A145910 - OEIS

%I #50 Nov 15 2024 19:05:46

%S 2,14,35,65,104,152,209,275,350,434,527,629,740,860,989,1127,1274,

%T 1430,1595,1769,1952,2144,2345,2555,2774,3002,3239,3485,3740,4004,

%U 4277,4559,4850,5150,5459,5777,6104,6440,6785,7139,7502,7874

%N a(n) = (1 + 3*n)*(4 + 3*n)/2.

%D R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., Vol. 58, No. 2 (2020), pp. 140-142.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = a(n-1) + 3*(3*n+1) = a(n-1) + A017197(n+1).

%F G.f.: (-2 - 8*x + x^2)/(x-1)^3. - _R. J. Mathar_, Jan 06 2011

%F a(n) = A144449(n)/8.

%F a(n) = 2*a(n-1) - a(n-2) + 9.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%F From _Amiram Eldar_, Mar 11 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 2/3.

%F Sum_{n>=0} (-1)^n/a(n) = 4*Pi/(9*sqrt(3)) + 4*log(2)/9 - 2/3. (End)

%F From _Elmo R. Oliveira_, Nov 15 2024: (Start)

%F E.g.f.: exp(x)*(4 + 24*x + 9*x^2)/2.

%F a(n) = A085001(n)/2. (End)

%p A145910:=n->(1+3*n)*(4+3*n)/2: seq(A145910(n), n=0..100); # _Wesley Ivan Hurt_, Jul 25 2017

%t Table[(1+3n)(4+3n)/2, {n,0,50}] (* _Harvey P. Dale_, Feb 23 2011 *)

%o (PARI) a(n)=(1+3*n)*(4+3*n)/2 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A017197, A085001, A144449.

%K nonn,easy

%O 0,1

%A _Paul Curtz_, Oct 24 2008

%E Terms a(11)-a(42) from _Vincenzo Librandi_, Nov 17 2009