A004736 -id:A004736 - cp's OEIS frontend

A128226 Triangle, A004736 * A127899 (unsigned).

1, 4, 2, 7, 7, 3, 10, 12, 10, 4, 13, 17, 17, 13, 5, 16, 22, 24, 22, 16, 6, 19, 27, 31, 31, 27, 19, 7, 22, 32, 38, 40, 38, 32, 22, 8, 25, 37, 45, 49, 49, 45, 37, 25, 9, 28, 42, 52, 58, 60, 58, 52, 42, 28, 10

Offset: 1

Gary W. Adamson, Feb 19 2007

Row sums = A084990: (1, 6, 17, 36, 65, 106, ...).

A128225 = A127899(unsigned) * A004736.

			First few row of the triangle are:
   1;
   4,  2;
   7,  7,  3;
  10, 12, 10,  4;
  13, 17, 17, 13,  5;
  16, 22, 24, 22, 16,  6;
  19, 27, 31, 31, 27, 19,  7;
  ...

Cf. A004736, A084990, A127899, A128225.

(* a127899U computes the unsigned version of A127899 *)
a127899U[n_, k_] := If[n==k||n-1==k, n, 0]/;(1<=k<=n)
a004736[n_, k_] := n-k+1/;(1<=k<=n+1)
a128225[n_, k_] := a127899U[n, n](a004736[n, k] + a004736[n-1, k])/;(1<=k<=n)
a128225[r_] := Table[a128225[n, k], {n, 1, r}, {k, 1, n}]
TableForm[a128225[7]] (* triangle *)
Flatten[a128225[10]] (* data *) (* Hartmut F. W. Hoft, Mar 13 2017 *)

A004736 * A127899 (unsigned). By columns, (3k+1), (5k+2), (7k+3), ...; k=0,1,2...

A131782 Triangle read by rows: (A004736 * A000012) + (A000012 * A004736) - A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 5, 1, 11, 5, 1, 19, 11, 5, 1, 29, 19, 11, 5, 1, 41, 29, 19, 11, 5, 1, 55, 41, 29, 19, 11, 5, 1, 71, 55, 41, 29, 19, 11, 5, 1, 89, 71, 55, 41, 29, 19, 11, 5, 1, 109, 89, 71, 55, 41, 29, 19, 11, 5, 1, 131, 109, 89, 71, 55, 41, 29, 19, 11, 5, 1, 155, 131, 109, 89, 71, 55, 41, 29, 19, 11, 5, 1

Offset: 1

Gary W. Adamson, Jul 14 2007

Row sums = A084990 starting (1, 6, 17, 36, 65, 106, ...).

			First few rows of the triangle:
   1;
   5,  1;
  11,  5,  1;
  19, 11,  5,  1;
  29, 19, 11,  5,  1;
  41, 29, 19, 11,  5,  1;
  55, 41, 29, 19, 11,  5,  1;
  ...

Cf. A004736, A028387, A084990.

n-th row = n descending terms of A028387: (1, 5, 11, 19, 29, 41, 55, ...). By columns, each column = A028387: (1, 5, 11, 19, 29, ...).

Sign in definition and a(41) corrected; more terms from Georg Fischer, Jun 05 2023

A134545 A051731 * A004736.

Original entry on oeis.org

1, 3, 1, 4, 2, 1, 7, 4, 2, 1, 6, 4, 3, 2, 1, 12, 8, 5, 3, 2, 1, 8, 6, 5, 4, 3, 2, 1, 15, 11, 8, 6, 4, 3, 2, 1, 13, 10, 8, 6, 5, 4, 3, 2, 1, 18, 14, 11, 9, 7, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, Oct 31 2007

Row sums = A007437: (1, 4, 7, 14, 16, 31, ...).

Left border = sigma(n), A000203: (1, 3, 4, 7, 6, 12, ...).

			First few rows of the triangle:
   1;
   3,  1;
   4,  2, 1;
   7,  4, 2, 1;
   6,  4, 3, 2, 1;
  12,  8, 5, 3, 2, 1;
   8,  6, 5, 4, 3, 2, 1;
  15, 11, 8, 6, 4, 3, 2, 1;
  ...

