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A270693 Alternating sum of centered 25-gonal numbers.

%I A270693 #15 Feb 16 2025 08:33:32
%S A270693 1,-25,51,-100,151,-225,301,-400,501,-625,751,-900,1051,-1225,1401,
%T A270693 -1600,1801,-2025,2251,-2500,2751,-3025,3301,-3600,3901,-4225,4551,
%U A270693 -4900,5251,-5625,6001,-6400,6801,-7225,7651,-8100,8551,-9025,9501,-10000,10501
%N A270693 Alternating sum of centered 25-gonal numbers.
%C A270693 The absolute value alternating sum of centered k-gonal numbers gives concentric k-gonal numbers.
%C A270693 More generally, the ordinary generating function for the alternating sum of centered k-gonal numbers is (1 - (k - 2)*x + x^2)/((1 - x)*(1 + x)^3).
%H A270693 OEIS Wiki, <a href="http://oeis.org/wiki/Centered_polygonal_numbers">Centered polygonal numbers</a>
%H A270693 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Number</a>
%H A270693 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-2,0,2,1).
%F A270693 G.f.: (1 - 23*x + x^2)/((1 - x)*(1 + x)^3).
%F A270693 E.g.f.: (1/8)*(-21*exp(x) + (29 - 150*x + 50*x^2)*exp(-x)).
%F A270693 a(n) = -2*a(n-1) + 2*a(n-3) + a(n-4).
%F A270693 a(n) = ((-1)^n*(50*n^2 + 100*n + 29) - 21)/8.
%p A270693 A270693:=n->((-1)^n*(50*n^2 + 100*n + 29) - 21)/8: seq(A270693(n), n=0..100); # _Wesley Ivan Hurt_, Sep 18 2017
%t A270693 LinearRecurrence[{-2, 0, 2, 1}, {1, -25, 51, -100}, 41]
%t A270693 Table[((-1)^n (50 n^2 + 100 n + 29) - 21)/8, {n, 0, 40}]
%o A270693 (PARI) x='x+O('x^100); Vec((1-23*x+x^2)/((1-x)*(1+x)^3)) \\ _Altug Alkan_, Mar 21 2016
%o A270693 (Magma) [((-1)^n*(50*n^2 + 100*n + 29) - 21)/8 : n in [0..40]]; // _Wesley Ivan Hurt_, Mar 21 2016
%Y A270693 Cf. A262221 (centered 25-gonal numbers).
%Y A270693 Cf. A032527, A032528, A077043, A077221, A195041, A195042, A195045, A195046, A195047, A195048, A195049, A195058, A195142, A195043, A195143, A195145, A195146, A195147, A195148, A195149, A195158.
%K A270693 easy,sign
%O A270693 0,2
%A A270693 _Ilya Gutkovskiy_, Mar 21 2016