A233328 -id:A233328 - cp's OEIS frontend

A279212 Fill an array by antidiagonals upwards; in the top left cell enter a(0)=1; thereafter, in the n-th cell, enter the sum of the entries of those earlier cells that can be seen from that cell.

1, 1, 2, 2, 6, 11, 4, 15, 39, 72, 8, 37, 119, 293, 543, 16, 88, 330, 976, 2364, 4403, 32, 204, 870, 2944, 8373, 20072, 37527, 64, 464, 2209, 8334, 26683, 74150, 176609, 331072, 128, 1040, 5454, 22579, 79534, 246035, 673156, 1595909, 2997466, 256, 2304, 13176, 59185, 226106, 762221, 2303159, 6231191, 14721429, 27690124

Offset: 0

N. J. A. Sloane, Dec 24 2016

"That can be seen from" means "that are on the same row, column, diagonal, or antidiagonal as".
Inspired by A279967.
Conjecture: Every column has a finite number of odd entries, and every row and diagonal have an infinite number of odd entries. - Peter Kagey, Mar 28 2020. The conjecture about columns is true, see that attached pdf file from Alec Jones.
The "look" keyword refers to Peter Kagey's bitmap. - N. J. A. Sloane, Mar 29 2020
The number of sequences of queen moves from (1, 1) to (n, k) in the first quadrant moving only up, right, diagonally up-right, or diagonally up-left. - Peter Kagey, Apr 12 2020
Column 0 gives A011782. In the column 1, the only powers of 2 occur at positions A233328(k) with value a(k(k+1)/2 + 1), k >=1 (see A335903). Conjecture: Those are the only multiple occurrences of numbers greater than 1 in this sequence (checked through the first 2000 antidiagonals). - Hartmut F. W. Hoft, Jun 29 2020

			The array begins:
i/j|  0    1    2     3     4      5      6       7       8
-------------------------------------------------------------
0  |  1    2   11    72   543   4403  37527  331072 2997466 ...
1  |  1    6   39   293  2364  20072 176609 1595909 ...
2  |  2   15  119   976  8373  74150 673156 ...
3  |  4   37  330  2944 26683 246035 ...
4  |  8   88  870  8334 79534 ...
5  | 16  204 2209 22579 ...
6  | 32  464 5454 ...
7  | 64 1040 ...
8  |128 ...
  ...
For example, when we get to the antidiagonal that reads 4, 15, 39, ..., the reason for the 39 is that from that cell we can see one cell that has been filled in above it (containing 11), one cell to the northwest (2), two cells to the west (1, 6), and two to the southwest (4, 15), for a total of a(8) = 39.
The next pair of duplicates greater than 2 is 2^20 = 1048576 = a(154) = a(231), located in antidiagonals 17 = A233328(2) and 21, respectively. For additional duplicate numbers in this sequence see A335903.  - _Hartmut F. W. Hoft_, Jun 29 2020

Alois P. Heinz, Antidiagonals n = 0..200, flattened (first 20 antidiagonals from Alec Jones)
Alec Jones, Proof that columns have finitely many odd entries.
Peter Kagey, Bitmap showing parity of first 1024 rows and 512 columns. (Odd values are white; even values are black.)
Peter Kagey, Animated example illustrating the first fifteen terms.

Cf. A064642 is analogous if a cell can only "see" its immediate neighbors.
Cf. A035002, A059450, A132439, A279966, A279967, A279211.
See A280026, A280027 for similar sequences based on a spiral.
Cf. A011782, A335903.

s[0, 0] = 1; s[i_, j_] := s[i, j] = Sum[s[k, j], {k, 0, i-1}] + Sum[s[i, k], {k, 0, j-1}] + Sum[s[i+j-k, k], {k, 0, j-1}] + Sum[s[i-k-1, j-k-1], {k, 0, Min[i, j] - 1}]
aDiag[m_] := Map[s[m-#, #]&, Range[0, m]]
a279212[n_] := Flatten[Map[aDiag, Range[0, n]]]
a279212[9] (* data - 10 antidiagonals;  Hartmut F. W. Hoft, Jun 29 2020 *)

T(0, 0) = 1; T(i, j) = Sum_{k=0..i-1} T(k, j) + Sum_{k=0..j-1} T(i, k) + Sum_{k=0..j-1} T(i+j-k, k) + Sum_{k=0..min(i, j)-1} T(i-k-1, j-k-1), with recursion upwards along antidiagonals. - Hartmut F. W. Hoft, Jun 29 2020

A226308 a(n) = a(n-1) + a(n-2) + 2*a(n-3) with a(0)=2, a(1)=1, a(2)=5.

