A084639 -id:A084639 - cp's OEIS frontend

A191663 Dispersion of A042948 (numbers >3, congruent to 0 or 1 mod 4), by antidiagonals.

1, 4, 2, 9, 5, 3, 20, 12, 8, 6, 41, 25, 17, 13, 7, 84, 52, 36, 28, 16, 10, 169, 105, 73, 57, 33, 21, 11, 340, 212, 148, 116, 68, 44, 24, 14, 681, 425, 297, 233, 137, 89, 49, 29, 15, 1364, 852, 596, 468, 276, 180, 100, 60, 32, 18, 2729, 1705, 1193, 937, 553

Offset: 1

Clark Kimberling, Jun 11 2011

Row 1: A084639.
For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion.  Each complement (beginning with its first term >1) also generates a dispersion.  The six sequences and dispersions are listed here:
...
A191663=dispersion of A042948 (0 or 1 mod 4 and >1)
A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)
A191664=dispersion of A014601 (0 or 3 mod 4 and >1)
A191665=dispersion of A042963 (1 or 2 mod 4 and >1)
A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)
A191666=dispersion of A042964 (2 or 3 mod 4)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191663 has 1st col A042964, all else A042948
A054582 has 1st col A005408, all else A005843
A191664 has 1st col A042963, all else A014601
A191665 has 1st col A014601, all else A042963
A191448 has 1st col A005843, all else A005408
A191666 has 1st col A042948, all else A042964
...
There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
   a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
   a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.

			Northwest corner:
1...4...9....20...41
2...5...12...25...52
3...8...17...36...73
6...13..28...57...116
7...16..33...68...137

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A042948, A042964, A191426.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a = 4; b = 5; m[n_] := If[Mod[n, 2] == 0, 1, 0];
f[n_] := a*m[n + 1] + b*m[n] + 4*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}]  (* A042948: (4+4k,5+4k) *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191663 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191663 *)

A265747 Numbers written in Jacobsthal greedy base.

Original entry on oeis.org

0, 1, 2, 10, 11, 100, 101, 102, 110, 111, 200, 1000, 1001, 1002, 1010, 1011, 1100, 1101, 1102, 1110, 1111, 10000, 10001, 10002, 10010, 10011, 10100, 10101, 10102, 10110, 10111, 10200, 11000, 11001, 11002, 11010, 11011, 11100, 11101, 11102, 11110, 11111, 20000, 100000, 100001, 100002, 100010, 100011, 100100

Offset: 0

Antti Karttunen, Dec 17 2015

These are called "Jacobsthal Representation Numbers" in Horadam's 1996 paper.
Sum_{i=0..} digit(i)*A001045(2+digit(i)) recovers n from such representation a(n), where digit(0) stands for the least significant digit (at the right), and A001045(k) gives the k-th Jacobsthal number.
No larger digits than 2 will occur, which allows representing the same sequence in a more compact form by base-3 coding in A265746.
Sequence A197911 gives the terms with no digit "2" in their representation, while its complement A003158 gives the terms where "2" occurs at least once.
Numbers beginning with digit "2" in this representation are given by A020988(n) [= 2*A002450(n) = 2*A001045(2n)].

			For n=7, when selecting the terms of A001045 with the greedy algorithm, we need terms A001045(4) + A001045(2) + A001045(2) = 5 + 1 + 1, thus a(7) = "102".
For n=10, we need A001045(4) + A001045(4) = 5+5, thus a(10) = "200".

Antti Karttunen, Table of n, a(n) for n = 0..10923
A. F. Horadam, Jacobsthal Representation Numbers, Fib Quart. 34 (1996), 40-54. (See especially page 50, which is page 11 in PDF.)

