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%I #32 Jan 22 2022 23:34:58
%S 14,19,31,38,54,63,83,94,118,131,159,174,206,223,259,278,318,339,383,
%T 406,454,479,531,558,614,643,703,734,798,831,899,934,1006,1043,1119,
%U 1158,1238,1279,1363,1406,1494,1539,1631,1678,1774,1823,1923,1974,2078,2131
%N Numbers n such that 4 consecutive terms of binomial(n,k) satisfy a quadratic relation for 0 <= k <= n/2.
%C From _Robert Israel_, Apr 28 2015: (Start)
%C Numbers n >= 14 such that 3*n + 7 is a square.
%C This is because
%C C(n,i+3) - 3*C(n,i+2) + 3*C(n,i+1) - C(n,i) = n!/((n-i)!*(i+3)!) * g(n,i)
%C where g(n,i) = (n-3-2*i) * ((n-3-2*i)^2 - 3*n - 7). (End)
%H Robert Israel, <a href="/A241199/b241199.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F a(n) = (55-7*(-1)^n-2*(-21+(-1)^n)*n+6*n^2)/8. G.f.: -x*(6*x^4-3*x^3-16*x^2+5*x+14) / ((x-1)^3*(x+1)^2). - _Colin Barker_, Apr 18 2014 and Apr 29 2015
%F The terms appear to satisfy a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), with initial terms 14, 19, 31, 38, 54. - _T. D. Noe_, Apr 18 2014
%F Numbers are of the form A200182(3n+1) and A200182(3n-1). - _Avi Friedlich_, Apr 25 2015
%F a(2*k-1) = 3*k^2 + 8*k + 3, a(2*k) = 3*k^2 + 10*k + 6. - _Robert Israel_, Apr 28 2015
%e Binomial(14,k) = (1, 14, 91, 364, 1001, 2002, 3003, 3432) for k = 0..7. The 4 terms beginning with 91 equal 182 - 273*x + 182*x^2 for x = 1..4.
%p map(k -> (3*k^2+8*k+3,3*k^2+10*k+6),[$1..100]); # _Robert Israel_, Apr 28 2015
%t Select[Range[2500], MemberQ[Differences[Binomial[#, Range[0, #/2]], 3], 0] &]
%t LinearRecurrence[{1,2,-2,-1,1},{14,19,31,38,54},50] (* _Harvey P. Dale_, Oct 29 2017 *)
%o (PARI) Vec(-x*(6*x^4-3*x^3-16*x^2+5*x+14)/((x-1)^3*(x+1)^2) + O(x^100)) \\ _Colin Barker_, Apr 29 2015
%o (PARI) a(n)=(6*n^2+42*n+55-(-1)^n*(2*n+7))/8 \\ _Charles R Greathouse IV_, Apr 15 2016
%Y Sequence A241200 gives the position of the first of the 4 terms. Sequence A008865 gives the terms greater than 2 for which 3 consecutive terms satisfy a linear relation.
%Y A014206 is a related sequence. - _Avi Friedlich_, Apr 28 2015
%Y Cf. A062730 (3 terms in arithmetic progression in Pascal's triangle).
%K nonn,easy
%O 1,1
%A _T. D. Noe_, Apr 17 2014
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