A001477 -id:A001477 - cp's OEIS frontend

A286145 Square array read by antidiagonals: A(n,k) = T(n XOR k, k), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

0, 4, 2, 12, 1, 5, 24, 18, 13, 9, 40, 17, 3, 8, 14, 60, 50, 11, 7, 26, 20, 84, 49, 61, 6, 42, 19, 27, 112, 98, 85, 73, 62, 52, 43, 35, 144, 97, 59, 72, 10, 51, 25, 34, 44, 180, 162, 83, 71, 22, 16, 41, 33, 64, 54, 220, 161, 181, 70, 38, 15, 23, 32, 88, 53, 65, 264, 242, 221, 201, 58, 48, 39, 31, 116, 102, 89, 77, 312, 241, 179, 200, 222, 47, 21, 30, 148, 101, 63, 76, 90

Offset: 0

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

			The top left 0 .. 12 x 0 .. 12 corner of the array:
   0,   4,  12,  24,  40,  60,  84, 112, 144, 180, 220, 264, 312
   2,   1,  18,  17,  50,  49,  98,  97, 162, 161, 242, 241, 338
   5,  13,   3,  11,  61,  85,  59,  83, 181, 221, 179, 219, 365
   9,   8,   7,   6,  73,  72,  71,  70, 201, 200, 199, 198, 393
  14,  26,  42,  62,  10,  22,  38,  58, 222, 266, 314, 366, 218
  20,  19,  52,  51,  16,  15,  48,  47, 244, 243, 340, 339, 240
  27,  43,  25,  41,  23,  39,  21,  37, 267, 315, 265, 313, 263
  35,  34,  33,  32,  31,  30,  29,  28, 291, 290, 289, 288, 287
  44,  64,  88, 116, 148, 184, 224, 268,  36,  56,  80, 108, 140
  54,  53, 102, 101, 166, 165, 246, 245,  46,  45,  94,  93, 158
  65,  89,  63,  87, 185, 225, 183, 223,  57,  81,  55,  79, 177
  77,  76,  75,  74, 205, 204, 203, 202,  69,  68,  67,  66, 197
  90, 118, 150, 186,  86, 114, 146, 182,  82, 110, 142, 178,  78

Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array
MathWorld, Pairing Function

Transpose: A286147.
Cf. A046092 (row 0), A000096 (column 0), A000217 (main diagonal).
Cf. A003987, A001477, A286108, A286109, A286150, A286151.

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=T[BitXor[n, k], k]; Table[A[k, n - k ], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, May 21 2017 *)

def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n^k, k)
for n in range(21): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, May 21 2017

(define (A286145 n) (A286145bi (A002262 n) (A025581 n)))
(define (A286145bi row col) (let ((a (A003987bi row col)) (b col)) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Where A003987bi implements bitwise-xor (A003987).

A(n,k) = T(A003987(n,k), k), where T(n,k) is sequence A001477 considered as a two-dimensional table, that is, as a pairing function from [0, 1, 2, 3, ...] x [0, 1, 2, 3, ...] to [0, 1, 2, 3, ...].

A286147 Square array read by antidiagonals: A(n,k) = T(n XOR k, n), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 2, 4, 5, 1, 12, 9, 13, 18, 24, 14, 8, 3, 17, 40, 20, 26, 7, 11, 50, 60, 27, 19, 42, 6, 61, 49, 84, 35, 43, 52, 62, 73, 85, 98, 112, 44, 34, 25, 51, 10, 72, 59, 97, 144, 54, 64, 33, 41, 16, 22, 71, 83, 162, 180, 65, 53, 88, 32, 23, 15, 38, 70, 181, 161, 220, 77, 89, 102, 116, 31, 39, 48, 58, 201, 221, 242, 264, 90, 76, 63, 101, 148, 30, 21, 47, 222, 200, 179, 241, 312