Cf. A051731, A004736, A007436, A000203.

A051731 * A004736 as infinite lower triangular matrices.

A218828 Reluctant sequence of reverse reluctant sequence A004736.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 3, 2, 1, 2, 1, 3, 2, 1, 1, 2, 1, 3, 2, 1, 4, 1, 2, 1, 3, 2, 1, 4, 3, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2

Offset: 1

Boris Putievskiy, Dec 15 2012

Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A.
Sequence B is called a reverse reluctant sequence of sequence A, if B is triangle array read by rows: row number k lists first k elements of the sequence A in reverse order.
Sequence A004736 is the reverse reluctant sequence of sequence 1,2,3,... (A000027).

			The start of the sequence as triangle array T(n,k) is:
  1;
  1,2;
  1,2,1;
  1,2,1,3;
  1,2,1,3,2;
  1,2,1,3,2,1;
  ...

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

Cf. A004736, A002260, A220280.

t=int((math.sqrt(8*n-7) - 1)/ 2)
n1=n-t*(t+1)/2
t1=int((math.sqrt(8*n1-7) - 1)/ 2)
m=(t1*t1+3*t1+4)/2-n1

T(n,k) = A004736(k) for every n.

As a linear array, the sequence is a(n) = (t1^2+3*t1+4)/2-n1, where n1=n-t(t+1)/2, t1=floor[(-1+sqrt(8*n1-7))/2], t=floor[(-1+sqrt(8*n-7))/2].

A220465 Reverse reluctant sequence of reverse reluctant sequence A004736.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 4, 1, 2, 3, 1, 2, 1, 3, 4, 1, 2, 3, 1, 2, 1, 2, 3, 4, 1, 2, 3, 1, 2, 1, 1, 2, 3, 4, 1, 2, 3, 1, 2, 1, 5, 1, 2, 3, 4, 1, 2, 3, 1, 2, 1, 4, 5, 1, 2, 3, 4, 1, 2, 3, 1, 2, 1, 3, 4, 5, 1, 2, 3, 4, 1, 2, 3, 1, 2, 1, 2, 3, 4, 5, 1, 2, 3, 4, 1, 2, 3, 1, 2, 1, 1, 2, 3, 4, 5, 1, 2, 3, 4, 1, 2, 3, 1, 2, 1, 6, 1, 2, 3

Offset: 1

Boris Putievskiy, Dec 15 2012

Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A.
Sequence B is called a reverse reluctant sequence of sequence A, if B is triangle array read by rows: row number k lists first k elements of the sequence A in reverse order.
Sequence A004736 is the reverse reluctant sequence of sequence 1,2,3,... (A000027).

			The start of the sequence as triangle array T(n,k) is:
  1;
  2,1;
  1,2,1;
  3,1,2,1;
  2,3,1,2,1;
  1,2,3,1,2,1;
  ...

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

Cf. A000027, A004736, A002260, A220280.

t=int((math.sqrt(8*n-7) - 1)/ 2)
n1=(t*t+3*t+4)/2-n
t1=int((math.sqrt(8*n1-7) - 1)/ 2)
m=(t1*t1+3*t1+4)/2-n1

T(n,k) = A004736(n-k+1).

As a linear array, the sequence is a(n) = (t1*t1+3*t1+4)/2-n1, where n1=(t*t+3*t+4)/2-n, t1=floor[(-1+sqrt(8*n1-7))/2], t=floor[(-1+sqrt(8*n-7))/2].

