A192398 a(n) = n^4 + 3*n^3 - 3*n.

1, 34, 153, 436, 985, 1926, 3409, 5608, 8721, 12970, 18601, 25884, 35113, 46606, 60705, 77776, 98209, 122418, 150841, 183940, 222201, 266134, 316273, 373176, 437425, 509626, 590409, 680428, 780361, 890910, 1012801, 1146784, 1293633, 1454146, 1629145, 1819476

Offset: 1

Author

Gary W. Adamson, Jun 30 2011

Keywords

• nonn
• easy

Comments

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,...).

The sequence is column 3 in the array of A162997.

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,...).

Examples

			a(5) = 5^4 + 3*5^3 - 3*5 = (625 + 375 - 15) = 985.
a(4) = 436 = (1, 3, 3, 1) dot (1, 33, 86, 78) = (1 + 99 + 258 + 78) = 436.
a(7) = 3409 = lower right term in M^4, M = {{1,6}{1,7}}.
a(4) = 436 = (3 + a) * (3 + b) * (3 + c) * (3 + d), = (5.347296...) * (3.120614...) * (4) * (6.532088...) = 436.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Peter Steinbach, Golden Fields: A Case for the Heptagon, Mathematics Magazine, Vol. 70, No. 1, Feb. 1997.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A028387, A054142, A123972, A162997.

Programs

Magma

[n^4 +3*n^3 -3*n: n in [1..45]]; // Vincenzo Librandi, Nov 25 2011

Maple

A192398:=n->n^4+3*n^3-3*n: seq(A192398(n), n=1..40); # Wesley Ivan Hurt, Sep 12 2014

Mathematica

LinearRecurrence[{5,-10,10,-5,1},{1,34,153,436,985},50] (* Vincenzo Librandi, Nov 25 2011 *)
Table[n^4+3n^3-3n,{n,40}] (* Harvey P. Dale, Feb 21 2023 *)

PARI

a(n) = n^4 +3*n^3 -3*n \\ Charles R Greathouse IV, Jun 30 2011

SageMath

[n*(n^3+3*n^2-3) for n in range(1,51)] # G. C. Greubel, Jul 11 2023

Formula

G.f.: (1 +29*x -7*x^2 +x^3) / (1-x)^5. - R. J. Mathar, Jul 08 2011

a(n) = binomial transform of [1, 33, 86, 78, 24, 0, 0, 0,...].

a(n) = lower right term in the 2 X 2 matrix M^4, M = {{1,n-1}, {1,n}}.

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.

E.g.f.: x*(1 + 16*x + 9*x^2 + x^3)*exp(x). - G. C. Greubel, Jul 11 2023