A246695 -id:A246695 - cp's OEIS frontend

A246694 Triangle read by rows: T(n,k) = T(n,k-2) + 1 if n > 1 and 2 <= k <= n; T(0,0) = 1, T(1,0) = 1, T(1,1) = 2; if n > 1 is odd, then T(n,0) = T(n-1,n-2) + 1 and T(n,1) = T(n-1,n-1) + 1; if n > 1 is even, then T(n,0) = T(n-1,n-1) + 1 and T(n,1) = T(n-1,n-2) + 1.

1, 1, 2, 3, 2, 4, 3, 5, 4, 6, 7, 5, 8, 6, 9, 7, 10, 8, 11, 9, 12, 13, 10, 14, 11, 15, 12, 16, 13, 17, 14, 18, 15, 19, 16, 20, 21, 17, 22, 18, 23, 19, 24, 20, 25, 21, 26, 22, 27, 23, 28, 24, 29, 25, 30, 31, 26, 32, 27, 33, 28, 34, 29, 35, 30, 36, 31, 37, 32

Offset: 0

Clark Kimberling, Sep 01 2014

As an array, for each m, row 2*m has m odd numbers and m+1 even numbers; row 2*m-1 has m odds and m evens. As a sequence, every positive integer n occurs exactly twice, separated by floor((n+1)/2) other numbers.

			First 8 rows:
  1
  1 ... 2
  3 ... 2 ... 4
  3 ... 5 ... 4 ... 6
  7 ... 5 ... 8 ... 6 ... 9
  7 .. 10 ... 8 .. 11 ... 9 .. 12
 13 .. 10 .. 14 .. 11 .. 15 .. 12 .. 16
 13 .. 17 .. 14 .. 18 .. 15 .. 19 .. 16 .. 20

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A246705, A246706, A246696.
Cf. A246695 (row sums), A174114 (central terms).
Cf. A002620 (main diagonal and first subdiagonal), A377802.

a246694 n k = a246694_tabl !! n !! k
a246694_row n = a246694_tabl !! n
a246694_tabl = [1] : [1,2] : f 1 2 [1,2] where
   f i z xs = ys : f j (z + 1) ys where
     ys = take (z + 1) $ map (+ 1) (xs !! (z - i) : xs !! (z - j) : ys)
     j = 3 - i
-- Reinhard Zumkeller, Sep 03 2014

z = 25; t[0, 0] = 1; t[1, 0] = 1; t[1, 1] = 2;
t[n_, 0] := If[OddQ[n], t[n - 1, n - 2] + 1, t[n - 1, n - 1] + 1];
t[n_, 1] := If[OddQ[n], t[n - 1, n - 1] + 1, t[n - 1, n - 2] + 1];
t[n_, k_] := t[n, k - 2] + 1; Flatten[Table[t[n, k], {n, 0, z}, {k, 0, n}]](*A246694*)

T(n, k) = 1 + floor(n/2) * (1+(-1)^k) / 2 + (floor(n/2))^2 + (2*k - 1 + (-1)^k) / 4 + (1-(-1)^n) * (1-(-1)^k) * n / 4. - Werner Schulte, Nov 16 2024
From Stefano Spezia, Nov 17 2024: (Start)
T(n, k) = (6 + (-1)^k + (-1)^(k+n) + 4*k + 2*n*(1 + (-1)^(k+n) + n))/8.
G.f.: (1 - x^3*y + x^7*y^3 + x^4*(1 - 2*y^2) - x^5*y*(1 - y^2))/((1 - x)^3*(1 + x)^2*(1 - x*y)^3*(1 + x*y)). (End)

Edited by M. F. Hasler, Nov 17 2014

A246697 Row sums of the triangular array A246696.

Original entry on oeis.org

1, 5, 16, 34, 67, 111, 178, 260, 373, 505, 676, 870, 1111, 1379, 1702, 2056, 2473, 2925, 3448, 4010, 4651, 5335, 6106, 6924, 7837, 8801, 9868, 10990, 12223, 13515, 14926, 16400, 18001, 19669, 21472, 23346, 25363, 27455, 29698, 32020, 34501, 37065, 39796

Offset: 0

Clark Kimberling, Sep 02 2014

			First 5 rows of A246694 preceded by sums
sum = 1: ...... 1;
sum = 5: ...... 2 ... 3;
sum = 16: ..... 5 ... 4 ... 7;
sum = 34: ..... 6 ... 9 ... 8 ... 11;
sum = 67: ..... 13 .. 10 .. 15 .. 12 .. 17.

