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A256645 25-gonal pyramidal numbers: a(n) = n*(n+1)*(23*n-20)/6. 9

%I #59 Sep 08 2022 08:46:11

%S 0,1,26,98,240,475,826,1316,1968,2805,3850,5126,6656,8463,10570,13000,

%T 15776,18921,22458,26410,30800,35651,40986,46828,53200,60125,67626,

%U 75726,84448,93815,103850,114576,126016,138193,151130,164850,179376,194731,210938

%N 25-gonal pyramidal numbers: a(n) = n*(n+1)*(23*n-20)/6.

%C If b(n,k) = n*(n+1)*((k-2)*n-(k-5))/6 is n-th k-gonal pyramidal number, then b(n,k) = A000292(n) + (k-3)*A000292(n-1) (see Deza in References section, p. 96).

%C Also, b(n,k) = b(n,k-1) + A000292(n-1) (see Deza in References section, p. 95). Some examples:

%C for k=4, A000330(n) = A000292(n) + A000292(n-1);

%C for k=5, A002411(n) = A000330(n) + A000292(n-1);

%C for k=6, A002412(n) = A002411(n) + A000292(n-1), etc.

%C This is the case k=25.

%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (23rd row of the table).

%H Colin Barker, <a href="/A256645/b256645.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F G.f.: x*(1 + 22*x)/(1 - x)^4.

%F a(n) = A000292(n) + 22*A000292(n-1) = A256716(n) + A000292(n-1), see comments.

%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3. - _Colin Barker_, Apr 07 2015

%t Table[n (n + 1) (23 n - 20)/6, {n, 0, 40}]

%t LinearRecurrence[{4, -6, 4, -1}, {0, 1, 26, 98}, 40] (* _Vincenzo Librandi_, Apr 08 2015 *)

%o (PARI) concat(0, Vec(x*(1 + 22*x)/(1 - x)^4 + O(x^100))) \\ _Colin Barker_, Apr 07 2015

%o (Magma) k:=25; [n*(n+1)*((k-2)*n-(k-5))/6: n in [0..40]]; // _Vincenzo Librandi_, Apr 08 2015

%Y Partial sums of A255184.

%Y Cf. similar sequences listed in A237616.

%Y Cf. A000292, A256716.

%K nonn,easy

%O 0,3

%A _Luciano Ancora_, Apr 07 2015

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