Offset: 0

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

			The top left 0 .. 12 x 0 .. 12 corner of the array:
    0,   2,   5,   9,  14,  20,  27,  35,  44,  54,  65,  77,  90
    4,   1,  13,   8,  26,  19,  43,  34,  64,  53,  89,  76, 118
   12,  18,   3,   7,  42,  52,  25,  33,  88, 102,  63,  75, 150
   24,  17,  11,   6,  62,  51,  41,  32, 116, 101,  87,  74, 186
   40,  50,  61,  73,  10,  16,  23,  31, 148, 166, 185, 205,  86
   60,  49,  85,  72,  22,  15,  39,  30, 184, 165, 225, 204, 114
   84,  98,  59,  71,  38,  48,  21,  29, 224, 246, 183, 203, 146
  112,  97,  83,  70,  58,  47,  37,  28, 268, 245, 223, 202, 182
  144, 162, 181, 201, 222, 244, 267, 291,  36,  46,  57,  69,  82
  180, 161, 221, 200, 266, 243, 315, 290,  56,  45,  81,  68, 110
  220, 242, 179, 199, 314, 340, 265, 289,  80,  94,  55,  67, 142
  264, 241, 219, 198, 366, 339, 313, 288, 108,  93,  79,  66, 178
  312, 338, 365, 393, 218, 240, 263, 287, 140, 158, 177, 197,  78

Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array
MathWorld, Pairing Function

Transpose: A286145.
Cf. A000096 (row 0), A046092 (column 0), A000217 (main diagonal).
Cf. A003987, A001477, A286108, A286109, A286150, A286151.

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=T[BitXor[n, k], k]; Table[A[n - k, k], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, May 21 2017 *)

def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n^k, k)
for n in range(21): print([A(n - k, k) for k in range(n + 1)]) # Indranil Ghosh, May 21 2017

(define (A286147 n) (A286147bi (A002262 n) (A025581 n)))
(define (A286147bi row col) (let ((a (A003987bi row col)) (b row)) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Where A003987bi implements bitwise-xor (A003987).

A(n,k) = T(A003987(n,k), n), where T(n,k) is sequence A001477 considered as a two-dimensional table, that is, as a pairing function from [0, 1, 2, 3, ...] x [0, 1, 2, 3, ...] to [0, 1, 2, 3, ...].

A286150 Square array read by antidiagonals: A(n,k) = T(n XOR k, min(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

Original entry on oeis.org

0, 2, 2, 5, 1, 5, 9, 13, 13, 9, 14, 8, 3, 8, 14, 20, 26, 7, 7, 26, 20, 27, 19, 42, 6, 42, 19, 27, 35, 43, 52, 62, 62, 52, 43, 35, 44, 34, 25, 51, 10, 51, 25, 34, 44, 54, 64, 33, 41, 16, 16, 41, 33, 64, 54, 65, 53, 88, 32, 23, 15, 23, 32, 88, 53, 65, 77, 89, 102, 116, 31, 39, 39, 31, 116, 102, 89, 77, 90, 76, 63, 101, 148, 30, 21, 30, 148, 101, 63, 76, 90

Offset: 0

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

			The top left 0 .. 12 x 0 .. 12 corner of the array:
   0,   2,   5,   9,  14,  20,  27,  35,  44,  54,  65,  77,  90
   2,   1,  13,   8,  26,  19,  43,  34,  64,  53,  89,  76, 118
   5,  13,   3,   7,  42,  52,  25,  33,  88, 102,  63,  75, 150
   9,   8,   7,   6,  62,  51,  41,  32, 116, 101,  87,  74, 186
  14,  26,  42,  62,  10,  16,  23,  31, 148, 166, 185, 205,  86
  20,  19,  52,  51,  16,  15,  39,  30, 184, 165, 225, 204, 114
  27,  43,  25,  41,  23,  39,  21,  29, 224, 246, 183, 203, 146
  35,  34,  33,  32,  31,  30,  29,  28, 268, 245, 223, 202, 182
  44,  64,  88, 116, 148, 184, 224, 268,  36,  46,  57,  69,  82
  54,  53, 102, 101, 166, 165, 246, 245,  46,  45,  81,  68, 110
  65,  89,  63,  87, 185, 225, 183, 223,  57,  81,  55,  67, 142
  77,  76,  75,  74, 205, 204, 203, 202,  69,  68,  67,  66, 178
  90, 118, 150, 186,  86, 114, 146, 182,  82, 110, 142, 178,  78

Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000096 (row 0 & column 0), A000217 (main diagonal).