Cf. A246694, A246695, A246696.

z = 25; t[0, 0] = 1; t[1, 0] = 2; t[1, 1] = 3; t[n_, 0] := t[n, 0] = If[OddQ[n], t[n - 1, n - 2] + 2, t[n - 1, n - 1] + 2]; t[n_, 1] := t[n, 1] = If[OddQ[n], t[n - 1, n - 1] + 2, t[n - 1, n - 2] + 2]
t[n_, k_] := t[n, k] = t[n, k - 2] + 2;
u = Flatten[Table[t[n, k], {n, 0, z}, {k, 0, n}]] (* A246696 *)
Table[Sum[t[n, k], {k, 0, n}], {n, 0, 2*z}] (* A246697 *)

Conjectured linear recurrence: a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6), with a(0) = 1, a(1) = 5, a(2) = 16, a(3) = 34, a(4) = 67, a(5) = 111, a(6) = 178.
Conjectured g.f.: (1 + 3*x + 5*x^2 + x^3 + 2*x^4)/((x - 1)^4*(x + 1)^2).
Conjecture: a(n) = (2*n^3+6*n^2+9*n+4+n*(-1)^n)/4. - Luce ETIENNE, Oct 16 2016
Conjectured e.g.f.: ((2 + 8*x + 6*x^2 + x^3)*cosh(x) + (2 + 9*x + 6*x^2 + x^3)*sinh(x))/2. - Stefano Spezia, May 10 2021

Corrected and edited by M. F. Hasler, Nov 17 2014

A257083 Partial sums of A257088.

Original entry on oeis.org

1, 2, 6, 9, 17, 22, 34, 41, 57, 66, 86, 97, 121, 134, 162, 177, 209, 226, 262, 281, 321, 342, 386, 409, 457, 482, 534, 561, 617, 646, 706, 737, 801, 834, 902, 937, 1009, 1046, 1122, 1161, 1241, 1282, 1366, 1409, 1497, 1542, 1634, 1681, 1777, 1826, 1926, 1977

Offset: 0

Reinhard Zumkeller, Apr 17 2015

Equivalently, numbers of the form m*(3*m+2)+1, where m = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Jan 05 2016

Also, numbers k such that 3*k-2 is a square. - Bruno Berselli, Jan 30 2018

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A246695 (partial sums), A257088.

Cf. A056109: numbers of the form m*(3*m+2)+1 for nonnegative m.

a257083 n = a257083_list !! n
a257083_list = scanl1 (+) a257088_list

[(6*n*(n+1) + (2*n+1)*(-1)^n + 7)/8 : n in [0..60]]; // Wesley Ivan Hurt, Oct 30 2022

Table[(6 n (n + 1) + (2 n + 1) (-1)^n + 7)/8, {n, 0, 60}] (* Bruno Berselli, Jan 05 2016 *)

vector(60, n, n--; (6*n*(n+1)+(2*n+1)*(-1)^n+7)/8) \\ Bruno Berselli, Jan 05 2016

From Bruno Berselli, Jan 05 2016: (Start)
G.f.: (1 + x + 2*x^2 + x^3 + x^4)/((1 + x)^2*(1 - x)^3).
a(n) = (6*n*(n+1) + (2*n+1)*(-1)^n + 7)/8. (End)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - Wesley Ivan Hurt, Oct 30 2022

A246705 Position of first n in A246694 (read as sequence with offset changed to 1); complement of A246706.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 11, 13, 15, 17, 19, 21, 22, 24, 26, 28, 30, 32, 34, 36, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 106, 108, 110, 112, 114, 116, 118

Offset: 1

Clark Kimberling, Sep 02 2014

			A246694 begins with 1, 1, 2, 3, 2, 4, 3, 5, 4, 6, 7, 5, 8, 6, 9, 7, so that, for a(n) = position of first n and b(n) = position of last n, we have a = (1,3,4,6,8,10,...) and b = (2,5,7,9,12,...).
Note, however, that A246694 has offset 0, so the values here are off by 1 from the positions as given by the indices (b-file or list) in that sequence. - _M. F. Hasler_, Nov 17 2014

Cf. A246694, A246695, A246706.

z = 25; t[0, 0] = 1; t[1, 0] = 1; t[1, 1] = 2;
t[n_, 0] := If[OddQ[n], t[n - 1, n - 2] + 1, t[n - 1, n - 1] + 1];
t[n_, 1] := If[OddQ[n], t[n - 1, n - 1] + 1, t[n - 1, n - 2] + 1];
t[n_, k_] := t[n, k - 2] + 1;
u = Flatten[Table[t[n, k], {n, 0, z}, {k, 0, n}]]; (*A246694*)
w[n_] := Flatten[Position[u, n]]; A246705 = Table[First[w[n]], {n, 1, 3*z}]

A246706 Position of last n in A246694 (read as a sequence, with offset changed to 1); complement of A246705.