Original entry on oeis.org

2, 1, 5, 10, 17, 37, 74, 145, 293, 586, 1169, 2341, 4682, 9361, 18725, 37450, 74897, 149797, 299594, 599185, 1198373, 2396746, 4793489, 9586981, 19173962, 38347921, 76695845, 153391690, 306783377, 613566757, 1227133514, 2454267025, 4908534053, 9817068106, 19634136209

Offset: 0

N. J. A. Sloane, Jun 07 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. Cerda-Morales, A note on modified third-order jacobsthal numbers, arXiv:1905.00725 [math.CO], 2019.
Charles K. Cook and Michael R. Bacon, Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations, Annales Mathematicae et Informaticae, 41 (2013) pp. 27-39.
Evren Eyican Polatlı and Yüksel Soykan, On generalized third-order Jacobsthal numbers, Asian Res. J. of Math. (2021) Vol. 17, No. 2, 1-19, Article No. ARJOM.66022.
Index entries for linear recurrences with constant coefficients, signature (1,1,2).

Cf. A046636, A047853, A233328.

A226308 := n -> 1/7*(2^(n+3) + 6*cos(2*Pi*n/3) - 4*sqrt(3)*sin(2*Pi*n/3)):
seq(A226308(n), n = 0 .. 34); # Mélika Tebni, Mar 09 2024

CoefficientList[Series[-(2 x^2 - x + 2) / ((2 x - 1) (x^2 + x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 18 2013 *)
LinearRecurrence[{1,1,2},{2,1,5},40] (* Harvey P. Dale, Nov 03 2024 *)

a(n)=([0,1,0; 0,0,1; 2,1,1]^n*[2;1;5])[1,1] \\ Charles R Greathouse IV, Jul 19 2016

a0, a1, a2 = 2, 1, 5
for n in range(77):
  a = a2 + a1 + 2*a0
  print(a0, end=', ')
  a0, a1, a2 = a1, a2, a # Alex Ratushnyak, Jun 08 2013

G.f.: -(2*x^2-x+2) / ((2*x-1)*(x^2 + x + 1)). - Colin Barker, Jun 08 2013
a(3*n) = A047853(n+1), a(3*n+1) = A233328(n), a(3*n+2) = A046636(n+1). - Philippe Deléham, Feb 24 2014
From Mélika Tebni, Mar 09 2024: (Start)
E.g.f.: (1/7)*(8*exp(2*x) + exp(-x/2)*(6*cos(sqrt(3)*x/2) - 4*sqrt(3)*sin(sqrt(3)*x/2))) (Charles K. Cook and Michael R. Bacon, 2013).
a(n) = (1/7)*(2^(n+3) + 6*cos(2*Pi*n/3) - 4*sqrt(3)*sin(2*Pi*n/3)). (End)

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A238303 Triangle T(n,k), 0<=k<=n, read by rows given by T(n,0) = 1, T(n,k) = 2 if k>0.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Philippe Deléham, Feb 24 2014

Row sums are A005408(n).
Diagonals sums are A109613(n).
Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000012(n), A005408(n), A036563(n+2), A058481(n+1), A083584(n), A137410(n), A233325(n), A233326(n), A233328(n), A211866(n+1), A165402(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A151575(n), A000012(n), A040000(n), A005408(n), A033484(n), A048473(n), A020989(n), A057651(n), A061801(n), A238275(n), A238276(n), A138894(n), A090843(n), A199023(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively.
Sum_{k=0..n} T(n,k)^x = A000027(n+1), A005408(n), A016813(n), A017077(n) for x = 0, 1, 2, 3 respectively.
Sum_{k=0..n} k*T(n,k) = A002378(n).
Sum_{k=0..n} A000045(k)*T(n,k) = A019274(n+2).
Sum_{k=0..n} A000142(k)*T(n,k) = A066237(n+1).

			Triangle begins:
1;
1, 2;
1, 2, 2;
1, 2, 2, 2;
1, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
...

Antti Karttunen, Table of n, a(n) for n = 0..22154; the first 210 rows of triangle

Cf. Diagonals: A040000.
Cf. Columns: A000012, A007395.
First differences of A001614.

A238303(n) = (2-ispolygonal(n,3)); \\ Antti Karttunen, Jan 19 2025

T(n,0) = A000012(n) = 1, T(n+k,k) = A007395(n) = 2 for k>0.

Data section extended to a(104) by Antti Karttunen, Jan 19 2025

A335903 Column 1 in the matrix of A279212 (whose indexing starts at 0).

Original entry on oeis.org

2, 6, 15, 37, 88, 204, 464, 1040, 2304, 5056, 11008, 23808, 51200, 109568, 233472, 495616, 1048576, 2211840, 4653056, 9764864, 20447232, 42729472, 89128960, 185597952, 385875968, 801112064, 1660944384, 3439329280, 7113539584, 14696841216, 30333206528, 62545461248, 128849018880, 265214230528, 545460846592

Offset: 1

Hartmut F. W. Hoft, Jun 29 2020

Indexing for this sequence starts at 1 since then the index is the same as the number of the antidiagonal in the matrix for A279212 in which a number in column 1 of A279212 occurs.