Cf. A003987, A001477, A286108, A286109, A286145, A286147, A286151.

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=T[BitXor[n, k],Min[n,  k]]; Table[A[k, n - k], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, May 21 2017 *)

def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n^k, min(n, k))
for n in range(21): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, May 21 2017

(define (A286150 n) (A286150bi (A002262 n) (A025581 n)))
(define (A286150bi row col) (let ((a (A003987bi row col)) (b (min col row))) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Where A003987bi implements bitwise-xor (A003987).

A(n,k) = T(A003987(n,k), min(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, that is, as a pairing function from [0, 1, 2, 3, ...] x [0, 1, 2, 3, ...] to [0, 1, 2, 3, ...].

A182439 Table a(k,i), read by antidiagonals, in which the n-th row comprises A214206(n) in position 0 followed by a second order recursive series G in which each product G(i)*G(i+1) lies in the same row of A001477 (interpreted as a square array - see below).

Original entry on oeis.org

0, 0, 4, 14, 1, 7, 110, 14, 2, 8, 672, 95, 14, 3, 10, 3948, 568, 84, 14, 4, 11, 23042, 3325, 492, 81, 14, 5, 12, 134330, 19394, 2870, 472, 74, 14, 6, 13, 782964, 113051, 16730, 2751, 424, 71, 14, 7, 14, 4563480, 658924, 97512, 16034, 2464, 404, 68, 14, 8, 15

Offset: 0

Kenneth J Ramsey, Apr 28 2012

This is a square array related to the square array of nonnegative integers, A001477. Each row k contains the positive argument of the largest triangular number equal to or less than 14*k in column 0 and a corresponding 2nd-order recursive sequence G(k) in the rest of the row. Each second-order recursive series term G(i) corresponds to a(k,i+1). If the product 14*k appears in row "r" of the square array A001477, then the product of adjacent terms G(i)*G(i+1), if greater than (r^2 + 3*r - 2)/2, is always in row "r" of square array A001477. If the product is less than (r^2 + 3*r -2)/2 then assuming the row can take negative indices, the product can still be said to lie in the same row r. For instance, 0, 1, 3, and 6 are each a triangular number and appear as the first 4 terms of row 0 of square array A001477. Note that in the next row and to the left of the 1, 3, and 6 are 2, 4 and 7 so going down a row and to the left in the square array increases the value by 1. Going down to the next row and to the left again would be 3, 5, and 8 so 3 which is 2 more than 1 would be in row 2 if that row were made to take the indices (2,-1).
A property of this table is that a(k+1,i)-a(k,i) directly depends on the value of a(k+1,0)-a(k,0) in the same manner regardless of the value of k. For example, a(k,2+n) - a(k,2+n) = A001652(n) for n=0,1,2,3,... whereever a(k+1,0) - a(k,0) = 1.
Also, a(k+1,2+n) - a(k,2+n) is divisible by A143608(n) for n>0 for all k.

			     0,     0,    14,   110,   672,  3948, 23042,134330,782964,
     4,     1,    14,    95,   568,  3325, 19394,113051,658924,
     7,     2,    14,    84,   492,  2870, 16730, 97512,568344,
     8,     3,    14,    81,   472,  2751, 16034, 93453,544684,
    10,     4,    14,    74,   424,  2464, 14354, 83654,487564,
    11,     5,    14,    71,   404,  2345, 13658, 79595,463904,
    12,     6,    14,    68,   384,  2226, 12962, 75536,440244.
Note that 0*14, 14*110, 110*672, etc. are all triangular numbers and thus appear in row 0 of square array A001477; while, 1*14, 14*95, 95*568, 568*3325, etc. are all 4 more than a triangular number and appear in row 4 of square array A001477.

Cf. A001108, A001652, A182431, A182440, A182441, A143608.

