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 A154153 Indices n such that 28 plus the n-th triangular number is a perfect square. 2
 6, 8, 47, 57, 278, 336, 1623, 1961, 9462, 11432, 55151, 66633, 321446, 388368, 1873527, 2263577, 10919718, 13193096, 63644783, 76895001, 370948982, 448176912, 2162049111, 2612166473, 12601345686, 15224821928, 73446025007, 88736765097, 428074804358, 517195768656 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 LINKS F. T. Adams-Watters, SeqFan Discussion, Oct 2009 David A. Corneth, Conjectured formula for a(n) FORMULA {k: 28+k*(k+1)/2 in A000290}. Conjectures: (Start) a(n) = +a(n-1) +6*a(n-2) -6*a(n-3) -a(n-4) +a(n-5). G.f.: x*(-6-2*x-3*x^2+2*x^3+7*x^4)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)). G.f.: ( 14 + 1/(x-1) + (14+29*x)/(x^2-2*x-1) + (-1-12*x)/(x^2+2*x-1) )/2. (End) See also the Corneth link - David A. Corneth, Mar 18 2019 EXAMPLE 6*(6+1)/2+28 = 7^2. 8*(8+1)/2+28 = 8^2. 47*(47+1)/2+28 = 34^2. 57*(57+1)/2+28 = 41^2. MATHEMATICA Join[{6, 8}, Select[Range[0, 10^5], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 28 &]] (* G. C. Greubel, Sep 03 2016 *) PROG (PARI) {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 28), print1(n, ", ") ) ); } (PARI) a(n)=(15*g((n+1)\2)-1)/2+(-1)^n*f((n+1)\2) g(n) = round((-1/2+sqrt(2)/2)*(3+sqrt(8))^n+(-1/2-sqrt(2)/2)*(3-sqrt(8))^n) f(n) = round(((2+sqrt(2))/4)*(3-sqrt(8))^n+((2-sqrt(2))/4)*(3+sqrt(8))^n) \\ based on conjectured formula \\ David A. Corneth, Mar 18 2019 CROSSREFS Cf. A000217, A000290. Cf. A001108 (0), A006451 (1), A154138 (3), A154139 (4), A154140 (6), A154141 (8), A154142 (9), A154143 (10), A154144 (13), A154145 (15), A154146 (16), A154147 (19), A154148 (21), A154149 (22), A154150(24), A154151 (25), A154151 (26), this sequence (28), A154154 (30). Sequence in context: A054102 A261063 A000380 * A164640 A223852 A227167 Adjacent sequences:  A154150 A154151 A154152 * A154154 A154155 A154156 KEYWORD nonn AUTHOR R. J. Mathar, Oct 18 2009 EXTENSIONS a(21)-a(30) from Amiram Eldar, Mar 18 2019 STATUS approved

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Last modified September 21 03:13 EDT 2020. Contains 337266 sequences. (Running on oeis4.)