A288533 Parse A004736 into distinct phrases [1], [2], [1,3], [2,1], [4], [3], [2,1,5], [4,3], [2,1,6], ...; a(n) is the length of the n-th phrase.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 3, 2, 3, 1, 3, 2, 1, 2, 2, 2, 1, 2, 4, 1, 1, 2, 3, 3, 2, 3, 5, 1, 3, 3, 3, 1, 1, 2, 2, 4, 3, 2, 3, 4, 4, 1, 3, 4, 4, 2, 1, 2, 2, 5, 5, 1, 2, 4, 3, 5, 1, 1, 2, 3, 4, 5, 2, 2, 3, 5, 5, 3, 1, 3, 3, 3, 4, 5, 1, 2, 2, 4, 5, 6, 1, 2, 4, 4, 6, 4, 1, 2, 3, 4, 4, 6, 2, 1, 2, 3, 3, 5, 5, 4, 1, 2, 3, 5, 6, 6, 1, 1, 2, 3, 4, 5, 7, 3, 2, 3, 4, 4, 7, 6, 1, 3, 3, 4, 5, 6, 5, 1, 2, 2

Offset: 1

Lewis Chen, Jun 11 2017

nonn

The phrases are formed by the Ziv-Lempel encoding described in A106182. - Neal Gersh Tolunsky, Nov 30 2023

			Consider the infinite sequence [1,2,1,3,2,1,4,3,2,1,5,4,3,2,1,...], i.e., A004736. We can first take [1] since we've never used it before. Then [2]. For the third term, we've already used [1], so we must instead take [1,3].

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Cf. A109337, A106182, A187180, A187181, A187182, A187183, A187184, A187185, A187186, A187187, A187188, A187199, A187200.

# you should use program from internal format
a = set()
i = 2
s = "1"
seq = ""
while i < 100:
    j = i
    while j > 0:
        if s not in a:
            seq = seq + "," + str(len(s)-len(s.replace(",",""))+1)
            a.add(s)
            s = str(j)
        else:
            s = s + "," + str(j)
        j -= 1
    i += 1
print(seq[1:])

A090988 a(n) = 2^A004736(n).

Original entry on oeis.org

2, 4, 2, 8, 4, 2, 16, 8, 4, 2, 32, 16, 8, 4, 2, 64, 32, 16, 8, 4, 2, 128, 64, 32, 16, 8, 4, 2, 256, 128, 64, 32, 16, 8, 4, 2, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 4096, 2048, 1024, 512, 256

Offset: 1

Alford Arnold, Feb 29 2004

			Triangle begins
   2
   4 2
   8 4 2
  16 8 4 2

Cf. A004736.

Flatten[Table[2^n,{i,15},{n,i,1,-1}]] (* Harvey P. Dale, Dec 27 2011 *)

a(n) = 2^A004736(n).

A127738 Triangle read by rows: the matrix product A004736 * A127701 of two triangular matrices.

Original entry on oeis.org

1, 3, 2, 5, 5, 3, 7, 8, 7, 4, 9, 11, 11, 9, 5, 11, 14, 15, 14, 11, 6, 13, 17, 19, 19, 17, 13, 7, 15, 20, 23, 24, 23, 20, 15, 8, 17, 23, 27, 29, 29, 27, 23, 17, 9, 19, 26, 31, 34, 35, 34, 31, 26, 19, 10

Offset: 1

Gary W. Adamson, Jan 27 2007

Left column = A028387: (1, 5, 11, 19, 29, 41, 55, ...).

			First few rows of the triangle:
   1;
   3,  2;
   5,  5,  3;
   7,  8,  7,  4;
   9, 11, 11,  9,  5;
  11, 14, 15, 14, 11,  6;
  13, 17, 19, 19, 17, 13,  7;
  ...

Cf. A004736, A127701, A008778 (row sums), A028387.

T(n,k) = Sum_{j=k..n} A004736(n,j)*A127701(j,k). - R. J. Mathar, Aug 31 2022

T(n,k) = k+(k+1)*(n-k) = n+k*(n-k) = n +A094053(n,k) = A059036(n,k). - R. J. Mathar, Aug 31 2022

A128177 A128174 * A004736 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 6, 4, 2, 1, 9, 6, 4, 2, 1, 12, 9, 6, 4, 2, 1, 16, 12, 9, 6, 4, 2, 1, 20, 16, 12, 9, 6, 4, 2, 1, 25, 20, 16, 12, 9, 6, 4, 2, 1, 30, 25, 20, 16, 12, 9, 6, 4, 2, 1, 36, 30, 25, 20, 16, 12, 9, 6, 4, 2, 1, 42, 36, 30, 25, 20, 16, 12, 9, 6, 4, 2, 1

Offset: 1

Gary W. Adamson, Feb 17 2007

n-th row has n nonzero terms of A002620: (1, 2, 4, 6, 9, 12, 16, ...) in reverse.

