A123972 -id:A123972 - cp's OEIS frontend

A299045 Rectangular array: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)binomial(k-floor((j+1)/2), floor(j/2))(-n)^(k-j), n >= 1, k >= 0, read by antidiagonals.

1, 1, 0, 1, -1, -1, 1, -2, 1, 1, 1, -3, 5, -1, 0, 1, -4, 11, -13, 1, -1, 1, -5, 19, -41, 34, -1, 1, 1, -6, 29, -91, 153, -89, 1, 0, 1, -7, 41, -169, 436, -571, 233, -1, -1, 1, -8, 55, -281, 985, -2089, 2131, -610, 1, 1, 1, -9, 71, -433, 1926, -5741, 10009, -7953, 1597, -1, 0

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Feb 01 2018

This array is used to compute A269252: A269252(n) = least k such that |A(n,k)| is a prime, or -1 if no such k exists.
For detailed theory, see [Hone].
The array can be extended to k<0 with A(n, k) = -A(n, -k-1) for all k in Z. - Michael Somos, Jun 19 2023

			Array begins:
1   0  -1     1     0      -1       1         0        -1           1
1  -1   1    -1     1      -1       1        -1         1          -1
1  -2   5   -13    34     -89     233      -610      1597       -4181
1  -3  11   -41   153    -571    2131     -7953     29681     -110771
1  -4  19   -91   436   -2089   10009    -47956    229771    -1100899
1  -5  29  -169   985   -5741   33461   -195025   1136689    -6625109
1  -6  41  -281  1926  -13201   90481   -620166   4250681   -29134601
1  -7  55  -433  3409  -26839  211303  -1663585  13097377  -103115431
1  -8  71  -631  5608  -49841  442961  -3936808  34988311  -310957991
1  -9  89  -881  8721  -86329  854569  -8459361  83739041  -828931049

Robert Price, Table of n, a(n) for n = 1..5050
Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269251, A269252, A269253, A269254, A294099, A298675, A298677, A298878, A299045, A299071.
Cf. A094954 (unsigned version of this array, but missing the first row).
Cf. Rows: A057078, A033999, A099496, A079935 (or A001835), A004253, A001653, A049685, A070997, A070998, A138288 (or A072256), ...
Cf. Columns: A000012, A001477 (A000027), A110331 (A165900), A123972, A192398, ...

(* Array: *)
Grid[Table[LinearRecurrence[{-n, -1}, {1, 1 - n}, 10], {n, 10}]]
(*Array antidiagonals flattened (gives this sequence):*)
A299045[n_, k_] := Sum[(-1)^(Floor[j/2]) Binomial[k - Floor[(j + 1)/2], Floor[j/2]] (-n)^(k - j), {j, 0, k}]; Flatten[Table[A299045[n - k, k], {n, 11}, {k, 0, n - 1}]]

{A(n, k) = sum(j=0, k, (-1)^(j\2)*binomial(k-(j+1)\2, j\2)*(-n)^(k-j))}; /* Michael Somos, Jun 19 2023 */

G.f. for row n: (1 + x)/(1 + n*x + x^2), n >= 1.

A(n, k) = B(-n, k) where B = A294099. - Michael Somos, Jun 19 2023

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

Original entry on oeis.org

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

Gary W. Adamson, Jun 30 2011

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,...).

			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.

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).

Cf. A028387, A054142, A123972, A162997.

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

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

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 *)

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

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

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

cp's OEIS Frontend

A299045 Rectangular array: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)binomial(k-floor((j+1)/2), floor(j/2))(-n)^(k-j), n >= 1, k >= 0, read by antidiagonals.

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

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cp's OEIS Frontend

A299045 Rectangular array: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)*binomial(k-floor((j+1)/2), floor(j/2))*(-n)^(k-j), n >= 1, k >= 0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

A299045 Rectangular array: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)binomial(k-floor((j+1)/2), floor(j/2))(-n)^(k-j), n >= 1, k >= 0, read by antidiagonals.

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