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A154153
Numbers k such that 28 plus the k-th triangular number is a perfect square.
3
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
OFFSET
1,1
LINKS
F. T. Adams-Watters, SeqFan Discussion, Oct 2009
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, 8, 47, and 57 are terms:
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, ", ") ) ); }
CROSSREFS
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
KEYWORD
nonn
AUTHOR
R. J. Mathar, Oct 18 2009
EXTENSIONS
a(21)-a(30) from Amiram Eldar, Mar 18 2019
STATUS
approved