Row sums = A002623: (1, 3, 7, 13, 22, 34, 50, ...).

			First few rows of the triangle:
   1;
   2, 1;
   4, 2, 1;
   6, 4, 2, 1;
   9, 6, 4, 2, 1;
  12, 9, 6, 4, 2, 1;
  ...

Cf. A128174, A004736, A002623, A002620.

seq(seq(floor((n-k+2)^2/4), k=1..n), n=1..20); # Ridouane Oudra, Mar 23 2024

T[n_,k_]:=Floor[(n-k+2)^2/4];Table[T[n,k],{n,12},{k,n}]//Flatten (* James C. McMahon, Jan 05 2025 *)

lista(nn) = {t128174 = matrix(nn, nn, n, k, (k<=n)*(1+(-1)^(n-k))/2); t004736 = matrix(nn, nn, n, k, (k<=n)*(n - k + 1)); t128177 = t128174*t004736; for (n = 1, nn, for (k = 1, n, print1(t128177[n, k], ", ");););} \\ Michel Marcus, Feb 11 2014

From Ridouane Oudra, Mar 23 2024: (Start)
T(n, k) = A002620(n-k+2), with 1 <= k <= n;
T(n, k) = floor((n-k+2)^2/4);
T(n, k) = (1/2)*floor((n-k+2)^2/2);
T(n, k) = (1/8)*(2*(n-k+2)^2 + (-1)^(n-k) - 1). (End)

Partially edited and more terms from Michel Marcus, Feb 11 2014

A128256 A004736(signed) * A007318.

Original entry on oeis.org

1, -1, 1, 2, 0, 1, -2, 2, 1, 1, 3, 0, 3, 2, 1, -3, 3, 3, 5, 3, 1, 4, 0, 6, 8, 8, 4, 1, -4, 4, 6, 14, 16, 12, 5, 1, 5, 0, 10, 20, 30, 28, 17, 6, 1, -5, 5, 10, 30, 50, 58, 45, 23, 7, 1

Offset: 1

Gary W. Adamson, Feb 21 2007

			First few rows of the triangle are:
  1;
  -1, 1;
  2, 0, 1;
  -2, 2, 1, 1;
  3, 0, 3, 2, 1;
  -3, 3, 3, 5, 3, 1;
  4, 0, 6, 8, 8, 4, 1;
  -4, 4, 6, 14, 16, 12, 5, 1;
  5, 0, 10, 10, 30, 28, 17, 6, 1;
  ...

Cf. A004736, A007318, A053088 (row sums).

A004736(with alternate signs: (1; -2,1; 3,-2,1;...)) * A007318, Pascal's triangle.

cp's OEIS Frontend

A128226 Triangle, A004736 * A127899 (unsigned).

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A131782 Triangle read by rows: (A004736 * A000012) + (A000012 * A004736) - A000012 as infinite lower triangular matrices.

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A134545 A051731 * A004736.

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A218828 Reluctant sequence of reverse reluctant sequence A004736.

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A220465 Reverse reluctant sequence of reverse reluctant sequence A004736.

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A288533 Parse A004736 into distinct phrases [1], [2], [1,3], [2,1], [4], [3], [2,1,5], [4,3], [2,1,6], ...; a(n) is the length of the n-th phrase.

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A090988 a(n) = 2^A004736(n).

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A127738 Triangle read by rows: the matrix product A004736 * A127701 of two triangular matrices.

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A128177 A128174 * A004736 as infinite lower triangular matrices.

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