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Shallow diagonal of triangular spiral in A051682.
2

%I #30 Nov 13 2024 15:58:48

%S 1,31,97,199,337,511,721,967,1249,1567,1921,2311,2737,3199,3697,4231,

%T 4801,5407,6049,6727,7441,8191,8977,9799,10657,11551,12481,13447,

%U 14449,15487,16561,17671,18817,19999,21217,22471,23761,25087,26449,27847,29281,30751,32257

%N Shallow diagonal of triangular spiral in A051682.

%C Reflection of A060544 in the horizontal A051682.

%C Binomial transform of (1, 30, 36, 0, 0, 0, ...).

%H Vincenzo Librandi, <a href="/A081275/b081275.txt">Table of n, a(n) for n = 0..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) = C(n,0) + 30*C(n,1) + 36*C(n,2).

%F a(n) = 18*n^2 + 12*n + 1.

%F G.f.: (1 + 28*x + 7*x^2)/(1-x)^3.

%F a(0)=1, a(1)=31, a(2)=97, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Jun 30 2011

%F E.g.f.: exp(x)*(1 + 30*x + 18*x^2). - _Elmo R. Oliveira_, Nov 13 2024

%t Table[30Binomial[n,1]+36Binomial[n,2]+1,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{1,31,97},40] (* _Harvey P. Dale_, Jun 30 2011 *)

%t CoefficientList[Series[(1 + 28 x + 7 x^2) / (1 - x)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 07 2013 *)

%o (PARI) a(n)=18*n^2+12*n+1 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A051682, A060544, A092296.

%K nonn,easy,changed

%O 0,2

%A _Paul Barry_, Mar 15 2003