A182439 := proc(n,k)
        if k = 0 then
                A003056(14*n) ;
        elif k = 1 then
                n;
        elif k = 2 then
                14;
        else
                6*procname(n,k-1)-procname(n,k-2)+ 28+2*n-2-4*procname(n,0) ;
        end if;
end proc: # R. J. Mathar, Jul 09 2012

highTri = Compile[{{S1,_Integer}}, Module[{xS0=0, xS1=S1}, While[xS1-xS0*(xS0+1)/2 > xS0, xS0++]; xS0]];
overTri = Compile[{{S2,_Integer}}, Module[{xS0=0, xS2=S2}, While[xS2-xS0*(xS0+1)/2 > xS0, xS0++]; xS2 - (xS0*(1+xS0)/2)]];
K1 = 0; m = 14; tab=Reap[While[K1<16,J1=highTri[m*K1]; X = 2*(m+K1-(J1*2+1)); K2 = (6 m - K1 + X); K3 = 6 K2 - m + X;
K4 = 6 K3 - K2 + X; K5 = 6 K4 -K3 + X; K6 = 6*K5 - K4 + X; K7 = 6*K6-K5+X; K8 = 6*K7-K6+X; Sow[J1,c]; Sow[K1,d]; Sow[m,e];
Sow[K2,f]; Sow[K3,g]; Sow[K4,h];
  Sow[K5,i]; Sow[K6,j]; Sow[K7,k]; Sow[K8,l]; K1++]][[2]]; a=1; list5 = Reap[While[a<11, b=a; While[b>0,
Sow[tab[[b,a+1-b]]]; b--]; a++]][[2,1]]; list5
(* Second program: *)
A003056[n_] := Floor[(Sqrt[1 + 8n] - 1)/2];
T[n_, k_] := Switch[k, 0, A003056[14n], 1, n, 2, 14, _, 6T[n, k-1] - T[n, k-2] + 28 + 2n - 2 - 4T[n, 0]];
Table[T[n-k, k], {n, 0, 9}, {k, n, 0, -1}] (* Jean-François Alcover, May 09 2023, after R. J. Mathar *)

a(k,0) equals the largest m such that m*(m+1)/2 is equal to or less than 14*k, A003056(14*k).
a(k,1) = k; a(k,2) = 14.
For i > 2, a(k,i) = 6*a(k,i-1) -a (k,i-2) + G_k where G_k = 28 + 2*k - 2 - 4*a(k,0).
a(k,i) = 7*a(k,i-1)-7*a(k,i-2)+a(k,i-3). - R. J. Mathar, Jul 09 2012

A182441 Table, read by antidiagonals, in which the n-th row comprises A214206(n) in position 0 followed by a second order recursive series G in which each product G(i)*G(i+1) lies in the same row of A001477 (interpreted as a square array - see below).

Original entry on oeis.org

0, 0, 4, 14, 1, 7, 114, 14, 2, 8, 700, 131, 14, 3, 10, 4116, 820, 144, 14, 4, 11, 24026, 4837, 912, 149, 14, 5, 12, 140070, 28250, 5390, 948, 158, 14, 6, 13, 816424, 164711, 31490, 5607, 1012, 163, 14, 7, 14, 4758504

Offset: 0

Kenneth J Ramsey, Apr 28 2012

This is a table related to the square array of the nonnegative integers (A001477). Each row k contains A003056(14*k) in column 0 and a corresponding 2nd order recursive sequence G(k) beginning at position a(k,1). That is each term G(i) is a(k,i+1). If A002262(14*n) is "r", the product of adjacent terms G(i)*G(i+1) if greater than (r^2 + 3*r - 2)/2, is always in row "r" of the square array A001477. If the product is less than (r^2 + 3*r -2)/2, then the product could still be said to lie in the same row r since the product is equal to the sum of a triangular number + r, which is a property of all numbers in row r of the square array A002262.

A property of this table is that a(k+1,i)-a(k,i) directly depends on the value of a(k+1,0)-a(k,0) in the same manner regardless of the value of k. For instance, wherever a(k+1,0)-a(k,0) = 0, a(k+1,i+1)-a(k,i+1) = A212329. Also, a(k+1,n+2)-a(k,n+2) is divisible by A143608(n).

			For i>0 a(0,i) * a(0,i+1) = 0*14,14*114,114*700,700*4116,etc. which are all triangular numbers and lie in row 0 of square array A001477, while a(1,i)*a(1.i+1) = 1*14, 14*131, 131*820, 820*4837 etc. which are all 4 more than a triangular number and lie in row 4 of square array A001477.

