A204008 -id:A204008 - cp's OEIS frontend

A204011 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{3i+j-3,i+3j-3} (A204008).

1, -1, -11, -6, 1, 40, 70, 15, -1, -116, -328, -240, -28, 1, 304, 1176, 1456, 610, 45, -1, -752, -3680, -6408, -4704, -1295, -66, 1, 1792, 10592, 23760, 25080, 12432, 2436, 91, -1, -4160, -28800, -79040

Offset: 1

Clark Kimberling, Jan 09 2012

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 for a guide to related sequences.

			Top of the array:
 1.....-1
-11....-6.....1
 40.....70....15....-1
-116...-328..-240....1

(For references regarding interlacing roots, see A202605.)

Cf. A204008, A202605.

f[i_, j_] := Max[3 i + j - 3, 3 j + i - 3];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]]   (* A204008 *)
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}]
Flatten[%]                (* A204011 *)
TableForm[Table[c[n], {n, 1, 10}]]

A204016 Symmetric matrix based on f(i,j) = max(j mod i, i mod j), by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 10 2012

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.
C(i+j,j)........................A007318...A045912
min(i,j)........................A003983...A202672
max(i,j)........................A051125...A203989
(i+j)*min(i,j)..................A203990...A203991
|i-j|...........................A049581...A203993
max(i-j+1,j-i+1)................A143182...A203992
min(i-j+1,j-i+1)................A203994...A203995
min(i(j+1),j(i+1))..............A203996...A203997
max(i(j+1)-1,j(i+1)-1)..........A203998...A203999
min(i(j+1)-1,j(i+1)-1)..........A204000...A204001
min(2i+j,i+2j)..................A204002...A204003
max(2i+j-2,i+2j-2)..............A204004...A204005
min(2i+j-2,i+2j-2)..............A204006...A204007
max(3i+j-3,i+3j-3)..............A204008...A204011
min(3i+j-3,i+3j-3)..............A204012...A204013
min(3i-2,3j-2)..................A204028...A204029
1+min(j mod i, i mod j).........A204014...A204015
max(j mod i, i mod j)...........A204016...A204017
1+max(j mod i, i mod j).........A204018...A204019
min(i^2,j^2)....................A106314...A204020
min(2i-1, 2j-1).................A157454...A204021
max(2i-1, 2j-1).................A204022...A204023
min(i(i+1)/2,j(j+1)/2)..........A106255...A204024
gcd(i,j)........................A003989...A204025
gcd(i+1,j+1)....................A204030...A204111
min(F(i+1),F(j+1)),F=A000045....A204026...A204027
gcd(F(i+1),F(j+1)),F=A000045....A204112...A204113
gcd(L(i),L(j)),L=A000032........A204114...A204115
gcd(2^i-1,2^j-2)................A204116...A204117
gcd(prime(i),prime(j))..........A204118...A204119
gcd(prime(i+1),prime(j+1))......A204120...A204121
gcd(2^(i-1),2^(j-1))............A144464...A204122
max(floor(i/j),floor(j/i))......A204123...A204124
min(ceiling(i/j),ceiling(j/i))..A204143...A204144
Delannoy matrix.................A008288...A204135
max(2i-j,2j-i)..................A204154...A204155
-1+max(3i-j,3j-i)...............A204156...A204157
max(3i-2j,3j-2i)................A204158...A204159
floor((i+1)/2)..................A204164...A204165
ceiling((i+1)/2)................A204166...A204167
i+j.............................A003057...A204168
i+j-1...........................A002024...A204169
i*j.............................A003991...A204170
..abbreviation below:  AOE means "all 1's except"
AOE f(i,i)=i....................A204125...A204126
AOE f(i,i)=A000045(i+1).........A204127...A204128
AOE f(i,i)=A000032(i)...........A204129...A204130
AOE f(i,i)=2i-1.................A204131...A204132
AOE f(i,i)=2^(i-1)..............A204133...A204134
AOE f(i,i)=3i-2.................A204160...A204161
AOE f(i,i)=floor((i+1)/2).......A204162...A204163
...
Other pairs (M, c.p.s.): (A204171, A204172) to (A204183, A204184)
See A202695 for a guide to choices of symmetric matrix M for which the zeros of the characteristic polynomials are all positive.

			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

Cf. A204017, A202453.

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}]
Flatten[%]               (* A204017 *)
TableForm[Table[c[n], {n, 1, 10}]]

A131229 Numbers congruent to {1,7} mod 10.

Original entry on oeis.org

1, 7, 11, 17, 21, 27, 31, 37, 41, 47, 51, 57, 61, 67, 71, 77, 81, 87, 91, 97, 101, 107, 111, 117, 121, 127, 131, 137, 141, 147, 151, 157, 161, 167, 171, 177, 181, 187, 191, 197, 201, 207, 211, 217, 221, 227, 231, 237, 241, 247, 251, 257, 261, 267, 271, 277, 281

Offset: 1

Gary W. Adamson, Jun 20 2007

Row sums of triangle A131228.
Take the sum of the squares of the first n triangular numbers and divide it by the sum of these n triangular numbers.  The sum evenly divides the sum of the squares for the n in this sequence. - J. M. Bergot, May 09 2012
a(n) = the difference between the sum of the terms in antidiagonal(n) and antidiagonal(n-1) in A204008. - J. M. Bergot, Jul 15 2013

			11 = a(3) = 1 mod 10.
11 = a(3) = row 4 sums, triangle A131228: (1 + 3 + 7).