Cf. A001045, A002450, A020988, A007089, A197911, A003158.
Cf. A265745 (sum of digits).
Cf. A265746 (same numbers interpreted in base-3, then shown in decimal).
Cf. A084639 (positions of repunits).
Cf. A007961, A014417, A014418, A244159 for analogous sequences.

jacob[n_] := (2^n - (-1)^n)/3; maxInd[n_] := Floor[Log2[3*n + 1]]; A265747[n_] := A265747[n] = 10^(maxInd[n] - 2) + A265747[n - jacob[maxInd[n]]]; A265747[0] = 0; Array[A265747, 100, 0] (* Amiram Eldar, Jul 21 2023 *)

A130249(n) = floor(log(3*n + 1) / log(2));
A001045(n) = (2^n - (-1)^n) / 3;
A265747(n) = {if(n==0, 0, my(d=n - A001045(A130249(n))); 10^(A130249(n)-2) + if(d == 0, 0, A265747(d)));} \\ Amiram Eldar, Jul 21 2023

def greedyJ(n): m = (3*n+1).bit_length() - 1; return (m, (2**m-(-1)**m)//3)
def a(n):
    if n == 0: return 0
    place, value = greedyJ(n)
    return 10**(place-2) + a(n - value)
print([a(n) for n in range(49)]) # Michael S. Branicky, Jul 11 2021

a(0) = 0; for n >= 1, a(n) = 10^(A130249(n)-2) + a(n-A001045(A130249(n))).

a(n) = A007089(A265746(n)).

A026646 a(n) = Sum_{i=0..n} Sum_{j=0..n} A026637(i,j).

Original entry on oeis.org

1, 3, 7, 17, 37, 79, 163, 333, 673, 1355, 2719, 5449, 10909, 21831, 43675, 87365, 174745, 349507, 699031, 1398081, 2796181, 5592383, 11184787, 22369597, 44739217, 89478459, 178956943, 357913913, 715827853, 1431655735

Offset: 0

Clark Kimberling

nonn

a(n) indexes the corner blocks on the Jacobsthal spiral built from blocks of unit area (using J(1) and J(2) as the sides of the first block). - Paul Barry, Mar 06 2008

Partial sums of A026644, which are the row sums of A026637. - Paul Barry, Mar 06 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Cf. A000126, A001045, A026637, A026644, A084639, A109613.
Cf. A026637, A026638, A026639, A026640, A026641, A026642, A026643.
Cf. A026644, A026966, A026967, A026968, A026969, A026970.

[(2^(n+4) -(6*n+9+(-1)^n))/6: n in [0..40]]; // G. C. Greubel, Jul 01 2024

CoefficientList[Series[(1-x^2+2x^3)/((1-x)(1-2x)(1-x^2)), {x, 0, 29}], x] (* Metin Sariyar, Sep 22 2019 *)
LinearRecurrence[{3,-1,-3,2}, {1,3,7,17}, 41] (* G. C. Greubel, Jul 01 2024 *)

[(2^(n+4) - (-1)^n -9 - 6*n)/6 for n in range(41)] # G. C. Greubel, Jul 01 2024

G.f.: (1 -x^2 +2*x^3)/((1-x)*(1-2*x)*(1-x^2)). - Ralf Stephan, Apr 30 2004
From Paul Barry, Mar 06 2008: (Start)
a(n) = A001045(n+3) - 2*floor((n+2)/2).
a(n) = -n + Sum_{k=0..n} A001045(k+2) = A084639(n+1) - n. (End)
a(n+1) = 2*a(n) + A109613(n), a(0)=1. - Paul Curtz, Sep 22 2019

A227190 a(n) = n minus (product of run lengths in binary representation of n).

Original entry on oeis.org

0, 1, 1, 2, 4, 4, 4, 5, 7, 9, 9, 8, 11, 11, 11, 12, 14, 16, 15, 18, 20, 20, 20, 18, 21, 24, 23, 22, 26, 26, 26, 27, 29, 31, 29, 32, 35, 34, 33, 37, 39, 41, 41, 40, 43, 43, 43, 40, 43, 46, 43, 48, 51, 50, 49, 47, 51, 55, 53, 52, 57, 57, 57, 58, 60, 62, 59, 62

Offset: 1

Antti Karttunen, Jul 04 2013

			For 8, "1000" in binary, the run lengths are 1 and 3, 1*3=3, and 8-3 = 5, thus a(8)=5. For 24, "11000" in binary, the run lengths are 2 and 3, 2*3=6, and 24-6 = 18, thus a(24)=18.

