A166146 - OEIS

%I #49 Mar 17 2024 02:12:45

%S 1,15,36,64,99,141,190,246,309,379,456,540,631,729,834,946,1065,1191,

%T 1324,1464,1611,1765,1926,2094,2269,2451,2640,2836,3039,3249,3466,

%U 3690,3921,4159,4404,4656,4915,5181,5454,5734,6021,6315,6616,6924,7239,7561

%N a(n) = (7*n^2 + 7*n - 12)/2.

%H Harvey P. Dale, <a href="/A166146/b166146.txt">Table of n, a(n) for n = 1..1000</a>

%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) + 7n, a(1)=1.

%F From _Harvey P. Dale_, Nov 01 2011: (Start)

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=15, a(3)=36.

%F G.f.: x*(1+12*x-6*x^2)/(1-x)^3. (End)

%F E.g.f.: (1/2)*((-12 + 14*x + 7*x^2)*exp(x) + 12). - _G. C. Greubel_, Apr 26 2016

%F Sum_{n>=1} 1/a(n) = 1/6 + (2*Pi/sqrt(385))*tan(sqrt(55/7)*Pi/2). - _Amiram Eldar_, Feb 20 2023

%F a(n) = T(n) + 12*T(n-1) - 6*T(n-2), where T(n) = A000217(n) is the n-th triangular number. - _Gary W. Adamson_, Mar 12 2024

%t RecurrenceTable[{a[1]==1,a[n]==a[n-1]+7n},a,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,15,36},50] (* _Harvey P. Dale_, Nov 01 2011 *)

%t Table[(7 n^2 + 7 n - 12)/2, {n, 46}] (* _Michael De Vlieger_, Apr 27 2016 *)

%o (PARI) a(n)=7*n*(n+1)/2-6 \\ _Charles R Greathouse IV_, Jan 11 2012

%Y Cf. A000217.

%K nonn,easy

%O 1,2

%A _Vincenzo Librandi_, Oct 08 2009

%E a(35) corrected by _Harvey P. Dale_, Nov 01 2011