A075362 -id:A075362 - cp's OEIS frontend

A193665 Q-residue of A075392, where Q=A075392. (See Comments.)

1, 6, 33, 208, 1505, 12330, 112973, 1145568, 12742389, 154308350, 2021296189, 28480485024, 429565218277, 6905903216562, 117891260108985, 2129869055824000, 40600135597843817, 814383095809997142, 17147155400516728601, 378137512431282658800

Offset: 0

Clark Kimberling, Aug 02 2011

nonn

The definition of Q-residue is given at A193649.

Cf. A075362, A193649.

q[n_, k_] := (k + 1) (n + 1);  (* A075362 *)
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}];
p[n_, k_] := (k + 1) (n + 1);  (* A075362 *)
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 20}]    (* A193665 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* A193665 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]

Conjecture: a(n) +(-n-4)*a(n-1) +(n+1)*a(n-2) -a(n-3)=0. - R. J. Mathar, Feb 19 2015

A223544 T(n, k) = n*k - 1.

Original entry on oeis.org

0, 1, 3, 2, 5, 8, 3, 7, 11, 15, 4, 9, 14, 19, 24, 5, 11, 17, 23, 29, 35, 6, 13, 20, 27, 34, 41, 48, 7, 15, 23, 31, 39, 47, 55, 63, 8, 17, 26, 35, 44, 53, 62, 71, 80, 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, 10, 21, 32, 43, 54, 65, 76, 87, 98, 109, 120, 11, 23, 35, 47, 59, 71, 83, 95, 107, 119, 131, 143

Offset: 1

Richard R. Forberg, Jul 19 2013

Previous name was: Triangle T(n,k), 0 < k <= n, T(n,1) = n - 1, T(n,k) = T(n,k-1) + n; read by rows.
This simple triangle arose analyzing f(x) = x/(n + e^(c/x)), for n <> 0. f(x) converges towards a rational number for large values of x, if x is rational. T(n+1,k)/(n+1)^2 equals the fractional portion of f(x) if x is large and restricted to the positive integers, c = 1 and n>=1, whereby the value of the fractional portion changes on a cycle with period n+1 (as k goes from 1 to n+1) for each n in the denominator of f(x). Other, somewhat similar triangles (or repeating fractional patterns) arise with other rational values of n or c, or other rational increments of x (even if a large irrational initial value of x is used).
Let S(n) = row sums = Sum(k>=1, T(n,k)), then:
S(n) = A077414(n);  S(n)/(n+2) = A000217(n); S(n)/n = A000096(n);
Let Sq(n) = sum of squares of row elements = Sum(k>=1, T(n,k)^2), then:
   Sq(n)/n^2 - 1/n = A058373(n)
Let D(n) = diagonal sums = Sum(k>=1, T(n-k+1, k)) then:
   D(2n) = A131423(n); D(2n-1) = 2/3*n^3 + 1/2*n^2 - 7/6*n;
   D(2n) - D(2n-1) = A000217(n); D(2n+1) - D(2n) = A115067(n);
   D(2n+2) - D(2n)= A056220(n+1);  D(2n+1) - D(2n -1) = A014106(n).
Equals A144204 with the first column of negative ones removed. - Georg Fischer, Jul 26 2023

			Triangle begins as:
0;
1,  3;
2,  5,  8;
3,  7, 11, 15;
4,  9  14, 19, 24;
5, 11, 17, 23, 29, 35;
6, 13, 20, 27, 34, 41, 48;
7, 15, 23, 31, 39, 47, 55, 63;
8, 17, 26, 35, 44, 53, 62, 71, 80;

Boris Putievskiy, Rows n = 1..140 of triangle, flattened

Cf. A075362, A144204.

Also note: T(n+1,k) = T(n,k)+ k, and T(n,n) = n^2 - 1.
a(n) = A075362(n)-1; a(n)=i(t+1)-1, where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Jul 24 2013
T(n, k) = n*k - 1. - Georg Fischer, Jul 26 2023

Simpler name from Georg Fischer, Jul 26 2023

A110749 Triangle read by rows with the n-th row containing the first n multiples of n with digits reversed.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 4, 8, 21, 61, 5, 1, 51, 2, 52, 6, 21, 81, 42, 3, 63, 7, 41, 12, 82, 53, 24, 94, 8, 61, 42, 23, 4, 84, 65, 46, 9, 81, 72, 63, 54, 45, 36, 27, 18, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 22, 33, 44, 55, 66, 77, 88, 99, 11, 121, 21, 42, 63, 84, 6, 27, 48, 69, 801, 21, 231, 441

Offset: 1

Amarnath Murthy, Aug 10 2005

			The triangle is:
1
2 4
3 6 9
4 8 21 61
5 1 15 2 52
etc

Harvey P. Dale, Table of n, a(n) for n = 1..1000

The non-reversed digit triangle is A075362.

