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A255184 25-gonal numbers: a(n) = n*(23*n-21)/2.

%I A255184 #39 Feb 06 2023 05:26:06
%S A255184 0,1,25,72,142,235,351,490,652,837,1045,1276,1530,1807,2107,2430,2776,
%T A255184 3145,3537,3952,4390,4851,5335,5842,6372,6925,7501,8100,8722,9367,
%U A255184 10035,10726,11440,12177,12937,13720,14526,15355,16207,17082,17980
%N A255184 25-gonal numbers: a(n) = n*(23*n-21)/2.
%C A255184 If b(n,k) = n*((k-2)*n-(k-4))/2 is n-th k-gonal number, then b(n,k) = A000217(n) + (k-3)* A000217(n-1) (see Deza in References section, page 21, where the formula is attributed to Bachet de Méziriac).
%C A255184 Also, b(n,k) = b(n,k-1) + A000217(n-1) (see Deza and Picutti in References section, page 20 and 137 respectively, where the formula is attributed to Nicomachus). Some examples:
%C A255184 for k=4, A000290(n) = A000217(n) + A000217(n-1);
%C A255184 for k=5, A000326(n) = A000290(n) + A000217(n-1);
%C A255184 for k=6, A000384(n) = A000326(n) + A000217(n-1), etc.
%C A255184 This is the case k=25.
%D A255184 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6 (23rd row of the table).
%D A255184 E. Picutti, Sul numero e la sua storia, Feltrinelli Economica (1977), pages 131-147.
%H A255184 Luciano Ancora, <a href="/A255184/b255184.txt">Table of n, a(n) for n = 0..1000</a>
%H A255184 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A255184 G.f.: x*(-1 - 22*x)/(-1 + x)^3.
%F A255184 a(n) = A000217(n) + 22*A000217(n-1) = A051876(n) + A000217(n-1), see comments.
%F A255184 Product_{n>=2} (1 - 1/a(n)) = 23/25. - _Amiram Eldar_, Jan 22 2021
%F A255184 E.g.f.: exp(x)*(x + 23*x^2/2). - _Nikolaos Pantelidis_, Feb 05 2023
%t A255184 Table[n (23 n - 21)/2, {n, 40}]
%o A255184 (Magma) k:=25; [n*((k-2)*n-(k-4))/2: n in [0..40]]; // _Bruno Berselli_, Apr 10 2015
%o A255184 (PARI) a(n)=n*(23*n-21)/2 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A255184 Cf. k-gonal numbers: A000217 (k=3), A000290 (k=4), A000326 (k=5), A000384 (k=6), A000566 (k=7), A000567 (k=8), A001106 (k=9), A001107 (k=10), A051682 (k=11), A051624 (k=12), A051865 (k=13), A051866 (k=14), A051867 (k=15), A051868 (k=16), A051869 (k=17), A051870 (k=18), A051871 (k=19), A051872 (k=20), A051873 (k=21), A051874 (k=22),  A051875 (k=23), A051876 (k=24), this sequence (k=25), A255185 (k=26), A255186 (k=27), A161935 (k=28), A255187 (k=29), A254474 (k=30).
%K A255184 nonn,easy
%O A255184 0,3
%A A255184 _Luciano Ancora_, Apr 03 2015