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A260260 a(n) = n*(16*n^2 - 21*n + 7)/2.

%I A260260 #46 Aug 09 2025 10:04:42
%S A260260 0,1,29,132,358,755,1371,2254,3452,5013,6985,9416,12354,15847,19943,
%T A260260 24690,30136,36329,43317,51148,59870,69531,80179,91862,104628,118525,
%U A260260 133601,149904,167482,186383,206655,228346,251504,276177,302413,330260,359766,390979
%N A260260 a(n) = n*(16*n^2 - 21*n + 7)/2.
%C A260260 Similar sequences, where P(s, m) = ((s-2)*m^2-(s-4)*m)/2 is the m-th s-gonal number:
%C A260260 A000578: P(3, m)*P( 3, m) - P(3, m-1)*P( 3, m-1);
%C A260260 A213772: P(3, m)*P( 4, m) - P(3, m-1)*P( 4, m-1) for m>0;
%C A260260 A005915: P(3, m)*P( 5, m) - P(3, m-1)*P( 5, m-1)   "    ;
%C A260260 A130748: P(3, m)*P( 6, m) - P(3, m-1)*P( 6, m-1) for m>1;
%C A260260 A027849: P(3, m)*P( 7, m) - P(3, m-1)*P( 7, m-1) for m>0;
%C A260260 A214092: P(3, m)*P( 8, m) - P(3, m-1)*P( 8, m-1)   "    ;
%C A260260 A100162: P(3, m)*P( 9, m) - P(3, m-1)*P( 9, m-1)   "    ;
%C A260260 A260260: P(3, m)*P(10, m) - P(3, m-1)*P(10, m-1), this sequence;
%C A260260 A100165: P(3, m)*P(11, m) - P(3, m-1)*P(11, m-1) for m>0.
%H A260260 Bruno Berselli, <a href="/A260260/b260260.txt">Table of n, a(n) for n = 0..1000</a>
%H A260260 OEIS Wiki, <a href="https://oeis.org/wiki/Figurate_numbers#Polygonal_numbers">Figurate numbers</a>.
%H A260260 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polygonal_number#Table_of_values">Polygonal numbers: Table of values</a>.
%H A260260 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A260260 G.f.: x*(1 + 25*x + 22*x^2)/(1 - x)^4. [corrected by _Georg Fischer_, May 10 2019]
%F A260260 a(n) = A000217(n)*A001107(n) - A000217(n-1)*A001107(n-1), with A000217(-1) = 0.
%F A260260 a(n) = A000292(n) + 25*A000292(n-1) + 22*A000292(n-2), with A000292(-2) = A000292(-1) = 0.
%F A260260 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
%F A260260 E.g.f.: exp(x)*x*(2 + 27*x + 16*x^2)/2. - _Elmo R. Oliveira_, Aug 08 2025
%t A260260 Table[n (16 n^2 - 21 n + 7)/2, {n, 0, 40}]
%t A260260 LinearRecurrence[{4,-6,4,-1},{0,1,29,132},40] (* _Harvey P. Dale_, May 08 2025 *)
%o A260260 (PARI) vector(40, n, n--; n*(16*n^2-21*n+7)/2)
%o A260260 (Sage) [n*(16*n^2-21*n+7)/2 for n in (0..40)]
%o A260260 (Magma) [n*(16*n^2-21*n+7)/2: n in [0..40]];
%Y A260260 Cf. A000217, A000292, A001107.
%Y A260260 Subsequence of A047275.
%Y A260260 Sequences of the same type (see comment): A000578, A005915, A027849, A100162, A100165, A130748, A213772, A214092.
%K A260260 nonn,easy
%O A260260 0,3
%A A260260 _Bruno Berselli_, Jul 21 2015