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A227191, A000975, A084639.

a227190 n = n - a167489 n  -- Reinhard Zumkeller, Jul 05 2013

Table[n-Times@@(Length/@Split[IntegerDigits[n,2]]),{n,70}] (* Harvey P. Dale, Aug 02 2013 *)

(define (A227190 n) (- n (A167489 n))) ;; The Scheme-program for A167489 is found under that entry.

a(n) = n - A167489(n).

A101622 A Horadam-Jacobsthal sequence.

Original entry on oeis.org

0, 1, 6, 13, 30, 61, 126, 253, 510, 1021, 2046, 4093, 8190, 16381, 32766, 65533, 131070, 262141, 524286, 1048573, 2097150, 4194301, 8388606, 16777213, 33554430, 67108861, 134217726, 268435453, 536870910, 1073741821, 2147483646, 4294967293, 8589934590

Offset: 0

Paul Barry, Dec 10 2004

Companion sequence to A084639.
This is the sequence A(0,1;1,2;5) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010
Except for the initial three terms, the decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 961", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Mar 27 2017
Named after the Australian mathematician Alwyn Francis Horadam (1923-2016) and the German mathematician Ernst Jacobsthal (1882-1965). - Amiram Eldar, Jun 10 2021

Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. F. Horadam, Jacobsthal Representation Numbers, Fib Quart., Vol. 34, No. 1 (1996), pp. 40-54.
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [_Wolfdieter Lang_, Oct 18 2010]
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
Stephen Wolfram, A New Kind of Science.
Wolfram Research, Wolfram Atlas of Simple Programs.
Index entries for sequences related to cellular automata.
Index to 2D 5-Neighbor Cellular Automata.
Index to Elementary Cellular Automata.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A131953.

[(2^(n+2)+(-1)^n-5)/2: n in [0..35]]; // Vincenzo Librandi, Aug 12 2011

LinearRecurrence[{2,1,-2},{0,1,6},40] (* Harvey P. Dale, Jul 08 2014 *)

concat(0, Vec(x*(1+4*x)/((1-x)*(1+x)*(1-2*x)) + O(x^30))) \\ Colin Barker, Mar 28 2017

a(n) = (2^(n+2) + (-1)^n - 5)/2.
G.f.: x*(1+4*x)/((1-x)*(1+x)*(1-2*x)).
a(n) = (A014551(n+2)-5)/2.
(1, 6, 13, 30, 61, ...) are the row sums of A131953. - Gary W. Adamson, Jul 31 2007
From Paul Curtz, Jan 01 2009: (Start)
a(n) = a(n-1) + 2*a(n-2) + 5.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
a(n) = A000079(n+1) - A010693(n).
a(n+1) = A141722(n) + 5 = A141722(n) + A010716(n).
a(2n+1) - a(2n) = 1, 7, 31, ... = A083420.
a(2n+1) - 2*a(2n) = 1.
a(2n) = A002446 = 6*A002450, a(2n+1) = A141725. (End)
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. - Colin Barker, Mar 28 2017
a(n) = (1/2) * Sum_{k=1..n} binomial(n+1,k) * (2+(-1)^k). - Wesley Ivan Hurt, Sep 23 2017

A137241 Number triples (k,3-k,2-2k), concatenated for k=0, 1, 2, 3,...

Original entry on oeis.org

0, 3, 2, 1, 2, 0, 2, 1, -2, 3, 0, -4, 4, -1, -6, 5, -2, -8, 6, -3, -10, 7, -4, -12, 8, -5, -14, 9, -6, -16, 10, -7, -18, 11, -8, -20, 12, -9, -22, 13, -10, -24, 14, -11, -26, 15, -12, -28, 16, -13, -30, 17, -14, -32, 18, -15, -34, 19, -16, -36, 20, -17, -38, 21, -18, -40

Offset: 0

Paul Curtz, Mar 09 2008

The entries are the coefficients in a family of Jacobsthal recurrences: a(n)=k*a(n-1)+(3-k)*a(n-2)+(2-2k)*a(n-3).
Examples for k=0 are in A001045 and A113954. Examples for k=1 are A001045, A078008.
Examples for k=2 are A000975, A087288, A084639, A000012 and A001045.
Examples for k=3 are A045883, A059570. Examples for k=4 are A094705 and A015518.

