|
%I #54 Sep 08 2022 08:46:01
%S 5,21,49,89,141,205,281,369,469,581,705,841,989,1149,1321,1505,1701,
%T 1909,2129,2361,2605,2861,3129,3409,3701,4005,4321,4649,4989,5341,
%U 5705,6081,6469,6869,7281,7705,8141,8589,9049,9521,10005,10501,11009,11529,12061
%N a(n) = 6n^2 + 10n + 5.
%C Numbers n where 6n-5 is a square of a number type 6n-1.
%C Also sequence found by reading the line from 5, in the direction 5, 21,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - _Omar E. Pol_, Jul 18 2012
%C The spiral mentioned above naturally appears on a "graphene" like lattice (planar net 6^3). The opposite diagonal is A080859. - _Yuriy Sibirmovsky_, Oct 04 2016
%C First differences of A048395. - _Leo Tavares_, Nov 24 2021 [Corrected by _Omar E. Pol_, Dec 26 2021]
%H Vincenzo Librandi, <a href="/A201279/b201279.txt">Table of n, a(n) for n = 0..10000</a>
%H Yuriy Sibirmovsky, <a href="/A201279/a201279.png">Diagonals of a 'graphene' number spiral</a>.
%H Leo Tavares, <a href="/A201279/a201279.jpg">Illustration: Diamond Frame Stars</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+x)*(5+x)/(1-x)^3. - _Colin Barker_, Jan 09 2012
%F a(n) = 1 + A033579(n+1). - _Omar E. Pol_, Jul 18 2012
%F a(n) = (n+1)*A001844(n+1)-n*A001844(n). [_Bruno Berselli_, Jan 15 2013]
%F From _Leo Tavares_, Nov 24 2021: (Start)
%F a(n) = A003154(n+2) - A022144(n+1). See Diamond Frame Stars illustration.
%F a(n) = A016754(n) + A046092(n+1). (End)
%t LinearRecurrence[{3,-3,1},{5,21,49},50] (* _Vincenzo Librandi_, Dec 01 2011 *)
%t Table[6 n^2 + 10 n + 5, {n, 0, 44}] (* or *)
%t CoefficientList[Series[(1 + x) (5 + x)/(1 - x)^3, {x, 0, 44}], x] (* _Michael De Vlieger_, Oct 04 2016 *)
%o (PARI) a(n)=6*n^2+10*n+5 \\ _Charles R Greathouse IV_, Nov 29 2011
%o (Magma) [6*n^2 + 10*n + 5: n in [0..60]]; // _Vincenzo Librandi_, Dec 01 2011
%Y Cf. A136392, A080859.
%Y Cf. A003154, A022144, A016754, A046092, A048395.
%K nonn,easy
%O 0,1
%A _Eleonora Echeverri-Toro_, Nov 29 2011
|