Cf. A212329, A182431, A182439, A182440, A143608.

highTri = Compile[{{S1,_Integer}}, Module[{xS0=0, xS1=S1}, While[xS1-xS0*(xS0+1)/2 > xS0, xS0++]; xS0]];
overTri = Compile[{{S2,_Integer}}, Module[{xS0=0, xS2=S2}, While[xS2-xS0*(xS0+1)/2 > xS0, xS0++]; xS2 - (xS0*(1+xS0)/2)]];
K1 = 0; m = 14; tab=Reap[While[K1<16,J1=highTri[m*K1]; X = 2*(m+K1+(J1*2+1)); K2 = (6 m - K1 + X); K3 = 6 K2 - m + X;
K4 = 6 K3 - K2 + X; K5 = 6 K4 -K3 + X; K6 = 6*K5 - K4 + X; K7 = 6*K6-K5+X; K8 = 6*K7-K6+X; Sow[J1,c]; Sow[K1,d]; Sow[m,e];
Sow[K2,f]; Sow[K3,g]; Sow[K4,h];
  Sow[K5,i]; Sow[K6,j]; Sow[K7,k]; Sow[K8,l]; K1++]][[2]]; a=1; list5 = Reap[While[a<11, b=a; While[b>0,
Sow[tab[[b,a+1-b]]]; b--]; a++]][[2,1]]; list5

a(k,0) equals the largest m such that m*(m+1)/2 <= 14*k (A003056(14*k)).
a(k,1) equals k; a(k,2) = 14.
For i > 2, a(k,i) = 6*a(k,i-1) -a (k,i-2) + G_k where G_k is a constant equal to 28 + 2*k + 2 + 4*A003056(14*k).

A286098 Square array read by antidiagonals: A(n,k) = T(n AND k, n OR k), where T(n,k) is sequence A001477 considered as a two-dimensional table, AND is bitwise-and (A004198) and OR is bitwise-or (A003986).

Original entry on oeis.org

0, 1, 1, 3, 4, 3, 6, 6, 6, 6, 10, 11, 12, 11, 10, 15, 15, 17, 17, 15, 15, 21, 22, 21, 24, 21, 22, 21, 28, 28, 28, 28, 28, 28, 28, 28, 36, 37, 38, 37, 40, 37, 38, 37, 36, 45, 45, 47, 47, 49, 49, 47, 47, 45, 45, 55, 56, 55, 58, 59, 60, 59, 58, 55, 56, 55, 66, 66, 66, 66, 70, 70, 70, 70, 66, 66, 66, 66, 78, 79, 80, 79, 78, 83, 84, 83, 78, 79, 80, 79, 78

Offset: 0

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

			The top left 0 .. 12 x 0 .. 12 corner of the array:
   0,  1,   3,   6,  10,  15,  21,  28,  36,  45,  55,  66,  78
   1,  4,   6,  11,  15,  22,  28,  37,  45,  56,  66,  79,  91
   3,  6,  12,  17,  21,  28,  38,  47,  55,  66,  80,  93, 105
   6, 11,  17,  24,  28,  37,  47,  58,  66,  79,  93, 108, 120
  10, 15,  21,  28,  40,  49,  59,  70,  78,  91, 105, 120, 140
  15, 22,  28,  37,  49,  60,  70,  83,  91, 106, 120, 137, 157
  21, 28,  38,  47,  59,  70,  84,  97, 105, 120, 138, 155, 175
  28, 37,  47,  58,  70,  83,  97, 112, 120, 137, 155, 174, 194
  36, 45,  55,  66,  78,  91, 105, 120, 144, 161, 179, 198, 218
  45, 56,  66,  79,  91, 106, 120, 137, 161, 180, 198, 219, 239
  55, 66,  80,  93, 105, 120, 138, 155, 179, 198, 220, 241, 261
  66, 79,  93, 108, 120, 137, 155, 174, 198, 219, 241, 264, 284
  78, 91, 105, 120, 140, 157, 175, 194, 218, 239, 261, 284, 312

Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000217 (row 0 & column 0), A084263 (seems to be row 1 & column 1), A046092 (main diagonal).
Cf. A003056, A003986, A004198.
Cf. also arrays A286099, A286101, A286102, A286108.

