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%I #37 Jul 11 2023 08:30:02
%S 1,34,153,436,985,1926,3409,5608,8721,12970,18601,25884,35113,46606,
%T 60705,77776,98209,122418,150841,183940,222201,266134,316273,373176,
%U 437425,509626,590409,680428,780361,890910,1012801,1146784,1293633,1454146,1629145,1819476
%N a(n) = n^4 + 3*n^3 - 3*n.
%C Related to the 9-gon (nonagon). Following Steinbach's strategy re: "Diagonal Product Formulas" and applied to the 9-gon (nonagon), we extract the constants (a, b, c, d) as e-vals of the 4 X 4 tridiagaonal matrix with (1's in the super and subdiagonals), (1,2,2,2), and the rest zeros. The charpoly of this matrix is row 4 of A054142, a Morgan-Voyce polynomial: x^4 - 7*x^3 + 15*x^2 + 10*x - 1 = 0. Following Steinbach's procedure, let the matrix = M; then find the first four rows of M^n * [1,0,0,0,...] getting (1; 1,1; 2,3,1; 5,9,5,1). Using the SIMULT operation, we equate each of these rows to successive powers of the constant c (largest e-val of matrix M), 3.5320888...as follows: SIMULT: [1,0,0,0] = 1; [1,1,0,0] = c; [2,3,1,0] = c^2; [5,9,5,1] = c^3. Solving, we obtain the four distinct diagonals of the 9-gon (nonagon) with edge = 1: (1, 2.5320888,..., 2.879385,..., and 1.879385,...).
%C The sequence is column 3 in the array of A162997.
%C Analogous sequences using the matrix M^k generator -M^2 generates A028387: (1, 5, 11, 19, 29, 41,...); M^3 generates A123972: (1, 13, 41, 91, 169,...).
%H Vincenzo Librandi, <a href="/A192398/b192398.txt">Table of n, a(n) for n = 1..10000</a>
%H Peter Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden Fields: A Case for the Heptagon</a>, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: (1 +29*x -7*x^2 +x^3) / (1-x)^5. - _R. J. Mathar_, Jul 08 2011
%F a(n) = binomial transform of [1, 33, 86, 78, 24, 0, 0, 0,...].
%F a(n) = lower right term in the 2 X 2 matrix M^4, M = {{1,n-1}, {1,n}}.
%F a(n) = ((n-1) + a) * ((n-1) + b) * ((n-1) + c) * ((n-1) + d), where a, b, c, d, = {k=1,2,3,4} 4*cos^2 (2*Pi*k)/9.
%F E.g.f.: x*(1 + 16*x + 9*x^2 + x^3)*exp(x). - _G. C. Greubel_, Jul 11 2023
%e a(5) = 5^4 + 3*5^3 - 3*5 = (625 + 375 - 15) = 985.
%e a(4) = 436 = (1, 3, 3, 1) dot (1, 33, 86, 78) = (1 + 99 + 258 + 78) = 436.
%e a(7) = 3409 = lower right term in M^4, M = {{1,6}{1,7}}.
%e a(4) = 436 = (3 + a) * (3 + b) * (3 + c) * (3 + d), = (5.347296...) * (3.120614...) * (4) * (6.532088...) = 436.
%p A192398:=n->n^4+3*n^3-3*n: seq(A192398(n), n=1..40); # _Wesley Ivan Hurt_, Sep 12 2014
%t LinearRecurrence[{5,-10,10,-5,1},{1,34,153,436,985},50] (* _Vincenzo Librandi_, Nov 25 2011 *)
%t Table[n^4+3n^3-3n,{n,40}] (* _Harvey P. Dale_, Feb 21 2023 *)
%o (PARI) a(n) = n^4 +3*n^3 -3*n \\ _Charles R Greathouse IV_, Jun 30 2011
%o (Magma) [n^4 +3*n^3 -3*n: n in [1..45]]; // _Vincenzo Librandi_, Nov 25 2011
%o (SageMath) [n*(n^3+3*n^2-3) for n in range(1,51)] # _G. C. Greubel_, Jul 11 2023
%Y Cf. A028387, A054142, A123972, A162997.
%K nonn,easy
%O 1,2
%A _Gary W. Adamson_, Jun 30 2011
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