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Array read by rows: row n lists the coefficients of the characteristic polynomial of the n-th principal submatrix of max(2i-j, 2j-i), as in A204154.
+20
3
1, -1, -7, -3, 1, 33, 39, 6, -1, -135, -255, -125, -10, 1, 513, 1323, 1092, 305, 15, -1, -1863, -6075, -7047, -3444, -630, -21, 1, 6561, 25839, 38610, 27135, 8946, 1162, 28, -1, -22599, -104247, -190593, -175230
COMMENTS
Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are real, and they interlace the zeros of p(n+1). See A202605 and A204016 for guides to related sequences.
REFERENCES
(For references regarding interlacing roots, see A202605.)
EXAMPLE
Top of the array:
1, -1;
-7, -3, 1;
33, 39, 6, -1;
-135, -255, -125, -10, 1;
MAPLE
f:= proc(n) local P, lambda, i;
P:= (-1)^n*LinearAlgebra:-CharacteristicPolynomial(Matrix(n, n, (i, j) -> max(2*i-j, 2*j-i)), lambda);
seq(coeff(P, lambda, i), i=0..n);
end proc:
MATHEMATICA
f[i_, j_] := Max[2 i - j, 2 j - i];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 15}, {i, 1, n}]] (* A204154 *) p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
TableForm[Table[c[n], {n, 1, 10}]]
s( A204154), where s(j)=(prime(j+1) + prime(j+2))/2.
+20
2
4, 4, 6, 4, 4, 6, 4, 4, 6, 6, 4, 6, 4, 4, 6, 18, 4, 12, 15, 6, 9, 4, 4, 6, 9, 4, 12, 6, 21, 4, 45, 18, 6, 26, 4, 6, 39, 4, 6, 6, 4, 18, 21, 6, 15, 4, 9, 12, 15, 6, 9, 4, 76, 6, 9, 4, 12, 6, 34, 4, 15, 72, 6, 12, 4, 6, 9, 4, 12, 6, 15, 4, 26, 12, 6, 26, 4, 15, 26, 6
COMMENTS
For a guide to related sequences, see A204892.
Permanent of the n-th principal submatrix of A204154.
+20
1
1, 11, 219, 8353, 501441, 43953387, 5289422019, 837277689897, 168675250704201, 42143281220724723, 12789052583333211963, 4633606346714879986017, 1975686665736618600155505, 979311369588813584600206491, 558412149326539306097214853299, 362948190926517939498607117046169
MATHEMATICA
f[i_, j_] := Max[2 i - j, 2 j - i]; (* A204154 *) m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Permanent[m_] :=
With[{a = Array[x, Length[m]]},
Coefficient[Times @@ (m.a), Times @@ a]];
Table[Permanent[m[n]], {n, 1, 14}] (* A204236 *)
Symmetric matrix based on f(i,j) = max(j mod i, i mod j), by antidiagonals.
+10
74
0, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 2, 0, 2, 1, 1, 2, 3, 3, 2, 1, 1, 2, 3, 0, 3, 2, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 2, 3, 4, 0, 4, 3, 2, 1, 1, 2, 3, 4, 5, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 0, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 0, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 7
COMMENTS
A204016 represents the matrix M given by f(i,j) = max{(j mod i), (i mod j)} for i >= 1 and j >= 1. See A204017 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
Guide to symmetric matrices M based on functions f(i,j) and characteristic polynomial sequences (c.p.s.) with interlaced zeros:
f(i,j)..........................M.........c.p.s.
..abbreviation below: AOE means "all 1's except"
...
See A202695 for a guide to choices of symmetric matrix M for which the zeros of the characteristic polynomials are all positive.
EXAMPLE
Northwest corner:
0 1 1 1 1 1 1 1
0 1 2 2 2 2 2 2
1 2 0 3 3 3 3 3
1 2 3 0 4 4 4 4
1 2 3 4 0 5 5 5
1 2 3 4 5 0 6 6
1 2 3 4 5 6 0 7
MATHEMATICA
f[i_, j_] := Max[Mod[i, j], Mod[j, i]];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]] (* A204016 *) p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
TableForm[Table[c[n], {n, 1, 10}]]
Octagonal pyramidal numbers: a(n) = n*(n+1)*(2*n-1)/2.