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=T[BitAnd[n, k],BitOr[n,  k]]; Table[A[n - k, k], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, May 21 2017 *)

def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n&k, n|k)
for n in range(0, 21): print([A(k, n - k) for k in range(0, n + 1)]) # Indranil Ghosh, May 21 2017

(define (A286098 n) (A286098bi (A002262 n) (A025581 n)))
(define (A286098bi row col) (let ((a (A004198bi row col)) (b (A003986bi row col))) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Here A003986bi and A004198bi implement bitwise-OR (A003986) and bitwise-AND (A004198).

A(n,k) = T(A004198(n,k), A003986(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, that is, as a pairing function from [0, 1, 2, 3, ...] x [0, 1, 2, 3, ...] to [0, 1, 2, 3, ...].

A286099 Square array read by antidiagonals: A(n,k) = T(n OR k, n AND k), where T(n,k) is sequence A001477 considered as a two-dimensional table, AND is bitwise-and (A004198) and OR is bitwise-or (A003986).

Original entry on oeis.org

0, 2, 2, 5, 4, 5, 9, 9, 9, 9, 14, 13, 12, 13, 14, 20, 20, 18, 18, 20, 20, 27, 26, 27, 24, 27, 26, 27, 35, 35, 35, 35, 35, 35, 35, 35, 44, 43, 42, 43, 40, 43, 42, 43, 44, 54, 54, 52, 52, 50, 50, 52, 52, 54, 54, 65, 64, 65, 62, 61, 60, 61, 62, 65, 64, 65, 77, 77, 77, 77, 73, 73, 73, 73, 77, 77, 77, 77, 90, 89, 88, 89, 90, 85, 84, 85, 90, 89, 88, 89, 90

Offset: 0

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), ...

			The top left 0 .. 12 x 0 .. 12 corner of the array:
   0,   2,   5,   9,  14,  20,  27,  35,  44,  54,  65,  77,  90
   2,   4,   9,  13,  20,  26,  35,  43,  54,  64,  77,  89, 104
   5,   9,  12,  18,  27,  35,  42,  52,  65,  77,  88, 102, 119
   9,  13,  18,  24,  35,  43,  52,  62,  77,  89, 102, 116, 135
  14,  20,  27,  35,  40,  50,  61,  73,  90, 104, 119, 135, 148
  20,  26,  35,  43,  50,  60,  73,  85, 104, 118, 135, 151, 166
  27,  35,  42,  52,  61,  73,  84,  98, 119, 135, 150, 168, 185
  35,  43,  52,  62,  73,  85,  98, 112, 135, 151, 168, 186, 205
  44,  54,  65,  77,  90, 104, 119, 135, 144, 162, 181, 201, 222
  54,  64,  77,  89, 104, 118, 135, 151, 162, 180, 201, 221, 244
  65,  77,  88, 102, 119, 135, 150, 168, 181, 201, 220, 242, 267
  77,  89, 102, 116, 135, 151, 168, 186, 201, 221, 242, 264, 291
  90, 104, 119, 135, 148, 166, 185, 205, 222, 244, 267, 291, 312

Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000096 (row 0 & column 0), A162761 (seems to be row 1 & column 1), A046092 (main diagonal).
Cf. A003056, A003986, A004198.
Cf. also arrays A286098, A286101, A286102, A286109.

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=T[BitOr[n, k],BitAnd[n,  k]]; Table[A[n - k, k], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, May 21 2017 *)

def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n|k, n&k)
for n in range(0, 21): print([A(k, n - k) for k in range(0, n + 1)]) # Indranil Ghosh, May 21 2017

(define (A286099 n) (A286099bi (A002262 n) (A025581 n)))
(define (A286099bi row col) (let ((a (A003986bi row col)) (b (A004198bi row col))) (/ (+ (expt (+ a b) 2) (* 3 a) b) 2))) ;; Here A003986bi and A004198bi implement bitwise-OR (A003986) and bitwise-AND (A004198).

A(n,k) = T(A003986(n,k), A004198(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, that is, as a pairing function from [0, 1, 2, 3, ...] x [0, 1, 2, 3, ...] to [0, 1, 2, 3, ...].

