1442*x + x^2)) + O(x^30)) CROSSREFS Cf. A000326, A279274, A279276. Sequence in context: A093250 A214242 A239046 * A097425 A320206 A320215 Adjacent sequences:  A279272 A279273 A279274 * A279276 A279277 A279278 KEYWORD nonn,easy AUTHOR Colin Barker, Dec 09 2016 STATUS approved

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Last modified July 1 22:23 EDT 2022. Contains 354984 sequences. (Running on oeis4.)