A081270 - OEIS

%I #27 Nov 13 2024 17:12:33

%S 3,16,38,69,109,158,216,283,359,444,538,641,753,874,1004,1143,1291,

%T 1448,1614,1789,1973,2166,2368,2579,2799,3028,3266,3513,3769,4034,

%U 4308,4591,4883,5184,5494,5813,6141,6478,6824,7179,7543,7916,8298,8689,9089,9498,9916

%N Diagonal of triangular spiral in A051682.

%H Vincenzo Librandi, <a href="/A081270/b081270.txt">Table of n, a(n) for n = 0..1000</a>

%H Hacène Belbachir, Toufik Djellal, Jean-Gabriel Luque, <a href="https://arxiv.org/abs/1703.00323">On the self-convolution of generalized Fibonacci numbers</a>, arXiv:1703.00323 [math.CO], 2017.

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

%F a(n) = A064226(n) + 2*n.

%F a(n) = 3*binomial(n,0) + 13*binomial(n,1) + 9*binomial(n,2); binomial transform of (3, 13, 9, 0, 0, 0, ...).

%F a(n) = (9*n^2 + 17*n + 6)/2.

%F G.f.: (3 + 7*x - x^2)/(1-x)^3.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Vincenzo Librandi_, Jul 08 2012

%F E.g.f.: exp(x)*(6 + 26*x + 9*x^2)/2. - _Elmo R. Oliveira_, Nov 13 2024

%t CoefficientList[Series[(3+7x-x^2)/(1-x)^3,{x,0,50}],x] (* _Vincenzo Librandi_, Jul 08 2012 *)

%o (Magma) [(9*n^2+17*n+6)/2: n in [0..50]]; // _Vincenzo Librandi_, Jul 08 2012

%o (PARI) a(n)=(9*n^2+17*n+6)/2 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A064226, A081268.

%K easy,nonn

%O 0,1

%A _Paul Barry_, Mar 15 2003