(Formerly M4609 N1966)
+10
50
1, 9, 30, 70, 135, 231, 364, 540, 765, 1045, 1386, 1794, 2275, 2835, 3480, 4216, 5049, 5985, 7030, 8190, 9471, 10879, 12420, 14100, 15925, 17901, 20034, 22330, 24795, 27435, 30256, 33264, 36465, 39865, 43470, 47286, 51319, 55575, 60060, 64780
COMMENTS
Number of ways of covering a 2n X 2n lattice with 2n^2 dominoes with exactly 2 horizontal dominoes. - Yong Kong (ykong@curagen.com), May 06 2000
Equals binomial transform of [0, 1, 7, 6, 0, 0, 0, ...]. - Gary W. Adamson, Jun 14 2008, corrected Oct 25 2012 Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF3 denominators of A156927. See A157704 for background information. - Johannes W. Meijer, Mar 07 2009 This sequence is related to A000326 by a(n) = n* A000326(n) - Sum_{i=0..n-1} A000326(i) and this is the case d=3 in the identity n*(n*(d*n-d+2)/2)-Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - Bruno Berselli, Apr 21 2010 Partial sums of the figurate octagonal numbers A000567. For each sequence with a linear recurrence with constant coefficients, the values reduced modulo some constant m generate a periodic sequence. For this sequence, these Pisano periods have length 1, 4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17, ... for m >= 1. - Ant King, Oct 26 2012 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. - Robert Price, May 23 2016 On a square grid of side length n+1, the number of embedded rectangles (where each side is greater than 1). For example, in a 2 X 2 square there is one rectangle, in a 3 X 3 square there are nine rectangles, etc. - Peter Woodward, Nov 26 2017 a(n) is the sum of the numbers in the n X n square array A204154(n). - Ali Sada, Jun 21 2019 Sum of all multiples of n and n+1 that are <= n^2. - Wesley Ivan Hurt, May 25 2023
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 2.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
B. Berselli, A description of the transform in Comments lines: website Matem@ticamente (in Italian).
FORMULA
a(n) = odd numbers * triangular numbers = (2*n-1)* binomial(n+1,2). - Xavier Acloque, Oct 27 2003
G.f.: x*(1+5*x)/(1-x)^4. [Conjectured by Simon Plouffe in his 1992 dissertation.] a(n) = a(n-1) + n*(3*n-2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = A000292(n) + 5* A000292(n-1) = binomial(n+2,3)+5*binomial(n+1,3). Sum_{n>=1} 1/a(n) = 2*(4*log(2)-1)/3 = 1.1817258148265...
(End)
a(n) = Sum_{i=0..n-1} (n-i)*(6*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014 Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(Pi + 1 - 4*log(2))/3. - Amiram Eldar, Jul 02 2020
EXAMPLE
a(2) = 9 since there are 9 ways to cover a 4 X 4 lattice with 8 dominoes, 2 of which is horizontal and the other 6 are vertical. - Yong Kong (ykong@curagen.com), May 06 2000
G.f. = x + 9*x^2 + 30*x^3 + 70*x^4 + 135*x^5 + 231*x^6 + 364*x^7 + 540*x^8 + 765*x^9 + ...
MATHEMATICA
LinearRecurrence[{4, -6, 4, -1}, {1, 9, 30, 70}, 40] (* Harvey P. Dale, Apr 12 2013 *)
PROG
(PARI) {a(n) = (2*n - 1) * n * (n + 1) / 2} \\ Michael Somos, Mar 17 2011
CROSSREFS
Cf. similar sequences listed in A237616. Cf. A260234 (largest prime factor of a(n+1)).
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), May 09 2000
Incorrect formula deleted by Ant King, Oct 04 2012
Alternating sum of heptagonal numbers.
+10
2
0, -1, 6, -12, 22, -33, 48, -64, 84, -105, 130, -156, 186, -217, 252, -288, 328, -369, 414, -460, 510, -561, 616, -672, 732, -793, 858, -924, 994, -1065, 1140, -1216, 1296, -1377, 1462, -1548, 1638, -1729, 1824, -1920, 2020, -2121, 2226, -2332, 2442, -2553
FORMULA
G.f.: -x*(1 - 4*x)/((1 - x)*(1 + x)^3).
a(n) = ((10*n^2 + 4*n - 3)*(-1)^n + 3)/8.
a(n) = Sum_{k = 0..n} (-1)^k* A000566(k). Lim_{n -> infinity} a(n + 1)/a(n) = -1.
E.g.f.: (1/8)*exp(-x)*(-3 + 3*exp(2*x) - 14*x + 10*x^2). - Stefano Spezia, Nov 13 2019
MATHEMATICA
Table[((10 n^2 + 4 n - 3) (-1)^n + 3)/8, {n, 0, 50}]
CoefficientList[Series[(x - 4 x^2)/(x^4 + 2 x^3 - 2 x - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *) LinearRecurrence[{-2, 0, 2, 1}, {0, -1, 6, -12}, 60] (* Harvey P. Dale, Jan 26 2023 *)
PROG
(Magma) [((10*n^2+4*n-3)*(-1)^n+3)/8: n in [0..50]]; // Vincenzo Librandi, Dec 21 2015 (Magma) R<x>:=PowerSeriesRing(Integers(), 50); [0] cat Coefficients(R!(-x*(1 - 4*x)/((1 - x)*(1 + x)^3))); // Marius A. Burtea, Nov 13 2019 (PARI) x='x+O('x^100); concat(0, Vec(-x*(1-4*x)/((1-x)*(1+x)^3))) \\ Altug Alkan, Dec 21 2015
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