A270693 - OEIS

%I #14 Sep 08 2022 08:46:16

%S 1,-25,51,-100,151,-225,301,-400,501,-625,751,-900,1051,-1225,1401,

%T -1600,1801,-2025,2251,-2500,2751,-3025,3301,-3600,3901,-4225,4551,

%U -4900,5251,-5625,6001,-6400,6801,-7225,7651,-8100,8551,-9025,9501,-10000,10501

%N Alternating sum of centered 25-gonal numbers.

%C The absolute value alternating sum of centered k-gonal numbers gives concentric k-gonal numbers.

%C 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 OEIS Wiki, <a href="http://oeis.org/wiki/Centered_polygonal_numbers">Centered polygonal numbers</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Number</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-2,0,2,1).

%F G.f.: (1 - 23*x + x^2)/((1 - x)*(1 + x)^3).

%F E.g.f.: (1/8)*(-21*exp(x) + (29 - 150*x + 50*x^2)*exp(-x)).

%F a(n) = -2*a(n-1) + 2*a(n-3) + a(n-4).

%F a(n) = ((-1)^n*(50*n^2 + 100*n + 29) - 21)/8.

%p 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 LinearRecurrence[{-2, 0, 2, 1}, {1, -25, 51, -100}, 41]

%t Table[((-1)^n (50 n^2 + 100 n + 29) - 21)/8, {n, 0, 40}]

%o (PARI) x='x+O('x^100); Vec((1-23*x+x^2)/((1-x)*(1+x)^3)) \\ _Altug Alkan_, Mar 21 2016

%o (Magma) [((-1)^n*(50*n^2 + 100*n + 29) - 21)/8 : n in [0..40]]; // _Wesley Ivan Hurt_, Mar 21 2016

%Y Cf. A262221 (centered 25-gonal numbers).

%Y Cf. A032527, A032528, A077043, A077221, A195041, A195042, A195045, A195046, A195047, A195048, A195049, A195058, A195142, A195043, A195143, A195145, A195146, A195147, A195148, A195149, A195158.

%K easy,sign

%O 0,2

%A _Ilya Gutkovskiy_, Mar 21 2016