			The triples (k,3-k,2-2k) are (0,3,2), (1,2,0), (2,1,-2), (3,0,-4),...

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

CoefficientList[Series[x*(3 + 2*x + x^2 - 4*x^3 - 4*x^4)/((x - 1)^2*(1 + x + x^2)^2), {x, 0, 50}], x] (* G. C. Greubel, Sep 28 2017 *)
Table[{n,3-n,2-2n},{n,0,30}]//Flatten (* or *) LinearRecurrence[ {0,0,2,0,0,-1},{0,3,2,1,2,0},100] (* Harvey P. Dale, Jun 23 2019 *)

x='x+O('x^50); Vec(x*(3+2*x+x^2-4*x^3-4*x^4)/((x-1)^2*(1+x +x^2 )^2)) \\ G. C. Greubel, Sep 28 2017

From R. J. Mathar, Feb 25 2009: (Start)
a(n) = 2*a(n-3) - a(n-6).
G.f.: x*(3+2*x+x^2-4*x^3-4*x^4)/((x-1)^2*(1+x+x^2)^2). (End)

Edited by R. J. Mathar, Jun 28 2008

A084640 Generalized Jacobsthal numbers.

Original entry on oeis.org

0, 1, 5, 11, 25, 51, 105, 211, 425, 851, 1705, 3411, 6825, 13651, 27305, 54611, 109225, 218451, 436905, 873811, 1747625, 3495251, 6990505, 13981011, 27962025, 55924051, 111848105, 223696211, 447392425, 894784851, 1789569705, 3579139411

Offset: 0

Paul Barry, Jun 06 2003

This is the sequence A(0,1;1,2;4) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [From _Wolfdieter Lang_, Oct 18 2010]
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A000975, A000225, A084639.

a084640 n = a084640_list !! n
a084640_list = 0 : 1 : (map (+ 4) $
   zipWith (+) (map (* 2) a084640_list) (tail a084640_list))
-- Reinhard Zumkeller, May 23 2013

[5*2^n/3+(-1)^n/3-2: n in [0..35]]; // Vincenzo Librandi, Jun 15 2011

LinearRecurrence[{2,1,-2},{0,1,5},40] (* Harvey P. Dale, Oct 27 2015 *)

x='x+O('x^50); Vec(x*(1+3*x)/((1-x^2)*(1-2*x))) \\ G. C. Greubel, Sep 26 2017

G.f.: x*(1+3*x)/((1-x^2)*(1-2*x)).
a(n) = a(n-1) + 2a(n-2) + 4, a(0)=0, a(1)=1.
a(n) = (5*2^n + (-1)^n - 6)/3.
a(n) = A001045(n+2) + 4*A000975(n-3).
a(n+1) - 2*a(n) = period 2: repeat 1, 3. - Paul Curtz, Apr 03 2008
Contribution from Paul Curtz, Dec 10 2009: (Start)
a(n+2) - a(n) = A020714(n).
Le the array D(n,k) of the first differences be defined via D(0,k) = a(k); D(n+1,k) = D(n,k+1)-D(n,k).
Then D(n,n) = 4*A131577(n); D(1,k) = A084214(k+1); D(2,k) = A115102(k-1) for k>0; D(3,k) = (-1)^(k+1)*A083581(k). (End)
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), a(0)=0, a(1)=1, a(2)=5. Observed by G. Detlefs. See the W. Lang link. - Wolfdieter Lang, Oct 18 2010

A173261 Array T(n,k) read by antidiagonals: T(n,2k)=1, T(n,2k+1)=n, n>=2, k>=0.