David Lovler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001622, A128174, A051340, A063097, A131228.

Select[Range[300],MemberQ[{1,7},Mod[#,10]]&] (* or *) LinearRecurrence[ {1,1,-1},{1,7,11},60] (* Harvey P. Dale, May 20 2012 *)

is(n)=n%10==1 || n%10==7 \\ Charles R Greathouse IV, Jul 16 2013

From R. J. Mathar, Apr 17 2008: (Start)
a(n) = a(n-2) + 10 = 5*n + ((-1)^n - 7)/2.
G.f.: x*(1 + 6*x + 3*x^2)/((-1+x)^2*(1+x)). (End)
a(n) = 10*n - a(n-1) - 12 with n > 1, a(1)=1. - Vincenzo Librandi, Nov 23 2010
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3; a(1)=1, a(2)=7, a(3)=11. - Harvey P. Dale, May 20 2012
a(n) = A063097(n) for n >= 2. - Georg Fischer, Nov 02 2018
E.g.f.: 3 + ((10*x - 7)*exp(x) + exp(-x))/2. - David Lovler, Sep 07 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt((5+sqrt(5))/2)*Pi/10 + 3*log(phi)/(2*sqrt(5)), where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023

More terms from R. J. Mathar, Apr 17 2008

A204004 Symmetric matrix based on f(i,j) = max{2i+j-2,i+2j-2}, by antidiagonals.

Original entry on oeis.org

1, 3, 3, 5, 4, 5, 7, 6, 6, 7, 9, 8, 7, 8, 9, 11, 10, 9, 9, 10, 11, 13, 12, 11, 10, 11, 12, 13, 15, 14, 13, 12, 12, 13, 14, 15, 17, 16, 15, 14, 13, 14, 15, 16, 17, 19, 18, 17, 16, 15, 15, 16, 17, 18, 19, 21, 20, 19, 18, 17, 16, 17, 18, 19, 20, 21, 23, 22, 21, 20, 19, 18

Offset: 1

Clark Kimberling, Jan 09 2012

A204004 represents the matrix M given by f(i,j)=max{2i+j,i+2j}for i>=1 and j>=1. See A204005 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
General case A206772. Let m be natural number. Table T(n,k)=max{m*n+k-m,n+m*k-m} read by antidiagonals.
  For m=1 the result is A002024,
  for m=2 the result is A204004,
  for m=3 the result is A204008,
  for m=4 the result is A206772. - Boris Putievskiy, Jan 24 2013

			Northwest corner:
  1,  3,  5,  7,  9
  3,  4,  6,  8, 10
  5,  6,  7,  9, 11
  7,  8,  9, 10, 12

Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.

Cf. A204005, A202453, A002024, A204008, A002260, A004736, A206772.

f[i_, j_] := Max[2 i + j - 2, 2 j + i - 2];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[6]] (* 6x6 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]]  (* A204004 *)
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}]
Flatten[%]   (* A204005 *)
TableForm[Table[c[n], {n, 1, 10}]]

From Boris Putievskiy, Jan 24 2013: (Start)
For the general case, a(n) = m*A002024(n) + (m-1)*max{-A002260(n),-A004736(n)}.
a(n) = m*(t+1) + (m-1)*max{t*(t+1)/2-n,n-(t*t+3*t+4)/2}, where t=floor((-1+sqrt(8*n-7))/2).
For m=2, a(n) = 2*(t+1) + max{t*(t+1)/2-n,n-(t*t+3*t+4)/2}, where t=floor((-1+sqrt(8*n-7))/2). (End)

A206772 Table T(n,k)=max{4n+k-4,n+4k-4} n, k > 0, read by antidiagonals.

Original entry on oeis.org

1, 5, 5, 9, 6, 9, 13, 10, 10, 13, 17, 14, 11, 14, 17, 21, 18, 15, 15, 18, 21, 25, 22, 19, 16, 19, 22, 25, 29, 26, 23, 20, 20, 23, 26, 29, 33, 30, 27, 24, 21, 24, 27, 30, 33, 37, 34, 31, 28, 25, 25, 28, 31, 34, 37, 41, 38, 35, 32, 29, 26, 29, 32, 35, 38, 41, 45

Offset: 1

Boris Putievskiy, Jan 15 2013

In general, let m be natural number. Table T(n,k)=max{m*n+k-m,n+m*k-m}. For m=1 the result is A002024, for m=2 the result is A204004, for m=3 the result is A204008. This sequence is result for m=4.

