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A193053 a(n) = (14*n*(n+3) + (2*n-5)*(-1)^n + 21)/16.

%I A193053 #51 Sep 08 2022 08:45:58
%S A193053 1,5,10,17,26,36,49,62,79,95,116,135,160,182,211,236,269,297,334,365,
%T A193053 406,440,485,522,571,611,664,707,764,810,871,920,985,1037,1106,1161,
%U A193053 1234,1292,1369,1430,1511,1575,1660,1727,1816,1886,1979,2052,2149,2225,2326
%N A193053 a(n) = (14*n*(n+3) + (2*n-5)*(-1)^n + 21)/16.
%C A193053 For an origin of this sequence, see the numerical spiral illustrated in the Links section.
%H A193053 Bruno Berselli, <a href="/A193053/b193053.txt">Table of n, a(n) for n = 0..1000</a>
%H A193053 Bruno Berselli, <a href="http://www.base5forum.it/upload/a193053.jpg">Illustration of initial terms</a>.
%H A193053 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A193053 O.g.f.: (1 + 4*x + 3*x^2 - x^3)/((1 + x)^2*(1 - x)^3).
%F A193053 E.g.f.: (1/16)*((21 + 56*x + 14*x^2)*exp(x) - (5 + 2*x)*exp(-x)). - _G. C. Greubel_, Aug 19 2017
%F A193053 a(n) = A195020(n) + n + 1.
%F A193053 a(n)   - a(-n-1)  = A047336(n+1).
%F A193053 a(n+1) - a(-n)    = A113804(n+1).
%F A193053 a(n+2) - a(n)     = A047385(n+3).
%F A193053 a(n+4) - a(n)     = A113803(n+4).
%F A193053 a(2*n) + a(2*n-1) = A069127(n+1).
%F A193053 a(2*n) - a(2*n-1) = A016813(n).
%F A193053 a(2*n+1) - a(2*n) = A016777(n+1).
%F A193053 a(n+2) + 2*a(n+1) + a(n) = A024966(n+2).
%t A193053 Table[(14*n*(n + 3) + (2*n - 5)*(-1)^n + 21)/16, {n, 0, 50}] (* _Vincenzo Librandi_, Mar 26 2013 *)
%t A193053 LinearRecurrence[{1,2,-2,-1,1},{1,5,10,17,26},60] (* _Harvey P. Dale_, Jun 19 2020 *)
%o A193053 (PARI) for(n=0, 50, print1((14*n*(n+3)+(2*n-5)*(-1)^n+21)/16", "));
%o A193053 (Magma) [(14*n*(n+3)+(2*n-5)*(-1)^n+21)/16: n in [0..50]];
%Y A193053 Cf. A195020 (vertices of the numerical spiral in link).
%Y A193053 Cf. A001106, A022264, A033572, A144555, A152760, A158482, A158485, A185019, A195021, A195023-A195030, A195320, A198017 [incomplete list].
%K A193053 nonn,easy
%O A193053 0,2
%A A193053 _Bruno Berselli_, Oct 20 2011 - based on remarks and sequences by _Omar E. Pol_