Original entry on oeis.org

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

Offset: 2

Paul Curtz, Feb 14 2010

One may define another array B(n,0) = -1, B(n,k) = T(n,k-1) + 2*B(n,k-1), n>=2, which also starts in columns k>=0, as follows:
  -1, -1, 0, 1,  4,  9, 20,  41,  84, 169,  340,  681, 1364 ...: A084639;
  -1, -1, 1, 3,  9, 19, 41,  83, 169, 339,  681, 1363, 2729;
  -1, -1, 2, 5, 14, 29, 62, 125, 254, 509, 1022, 2045, 4094;
  -1, -1, 3, 7, 19, 39, 83, 167, 339, 679, 1363, 2727, 5459 ...: -A173114;
B(n,k) = (n-1)*A001045(k) - T(n,k).
First differences are B(n,k+1) - B(n,k) = (n-1)*A001045(k).

			The array T(n,k) starts in row n=2 with columns k>=0 as:
  1,  2, 1,  2, 1,  2, 1,  2, 1,  2, 1,  2 ... A000034;
  1,  3, 1,  3, 1,  3, 1,  3, 1,  3, 1,  3 ... A010684;
  1,  4, 1,  4, 1,  4, 1,  4, 1,  4, 1,  4 ... A010685;
  1,  5, 1,  5, 1,  5, 1,  5, 1,  5, 1,  5 ... A010686;
  1,  6, 1,  6, 1,  6, 1,  6, 1,  6, 1,  6 ... A010687;
  1,  7, 1,  7, 1,  7, 1,  7, 1,  7, 1,  7 ... A010688;
  1,  8, 1,  8, 1,  8, 1,  8, 1,  8, 1,  8 ... A010689;
  1,  9, 1,  9, 1,  9, 1,  9, 1,  9, 1,  9 ... A010690;
  1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10 ... A010691.
Antidiagonal triangle begins as:
  1;
  1,  2;
  1,  3,  1;
  1,  4,  1,  2;
  1,  5,  1,  3,  1;
  1,  6,  1,  4,  1,  2;
  1,  7,  1,  5,  1,  3,  1;
  1,  8,  1,  6,  1,  4,  1,  2;
  1,  9,  1,  7,  1,  5,  1,  3,  1;
  1, 10,  1,  8,  1,  6,  1,  4,  1,  2;
  1, 11,  1,  9,  1,  7,  1,  5,  1,  3,  1;
  1, 12,  1, 10,  1,  8,  1,  6,  1,  4,  1,  2;
  1, 13,  1, 11,  1,  9,  1,  7,  1,  5,  1,  3,  1;
  1, 14,  1, 12,  1, 10,  1,  8,  1,  6,  1,  4,  1,  2;

G. C. Greubel, Antidiagonals n = 0..50 of the array, flattened

Cf. A000034, A010684, A010685, A010686, A010687, A010688, A010689, A010690, A010691.

Cf. A001045, A024206, A084639, A093178, A173114.

T[n_, k_]:= (1/2)*((n+3) - (n+1)*(-1)^k);
Table[T[n-k, k], {n,2,17}, {k,2,n}]//Flatten (* G. C. Greubel, Dec 03 2021 *)

flatten([[(1/2)*((n-k+3) - (n-k+1)*(-1)^k) for k in (2..n)] for n in (2..17)]) # G. C. Greubel, Dec 03 2021

From G. C. Greubel, Dec 03 2021: (Start)
T(n, k) = (1/2)*((n+3) - (n+1)*(-1)^k).
Sum_{k=0..n} T(n-k, k) = A024206(n).
Sum_{k=0..floor((n+2)/2)} T(n-2*k+2, k) = (1/16)*(2*n^2 4*n -5*(1 +(-1)^n) + 4*sin(n*Pi/2)) (diagonal sums).
T(2*n-2, n) = A093178(n). (End)

A364378 Numbers whose representation in Jacobsthal greedy base (A265747) is palindromic.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 20, 22, 27, 36, 41, 44, 60, 68, 84, 86, 97, 112, 123, 132, 143, 158, 169, 172, 204, 220, 252, 260, 292, 308, 340, 342, 363, 396, 417, 432, 453, 486, 507, 516, 537, 570, 591, 606, 627, 660, 681, 684, 748, 780, 844, 860, 924, 956, 1020, 1028

Offset: 1

Amiram Eldar, Jul 21 2023

A128209(n) = A001045(n) + 1 is a term for n >= 3, since its representation is two 1's with n-3 0's between them.
A084639(n) is a term for n >= 1 since its representation is n 1's.
A014825(n) is a term for n >= 1 since its representation is n-1 0's interleaved with n 1's.

