%I #40 Sep 08 2022 08:46:13

%S 0,1,29,132,358,755,1371,2254,3452,5013,6985,9416,12354,15847,19943,

%T 24690,30136,36329,43317,51148,59870,69531,80179,91862,104628,118525,

%U 133601,149904,167482,186383,206655,228346,251504,276177,302413,330260,359766,390979

%N a(n) = n*(16*n^2 - 21*n + 7)/2.

%C Similar sequences, where P(s, m) = ((s-2)*m^2-(s-4)*m)/2 is the m-th s-gonal number:

%C A000578: P(3, m)*P( 3, m) - P(3, m-1)*P( 3, m-1);

%C A213772: P(3, m)*P( 4, m) - P(3, m-1)*P( 4, m-1) for m>0;

%C A005915: P(3, m)*P( 5, m) - P(3, m-1)*P( 5, m-1) " ;

%C A130748: P(3, m)*P( 6, m) - P(3, m-1)*P( 6, m-1) for m>1;

%C A027849: P(3, m)*P( 7, m) - P(3, m-1)*P( 7, m-1) for m>0;

%C A214092: P(3, m)*P( 8, m) - P(3, m-1)*P( 8, m-1) " ;

%C A100162: P(3, m)*P( 9, m) - P(3, m-1)*P( 9, m-1) " ;

%C A260260: P(3, m)*P(10, m) - P(3, m-1)*P(10, m-1), this sequence;

%C A100165: P(3, m)*P(11, m) - P(3, m-1)*P(11, m-1) for m>0.

%H Bruno Berselli, <a href="/A260260/b260260.txt">Table of n, a(n) for n = 0..1000</a>

%H OEIS Wiki, <a href="https://oeis.org/wiki/Figurate_numbers#Polygonal_numbers">Figurate numbers</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polygonal_number#Table_of_values">Polygonal numbers: Table of values</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 + 25*x + 22*x^2)/(1 - x)^4. [corrected by _Georg Fischer_, May 10 2019]

%F a(n) = A000217(n)*A001107(n) - A000217(n-1)*A001107(n-1), with A000217(-1) = 0.

%F a(n) = A000292(n) + 25*A000292(n-1) + 22*A000292(n-2), with A000292(-2) = A000292(-1) = 0.

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - _Wesley Ivan Hurt_, Dec 18 2020

%t Table[n (16 n^2 - 21 n + 7)/2, {n, 0, 40}]

%o (PARI) vector(40, n, n--; n*(16*n^2-21*n+7)/2)

%o (Sage) [n*(16*n^2-21*n+7)/2 for n in (0..40)]

%o (Magma) [n*(16*n^2-21*n+7)/2: n in [0..40]];

%Y Cf. A000217, A000292, A001107.

%Y Subsequence of A047275.

%Y Sequences of the same type (see comment): A000578, A005915, A027849, A100162, A100165, A130748, A213772, A214092.

%K nonn,easy

%O 0,3

%A _Bruno Berselli_, Jul 21 2015