Table[IntegerReverse[n*Range[n]],{n,20}]//Flatten (* Harvey P. Dale, Jul 14 2024 *)

T(n, k) = subst(Polrev(digits(k*n)), x, 10); \\ Michel Marcus, Sep 16 2013

More terms from Bruce Corrigan (scentman(AT)myfamily.com), Aug 10 2005

A377133 Triangle read by rows: T(n,k) is the maximum volume of an integer-sided box that can be made from a piece of paper of size n X k by cutting away identical squares at each corner and folding up the sides, n >= 3, 3 <= k <= n.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 4, 8, 12, 16, 5, 10, 15, 20, 25, 6, 12, 18, 24, 30, 36, 7, 14, 21, 28, 35, 42, 50, 8, 16, 24, 32, 40, 48, 60, 72, 9, 18, 27, 36, 45, 56, 70, 84, 98, 10, 20, 30, 40, 50, 64, 80, 96, 112, 128, 11, 22, 33, 44, 55, 72, 90, 108, 126, 144, 162, 12, 24

Offset: 3

Felix Huber, Oct 25 2024

For a sketch see linked illustration "Box made from nXk-paper".

The first few rows follow (n-2) * (k-2), so the initial terms are the same as in A075362. The first difference is at T(9,9) = 50 which is greater than 7 * 7.

			Triangle T(n,k) begins:
   n\k   3     4     5     6     7     8     9    10    11    12    13 ...
   3     1
   4     2     4
   5     3     6     9
   6     4     8    12    16
   7     5    10    15    20    25
   8     6    12    18    24    30    36
   9     7    14    21    28    35    42    50
  10     8    16    24    32    40    48    60    72
  11     9    18    27    36    45    56    70    84    98
  12    10    20    30    40    50    64    80    96   112   128
  13    11    22    33    44    55    72    90   108   126   144   162

Felix Huber, Rows n = 3..142 of triangle, flattened
Felix Huber, Box made from nXk-paper

Cf. A075362, A355880, A375580, A375785, A375785.

A377113:=proc(n,k)
   local a,x,V;
   a:=0;
   for x to (k-1)/2 do
      V:=x*(n-2*x)*(k-2*x);
      if V>a then
         a:=V
      fi
   od;
   return a
end proc;
seq(seq(A377113(n,k),k=3..n),n=3..14);

T(n,k) = (n-2*x)*(k-2*x)*x with x = round((n+k-(sqrt(n^2+k^2-n*k)))/6).

A360373 Triangular array T read by rows related to the multiplication table.

Original entry on oeis.org

1, 2, 4, 2, 3, 6, 9, 6, 3, 4, 8, 12, 16, 12, 8, 4, 5, 10, 15, 20, 25, 20, 15, 10, 5, 6, 12, 18, 24, 30, 36, 30, 24, 18, 12, 6, 7, 14, 21, 28, 35, 42, 49, 42, 35, 28, 21, 14, 7, 8, 16, 24, 32, 40, 48, 56, 64, 56, 48, 40, 32, 24, 16, 8, 9, 18, 27, 36, 45, 54, 63, 72, 81

Offset: 1

Philippe Deléham, Feb 04 2023

			Table T(n, k) , n>=1 , 1<=k<=2*n-1.
n = 1 : 1 ;
n = 2 : 2,  4,  2 ;
n = 3 : 3,  6,  9,  6,  3 ;
n = 4 : 4,  8, 12, 16, 12,  8,  4 ;
n = 5 : 5, 10, 15, 20, 25, 20, 15, 10,  5 ;
n = 6 : 6, 12, 18, 24, 30, 36, 30, 24, 18, 12,  6 ;
n = 7 : 7, 14, 21, 28, 35, 42, 49, 42, 35, 28, 21, 14,  7 ;
n = 8 : 8, 16, 24, 32, 40, 48, 56, 64, 56, 48, 40, 32, 24, 16,  8 ;
...

Cf. A000290 (central terms), A000578 (row sums), A060747 (row lengths).

Cf. A003991, A004737, A075362, A272125 .

T:= (n, k)-> n*min(k, 2*n-k):
seq(seq(T(n,k), k=1..2*n-1), n=1..10);  # Alois P. Heinz, Feb 04 2023

T(n, k) = T(n, 2*n-k) = n*k for 1<=k<=n .
Sum_{k=1..2*n-1} T(n, k) = n^3.
Sum_{k=1..2*n-1} T(n, k)^2 = n^3*(2*n^2 + 1)/3 = A272125(n).
T(n, k) = n * A004737(n,k).

cp's OEIS Frontend

A193665 Q-residue of A075392, where Q=A075392. (See Comments.)

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A223544 T(n, k) = n*k - 1.

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A110749 Triangle read by rows with the n-th row containing the first n multiples of n with digits reversed.

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A377133 Triangle read by rows: T(n,k) is the maximum volume of an integer-sided box that can be made from a piece of paper of size n X k by cutting away identical squares at each corner and folding up the sides, n >= 3, 3 <= k <= n.

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A360373 Triangular array T read by rows related to the multiplication table.

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