Original entry on oeis.org

2, 5, 7, 9, 12, 14, 16, 18, 20, 23, 25, 27, 29, 31, 33, 35, 38, 40, 42, 44, 46, 48, 50, 52, 54, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131

Offset: 1

Clark Kimberling, Sep 02 2014

Note that A246694 has offset 0, so the values here are off by 1 from the positions as given by the indices (b-file or list) in that sequence. - M. F. Hasler, Nov 17 2014

			A246694 begins with 1, 1, 2, 3, 2, 4, 3, 5, 4, 6, 7, 5, 8, 6, 9, 7, so that, for a(n) = position of first n and b(n) = position of last n, we have a = (1,3,4,6,8,10,...) and b = (2,5,7,9,12,...).

Cf. A246694, A246695, A246705.

z = 25; t[0, 0] = 1; t[1, 0] = 1; t[1, 1] = 2;
t[n_, 0] := If[OddQ[n], t[n - 1, n - 2] + 1, t[n - 1, n - 1] + 1];
t[n_, 1] := If[OddQ[n], t[n - 1, n - 1] + 1, t[n - 1, n - 2] + 1];
t[n_, k_] := t[n, k - 2] + 1;
u = Flatten[Table[t[n, k], {n, 0, z}, {k, 0, n}]]; (*A246694*)
w[n_] := Flatten[Position[u, n]]; A246706 = Table[Last[w[n]], {n, 1, 3*z}]

A257088 a(2n) = 4n if n>0, a(2n + 1) = 2n + 1, a(0) = 1.

Original entry on oeis.org

1, 1, 4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17, 36, 19, 40, 21, 44, 23, 48, 25, 52, 27, 56, 29, 60, 31, 64, 33, 68, 35, 72, 37, 76, 39, 80, 41, 84, 43, 88, 45, 92, 47, 96, 49, 100, 51, 104, 53, 108, 55, 112, 57, 116, 59, 120, 61, 124, 63, 128

Offset: 0

Michael Somos, Apr 16 2015

			G.f. = 1 + x + 4*x^2 + 3*x^3 + 8*x^4 + 5*x^5 + 12*x^6 + 7*x^7 + 16*x^8 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A001082, A005408, A007310, A008574, A040001, A215947.

CF. A257083 (partial sums), A246695.

import Data.List (transpose)
a257088 n = a257088_list !! n
a257088_list = concat $ transpose [a008574_list, a005408_list]
-- Reinhard Zumkeller, Apr 17 2015

a[ n_] := Which[ n < 1, Boole[n == 0], OddQ[n], n, True, 2 n];
a[ n_] := SeriesCoefficient[ (1 + x + 2*x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4), {x, 0, n}];

PARI
```
{a(n) = if( n<1, n==0, n%2, n, 2*n)};
```

{a(n) = if( n<0, 0, polcoeff( (1 + x + 2*x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4) + x * O(x^n), n))};

Euler transform of length 4 sequence [ 1, 3, -1, -1].
a(n) is multiplicative with a(2^e) = 2^(e+1) if e>0, otherwise a(p^e) = p^e.
G.f.: (1 + x + 2*x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4).
G.f.: (1 - x^3) * (1 - x^4) / ((1 - x) * (1 - x^2)^3).
MOBIUS transform of A215947 is [1, 4, 3, 8, 5, ...].
a(n) = n * A040001(n) if n>0.
a(n) + a(n-1) = A007310(n) if n>0.
a(n) = A001082(n+1) - A001082(n) if n>0.
Binomial transform with a(0)=0 is A128543 if n>0.
a(2*n) = A008574(n). a(2*n + 1) = A005408(n).
a(n) = A022998(n) if n>0. - R. J. Mathar, Apr 19 2015
From Amiram Eldar, Jan 28 2025: (Start)
Dirichlet g.f.: (1+2^(1-s)) * zeta(s-1).
Sum_{k=1..n} a(k) ~ (3/4) * n^2. (End)
a(n) = gcd(n^n, 2*n). - Mia Boudreau, Jun 27 2025

cp's OEIS Frontend

A246694 Triangle read by rows: T(n,k) = T(n,k-2) + 1 if n > 1 and 2 <= k <= n; T(0,0) = 1, T(1,0) = 1, T(1,1) = 2; if n > 1 is odd, then T(n,0) = T(n-1,n-2) + 1 and T(n,1) = T(n-1,n-1) + 1; if n > 1 is even, then T(n,0) = T(n-1,n-1) + 1 and T(n,1) = T(n-1,n-2) + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

Extensions

A246697 Row sums of the triangular array A246696.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A257083 Partial sums of A257088.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Formula

A246705 Position of first n in A246694 (read as sequence with offset changed to 1); complement of A246706.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A246706 Position of last n in A246694 (read as a sequence, with offset changed to 1); complement of A246705.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A257088 a(2*n) = 4*n if n>0, a(2*n + 1) = 2*n + 1, a(0) = 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

A257088 a(2n) = 4n if n>0, a(2n + 1) = 2n + 1, a(0) = 1.