			a(17) = a(A233328(2)) = 1048576 = 2^20 = T(16, 1) = T(21, 0) in terms of matrix T of A279212; 2^20 is in column 1 of the 17th antidiagonal and in column 0 of the 21st antidiagonal of the matrix of A279212.
A search for duplicates in A279212 through antidiagonal 2000 produced only pairs of powers of 2 in columns 0 and 1 of the matrix of A279212. Let k_0 and k_1 be the antidiagonals in columns 0 and 1, respectively, for the pair of the n-th duplicates. Since k_0 = 2 and k_1 = 1 for the duplicates of 2, the first pair in both columns, then k_0 = k_1 + 3*n - 2 for the n-th pair, n >=1.
Table of duplicates in column 1 of the matrix of A279212 (the values for k_0 are one larger than the exponents in the left column of the table below because column 0 is sequence A011782):
value of    number of        index in
number      antidiagonal     A279212
-------------------------------------
2^1               1               2
2^20             17             154
2^151           145           10586
2^1178         1169          683866
2^9373         9361        43818842
2^74912       74897      2804817754
2^599203     599185    179511631706
... ... ...
The central column of the table is A233328. The values for the first 4 antidiagonals were computed using sequence A279212, the ones larger than antidiagonal 2000 were determined by computing those n for which 7*n + 9 is a power of 2.
The right column is n*(n+1)/2 + 1, where n is the number in the central column.

Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A011782, A233328, A279212.

a335903[1] = 2; a335903[2] = 6; a335903[n_] := (7n+9)*2^(n-4)
Map[a335903, Range[35]]  (* data  *)

Vec(x*(1 - x)*(2 - x^2) / (1 - 2*x)^2 + O(x^30)) \\ Colin Barker, Jun 29 2020

a(1) = 2, a(2) = 6, a(3) = 15, a(n) = 2 * a(n-1) + 7 * 2^(n-4), for n >= 4 (recursion for column 1 in the matrix of A279212).
a(1) = 2, a(2) = 6, a(n) = (7*n + 9) * 2^(n - 4), for n >= 3.
From Colin Barker, Jun 29 2020: (Start)
G.f.: x*(1 - x)*(2 - x^2) / (1 - 2*x)^2.
a(n) = 4*a(n-1) - 4*a(n-2) for n > 4.
(End)

A238339 Square number array read by ascending antidiagonals: T(1,k) = 2k + 1, and T(n,k) = (2n^(k+1)-n-1)/(n-1) otherwise.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 13, 7, 1, 1, 9, 25, 29, 9, 1, 1, 11, 41, 79, 61, 11, 1, 1, 13, 61, 169, 241, 125, 13, 1, 1, 15, 85, 311, 681, 727, 253, 15, 1, 1, 17, 113, 517, 1561, 2729, 2185, 509, 17, 1, 1, 19, 145, 799, 3109, 7811, 10921, 6559, 1021, 19, 1

Offset: 0

Philippe Deléham, Feb 24 2014

			Square array begins:
1..1...1.....1......1.......1........1........1...
1..3...5.....7......9......11.......13.......15...
1..5..13....29.....61.....125......253......509...
1..7..25....79....241.....727.....2185.....6559...
1..9..41...169....681....2729....10921....43689...
1.11..61...311...1561....7811....39061...195311...
1.13..85...517...3109...18661...111973...671845...
1.15.113...799...5601...39215...274513..1921599...
1.17.145..1169...9361...74897...599185..4793489...
1.19.181..1639..14761..132859..1195741.10761679...
1.21.221..2221..22221..222221..2222221.22222221...

Cf. A238303.

T:= proc(n, k); if n=1 then 2*k+1 else (2*n^(k+1)-n-1)/(n-1) fi end:
seq(seq(T(n-k, k), k=0..n), n=0..10); # Georg Fischer, Oct 14 2023

T(0,k) = A000012(k) = 1;
T(1,k) = A005408(k) = 2k+1;
T(2,k) = A036563(k+2);
T(3,k) = A058481(k+1);
T(4,k) = A083584(k);
T(5,k) = A137410(k);
T(6,k) = A233325(k);
T(7,k) = A233326(k);
T(8,k) = A233328(k);
T(9,k) = A211866(k+1);
T(10,k) = A165402(k+1);
T(n,0) = A000012(n) = 1;
T(n,1) = A005408(n) = 2*n+1;
T(n,2) = A001844(n) = 2*n^2 + 2*n + 1.

Definition amended by Georg Fischer, Oct 14 2023

cp's OEIS Frontend

A279212 Fill an array by antidiagonals upwards; in the top left cell enter a(0)=1; thereafter, in the n-th cell, enter the sum of the entries of those earlier cells that can be seen from that cell.

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A226308 a(n) = a(n-1) + a(n-2) + 2*a(n-3) with a(0)=2, a(1)=1, a(2)=5.

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A238303 Triangle T(n,k), 0<=k<=n, read by rows given by T(n,0) = 1, T(n,k) = 2 if k>0.

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A335903 Column 1 in the matrix of A279212 (whose indexing starts at 0).

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A238339 Square number array read by ascending antidiagonals: T(1,k) = 2*k + 1, and T(n,k) = (2*n^(k+1)-n-1)/(n-1) otherwise.

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Extensions

A238339 Square number array read by ascending antidiagonals: T(1,k) = 2k + 1, and T(n,k) = (2n^(k+1)-n-1)/(n-1) otherwise.