			The start of the sequence as table for general case:
  1........m+1..2*m+1..3*m+1..4*m+1..5*m+1..6*m+1 ...
  m+1......m+2..2*m+2..3*m+2..4*m+2..5*m+2..6*m+2 ...
  2*m+1..2*m+2..2*m+3..3*m+3..4*m+3..5*m+3..6*m+3 ...
  3*m+1..3*m+2..3*m+3..3*m+4..4*m+4..5*m+4..6*m+4 ...
  4*m+1..4*m+2..4*m+3..4*m+4..4*m+5..5*m+5..6*m+5 ...
  5*m+1..5*m+2..5*m+3..5*m+4..5*m+5..5*m+6..6*m+6 ...
  6*m+1..6*m+2..6*m+3..6*m+4..6*m+5..6*m+6..6*m+7 ...
  . . .
The start of the sequence as triangle array read by rows for general case:
  1;
  m+1,     m+1;
  2*m+1,   m+2, 2*m+1;
  3*m+1, 2*m+2, 2*m+2, 3*m+1;
  4*m+1, 3*m+2, 2*m+3, 3*m+2, 4*m+1;
  5*m+1, 4*m+2, 3*m+3, 2*m+4, 3*m+3, 4*m+2; 5*m+1;
  6*m+1, 5*m+2, 4*m+3, 3*m+4, 2*m+5, 3*m+4, 4*m+3; 5*m+2, 6*m+1;
  . . .
Row number r contains r numbers: r*m+1, (r-1)*m+2, ... (r-1)*m+2, r*m+1.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A002024, A204004, A204008, A002260, A004736.

t=int((math.sqrt(8*n-7)-1)/2)
result=4*(t+1)+3*max(t*(t+1)/2-n,n-(t*t+3*t+4)/2)

For the general case
a(n) = m*A002024(n) + (m-1)*max{-A002260(n),-A004736(n)}.
a(n) = m*(t+1) + (m-1)*max{t*(t+1)/2-n,n-(t*t+3*t+4)/2}
where t=floor((-1+sqrt(8*n-7))/2).
For m=4
a(n) = 4*(t+1) + 3*max{t*(t+1)/2-n,n-(t*t+3*t+4)/2}
where t=floor((-1+sqrt(8*n-7))/2).

A214659 a(n) = n(7n^2 - 3*n - 1)/3.

Original entry on oeis.org

0, 1, 14, 53, 132, 265, 466, 749, 1128, 1617, 2230, 2981, 3884, 4953, 6202, 7645, 9296, 11169, 13278, 15637, 18260, 21161, 24354, 27853, 31672, 35825, 40326, 45189, 50428, 56057, 62090, 68541, 75424, 82753, 90542, 98805, 107556, 116809, 126578, 136877

Offset: 0

Reinhard Zumkeller, Jul 25 2012

a(n) = the sum of the n X n matrices of A204008. For example, for n = 3, the sum of the 9 elements of the 3 X 3 submatrix of A204008 is 1 + 4 + 7 + 4 + 5 + 8 + 7 + 8 + 9 = 53. - J. M. Bergot, Jul 15 2013

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002378, A051673, A204008, A214604, A214675.

a214659 n = ((7 * n - 3) * n - 1) * n `div` 3

[(7*n^3-3*n^2-n)/3 : n in [0..50]]; // Wesley Ivan Hurt, Apr 11 2015

A214659:=n->(7*n^3-3*n^2-n)/3: seq(A214659(n), n=0..50); # Wesley Ivan Hurt, Apr 11 2015

Table[(7 n^3 -3 n^2 -n)/3, {n,0,50}] (* Wesley Ivan Hurt, Apr 11 2015 *)
LinearRecurrence[{4,-6,4,-1}, {0,1,14,53}, 51] (* G. C. Greubel, Mar 09 2024 *)

[(7*n^3-3*n^2-n)/3 for n in range(51)] # G. C. Greubel, Mar 09 2024

a(n) = Sum_{k=0..n} A214604(n, k) for n > 0 (row sums).
a(n) = A002378(n) + A051673(n).
From Wesley Ivan Hurt, Apr 11 2015: (Start)
a(n) = (7*n^3 - 3*n^2 - n)/3.
G.f.: x*(1+10*x+3*x^2)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
E.g.f.: (x/3)*(3 + 18*x + 7*x^2)*exp(x). - G. C. Greubel, Mar 09 2024

cp's OEIS Frontend

A204011 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{3i+j-3,i+3j-3} (A204008).

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A204016 Symmetric matrix based on f(i,j) = max(j mod i, i mod j), by antidiagonals.

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A131229 Numbers congruent to {1,7} mod 10.

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A204004 Symmetric matrix based on f(i,j) = max{2i+j-2,i+2j-2}, by antidiagonals.

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A206772 Table T(n,k)=max{4*n+k-4,n+4*k-4} n, k > 0, read by antidiagonals.

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A214659 a(n) = n*(7*n^2 - 3*n - 1)/3.

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A206772 Table T(n,k)=max{4n+k-4,n+4k-4} n, k > 0, read by antidiagonals.

A214659 a(n) = n(7n^2 - 3*n - 1)/3.