A085473 - OEIS

%I #40 Oct 19 2024 09:34:27

%S 1,10,31,64,109,166,235,316,409,514,631,760,901,1054,1219,1396,1585,

%T 1786,1999,2224,2461,2710,2971,3244,3529,3826,4135,4456,4789,5134,

%U 5491,5860,6241,6634,7039,7456,7885,8326,8779,9244,9721,10210,10711,11224,11749,12286,12835

%N a(n) = 6*n^2 + 3*n + 1.

%C T(n,3) of A085475.

%C Sequence found by reading the line from 1, in the direction 1, 10,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - _Omar E. Pol_, Sep 09 2011

%C Sums of the triangular numbers from A000217(2*n-1) to A000217(2*n+1), with A000217(-1) = 0. - _Bruno Berselli_, Sep 04 2018

%H G. C. Greubel, <a href="/A085473/b085473.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 G.f.: (1 + 7*x + 4*x^2)/(1 - x)^3.

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

%F a(n) = 12*n + a(n-1) - 3 for n > 0, a(0)=1. - _Vincenzo Librandi_, Aug 08 2010

%F a(0)=1, a(1)=10, a(2)=31; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Nov 15 2011

%F E.g.f.: exp(x)*(1 + 9*x + 6*x^2). - _Elmo R. Oliveira_, Oct 18 2024

%t Table[3 n (2 n + 1) + 1, {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 06 2011 *)

%t Table[Binomial[2 n + 3, 3] - Binomial[2 n, 3], {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 10, 31}, 50] (* _Harvey P. Dale_, Nov 15 2011 *)

%o (PARI) x='x+O('x^50); Vec((1+7*x+4*x^2)/(1-x)^3) \\ _G. C. Greubel_, Jun 13 2017

%o (PARI) for(n=0,25, print1(6*n^2 + 3*n + 1, ", ")) \\ _G. C. Greubel_, Jun 13 2017

%Y Cf. A000217, A001318, A016813, A085474, A085475.

%K nonn,easy

%O 0,2

%A _Paul Barry_, Jul 01 2003