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 A165717 Integers of the form k*(5+k)/4. 6
 6, 9, 21, 26, 44, 51, 75, 84, 114, 125, 161, 174, 216, 231, 279, 296, 350, 369, 429, 450, 516, 539, 611, 636, 714, 741, 825, 854, 944, 975, 1071, 1104, 1206, 1241, 1349, 1386, 1500, 1539, 1659, 1700, 1826, 1869, 2001, 2046, 2184, 2231, 2375, 2424, 2574, 2625 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Integers of the form k+k*(k+1)/4 = k+A000217(k)/2; for k see A014601, for A000217(k)/2 see A074378. Are all terms composite? Yes, because a(2*k) = k*(4*k+5) and a(2*k-1) = (k+1)*(4*k-1). - Bruno Berselli, Apr 07 2013 Numbers m such that 16*m + 25 is a square. - Vincenzo Librandi, Apr 07 2013 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1). FORMULA From R. J. Mathar, Sep 25 2009: (Start) a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). G.f.: x*(-6-3*x+x^3)/( (1+x)^2 * (x-1)^3 ). (End) Sum_{n>=1} 1/a(n) = 29/25 - Pi/5. - Amiram Eldar, Jul 26 2024 EXAMPLE For k =1,2,3,.. the value of k*(k+5)/4 is 3/2, 7/2, 6, 9, 25/2, 33/2, 21, 26, 63/2, 75/2, 44, 51,.. and the integer values define the sequence. MATHEMATICA q=2; s=0; lst={}; Do[s+=((n+q)/q); If[IntegerQ[s], AppendTo[lst, s]], {n, 6!}]; lst Select[Table[k*(5+k)/4, {k, 100}], IntegerQ] (* or *) LinearRecurrence[ {1, 2, -2, -1, 1}, {6, 9, 21, 26, 44}, 60] (* Harvey P. Dale, Aug 11 2011 *) Select[Range[1, 3000], IntegerQ[Sqrt[16 # + 25]]&] (* Vincenzo Librandi, Apr 07 2013 *) PROG (Magma) [n: n in [1..3000] | IsSquare(16*n+25)]; // Vincenzo Librandi, Apr 07 2013 CROSSREFS Cf. A000217, A014601, A074378. Sequence in context: A357840 A355484 A358222 * A239874 A043103 A242756 Adjacent sequences: A165714 A165715 A165716 * A165718 A165719 A165720 KEYWORD nonn,easy,changed AUTHOR Vladimir Joseph Stephan Orlovsky, Sep 24 2009 EXTENSIONS Definition simplified by R. J. Mathar, Sep 25 2009 STATUS approved

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Last modified August 7 08:38 EDT 2024. Contains 375008 sequences. (Running on oeis4.)