			The first 10 terms are:
   n  a(n)  A265747(a(n))
  --  ----  -------------
   1     0              0
   2     1              1
   3     2              2
   4     4             11
   5     6            101
   6     9            111
   7    12           1001
   8    20           1111
   9    22          10001
  10    27          10101

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001045, A014825, A084639, A128209, A265747.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341, A364214.

palJacobQ[n_] := PalindromeQ[A265747[n]]; Select[Range[0, 1000], palJacobQ] (* using A265747[n] *)

is(n) = {my(dig = digits(A265747(n))); dig == Vecrev(dig);} \\ using A265747(n)

A116588 Array read by antidiagonals: T(n,k) = max(2^(n - k), 2^(k - n)).

Original entry on oeis.org

1, 2, 2, 4, 1, 4, 8, 2, 2, 8, 16, 4, 1, 4, 16, 32, 8, 2, 2, 8, 32, 64, 16, 4, 1, 4, 16, 64, 128, 32, 8, 2, 2, 8, 32, 128, 256, 64, 16, 4, 1, 4, 16, 64, 256, 512, 128, 32, 8, 2, 2, 8, 32, 128, 512, 1024, 256, 64, 16, 4, 1, 4, 16, 64, 256, 1024, 2048, 512, 128

Offset: 0

Roger L. Bagula, Mar 27 2006

This array is an infinite symmetric Toeplitz matrix whose first row is the powers of two A000079. - Franck Maminirina Ramaharo, Sep 08 2018

			Array begins:
    1   2   4   8  16  32  64  128 ...
    2   1   2   4   8  16  32   64 ...
    4   2   1   2   4   8  16   32 ...
    8   4   2   1   2   4   8   16 ...
   16   8   4   2   1   2   4    8 ...
   32  16   8   4   2   1   2    4 ...
   64  32  16   8   4   2   1    2 ...
  128  64  32  16   8   4   2    1 ...
  ... reformatted and extended. - _Franck Maminirina Ramaharo_, Sep 08 2018

M. Rosenblum and J. Rovnyak, Hardy Classes and Operator Theory, Oxford University Press, New York, 1985, p. 62.

A. Böttcher and S. M. Grudsky, Toeplitz Matrices, Asymptotic Linear Algebra and Functional Analysis, Texts and Readings in Mathematics Vol. 67, 2000.
Wikipedia, Toeplitz matrix.

Antidiagonal sums: A084639.

Cf. A082693, A130321.

row[n_] := Table[Max[2^(r - q), 2^(q - r)], {r, 1, n}, {q, 1, n}];
TableForm[row[10]] (* Franck Maminirina Ramaharo, Sep 08 2018 *)

T(n, k) := max(2^(n - k), 2^(k - n))$
for n:0 thru 10 do (print(makelist(T(n,k), k, 0, 10))); /* Franck Maminirina Ramaharo, Sep 08 2018 */

From Franck Maminirina Ramaharo, Sep 08 2018: (Start)
T(n,k) = A130321(n,k) for 0 <= k <= n and A130321(k,n) otherwise.
G.f.: (1 - 4*x*y)/((1 - 2*x)*(1 - 2*y)*(1 - x*y)). (End)

Edited, new name and extended by Franck Maminirina Ramaharo, Sep 08 2018

cp's OEIS Frontend

A191663 Dispersion of A042948 (numbers >3, congruent to 0 or 1 mod 4), by antidiagonals.

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A265747 Numbers written in Jacobsthal greedy base.

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A026646 a(n) = Sum_{i=0..n} Sum_{j=0..n} A026637(i,j).

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A227190 a(n) = n minus (product of run lengths in binary representation of n).

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A101622 A Horadam-Jacobsthal sequence.

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A137241 Number triples (k,3-k,2-2k), concatenated for k=0, 1, 2, 3,...

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A084640 Generalized Jacobsthal numbers.

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