A246706 -id:A246706 - cp's OEIS frontend

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

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}]

A128174 Transform, (1,0,1,...) in every column.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 1

Gary W. Adamson, Feb 17 2007

Inverse of the triangle = a tridiagonal matrix with (1,1,1,...) in the superdiagonal, (0,0,0,...) in the main diagonal and (-1,-1,-1,...) in the subdiagonal.
Riordan array (1/(1-x^2), x) with inverse (1-x^2,x). - Paul Barry, Sep 10 2008
The position of 1's in this sequence is equivalent to A246705, and the position of 0's is equivalent to A246706. - Bernard Schott, Jun 05 2019

			First few rows of the triangle are:
  1;
  0, 1;
  1, 0, 1;
  0, 1, 0, 1;
  1, 0, 1, 0, 1; ...

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A004526 (row sums).

a128174 n k = a128174_tabl !! (n-1) !! (k-1)
a128174_row n = a128174_tabl !! (n-1)
a128174_tabl = iterate (\xs@(x:_) -> (1 - x) : xs) [1]
-- Reinhard Zumkeller, Aug 01 2014

[[(1+(-1)^(n-k))/2: k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jun 05 2019

A128174 := proc(n,k)
    if k > n or k < 1 then
        0;
    else
        modp(k+n+1,2) ;
    end if;
end proc: # R. J. Mathar, Aug 06 2016

a128174[r_] := Table[If[EvenQ[n+k], 1, 0], {n, 1, r}, {k, 1, n}]
TableForm[a128174[5]] (* triangle *)
Flatten[a128174[10]] (* data *) (* Hartmut F. W. Hoft, Mar 15 2017 *)
Table[(1+(-1)^(n-k))/2, {n,1,12}, {k,1,n}]//Flatten (* G. C. Greubel, Sep 26 2017 *)

for(n=1,12, for(k=1,n, print1((1+(-1)^(n-k))/2, ", "))) \\ G. C. Greubel, Sep 26 2017

[[(1+(-1)^(n-k))/2 for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jun 05 2019

A lower triangular matrix transform, (1, 0, 1, ...) in every column; n terms of (1, 0, 1, ...) in odd rows; n terms of (0, 1, 0, ...) in even rows.
T(n,k) = [k<=n]*(1+(-1)^(n-k))/2. - Paul Barry, Sep 10 2008
With offset n=1, k=0: Sum_{k=0..n} {T(n,k)*x^k} = A000035(n), A004526(n+1), A000975(n), A033113(n), A033114(n), A033115(n), A033116(n), A033117(n), A033118(n), A033119(n), A056830(n+1) for x=0,1,2,3,4,5,6,7,8,9,10 respectively. - Philippe Deléham, Oct 17 2011
T(n+1,1) = 1 - T(n,1); T(n+1,k) = T(n,k-1), 1 < k <= n+1. - Reinhard Zumkeller, Aug 01 2014

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.

Original entry on oeis.org

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

A246695 Row sums of the triangular array A246694.

Original entry on oeis.org

1, 3, 9, 18, 35, 57, 91, 132, 189, 255, 341, 438, 559, 693, 855, 1032, 1241, 1467, 1729, 2010, 2331, 2673, 3059, 3468, 3925, 4407, 4941, 5502, 6119, 6765, 7471, 8208, 9009, 9843, 10745, 11682, 12691, 13737, 14859, 16020, 17261, 18543, 19909, 21318, 22815

Offset: 0

Clark Kimberling, Sep 01 2014

Also partial sums of A257083. - Reinhard Zumkeller, Apr 17 2015

			First 5 rows of A246694 preceded by sums
sum = 1: ...... 1
sum = 3: ...... 1 ... 2
sum = 9: ...... 3 ... 2 ... 4
sum = 18: ..... 3 ... 5 ... 4 ... 6
sum = 35: ..... 7 ... 5 ... 8 ... 6 ... 9

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A246694, A246705, A246706.

Cf. A257083, A377802.

a246695 n = a246695_list !! n
a246695_list = scanl1 (+) a257083_list
-- Reinhard Zumkeller, Apr 17 2015

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; A246695 = Table[Sum[t[n, k], {k, 0, n}], {n, 0, z}]

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) = 3, a(2) = 9, a(3) = 18, a(4) = 35, a(5) = 57, a(6) = 91.
Conjectured g.f.: (1 + x + 2*x^2 + x^3 + x^4)/((x - 1)^4*(x + 1)^2).
Conjecture: a(n) = (1/8)*(n + 1)*((-1)^n + 2*n^2 + 4*n + 7). - Eric Simon Jacob, Jul 19 2023 [This conjecture is correct; compare A377802, note offset 1. - Werner Schulte, Nov 22 2024]

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

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

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A128174 Transform, (1,0,1,...) in every column.

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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.

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A246695 Row sums of the triangular array A246694.

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