A286153 Square array read by descending antidiagonals A(1,1), A(1,2), A(2,1), ...: If n > k, A(n,k) = T(n XOR k, k), and otherwise A(n,k) = T(n, n XOR k), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

Original entry on oeis.org

2, 11, 13, 7, 5, 8, 22, 8, 7, 26, 16, 38, 9, 42, 19, 37, 47, 58, 62, 52, 43, 29, 23, 48, 14, 51, 25, 34, 56, 30, 39, 19, 16, 41, 33, 64, 46, 80, 31, 25, 20, 23, 32, 88, 53, 79, 93, 108, 32, 41, 39, 31, 116, 102, 89, 67, 57, 94, 140, 33, 27, 30, 148, 101, 63, 76, 106, 68, 81, 157, 176, 34, 29, 184, 166, 87, 75, 118, 92, 138, 69, 175, 158, 216, 35, 224, 165, 185, 74, 150, 103

Offset: 1

Antti Karttunen, May 03 2017

			The top left 1 .. 12 x 1 .. 12 corner of the array:
    2,  11,   7,  22,  16,  37,  29,  56,  46,  79,  67, 106
   13,   5,   8,  38,  47,  23,  30,  80,  93,  57,  68, 138
    8,   7,   9,  58,  48,  39,  31, 108,  94,  81,  69, 174
   26,  42,  62,  14,  19,  25,  32, 140, 157, 175, 194,  82
   19,  52,  51,  16,  20,  41,  33, 176, 158, 215, 195, 110
   43,  25,  41,  23,  39,  27,  34, 216, 237, 177, 196, 142
   34,  33,  32,  31,  30,  29,  35, 260, 238, 217, 197, 178
   64,  88, 116, 148, 184, 224, 268,  44,  53,  63,  74,  86
   53, 102, 101, 166, 165, 246, 245,  46,  54,  87,  75, 114
   89,  63,  87, 185, 225, 183, 223,  57,  81,  65,  76, 146
   76,  75,  74, 205, 204, 203, 202,  69,  68,  67,  77, 182
  118, 150, 186,  86, 114, 146, 182,  82, 110, 142, 178,  90

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array

Array A286151 without its topmost row and leftmost column.

Cf. A003987, A091255, A286155.

T[a_, b_]:=((a + b)^2 + 3a + b)/2; A[n_, k_]:=If[n>k,T[BitXor[n, k], k], T[n, BitXor[n, k]]]; Table[A[k, n - k + 1], {n, 20}, {k, n}] // Flatten (* Indranil Ghosh, May 21 2017 *)

def T(a, b): return ((a + b)**2 + 3*a + b)//2
def A(n, k): return T(n^k, k) if n>k else T(n, n^k)
for n in range(1, 21): print([A(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, May 21 2017

(define (A286153 n) (A286151bi (A002260 n) (A004736 n))) ;; For A286151bi see A286151.

A(n,k) = A286151(n,k), for n >= 1, k >= 1.

If n > k, A(n,k) = T(A003987(n,k),k), otherwise A(n,k) = T(n,A003987(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

A071653 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using the bivariate form of A001477 as the packing bijection N x N -> N.

Original entry on oeis.org

0, 1, 3, 2, 10, 6, 5, 7, 4, 66, 28, 21, 36, 15, 14, 9, 12, 56, 22, 8, 16, 29, 11, 2278, 435, 253, 703, 136, 120, 55, 91, 1653, 276, 45, 153, 465, 78, 77, 35, 27, 44, 20, 25, 18, 68, 2212, 407, 30, 232, 667, 121, 19, 13, 23, 106, 46, 38, 79, 1597, 254, 17, 37, 137, 436

Offset: 0

Antti Karttunen, May 30 2002

nonn

A071653(A014137(n-1)) = A072638(n) for all n > 0. - Paul D. Hanna, Jan 04 2007

Also seems that A071653(A014137(n)-1) = A006894(n) for all n > 0. - Antti Karttunen, Jul 30 2012

Antti Karttunen, Table of n, a(n) for n = 0..2055
A. Karttunen, Alternative Catalan Orderings
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A071654. Cf. also A014486, A001477, A071651, A071652.

A071672 The binary encoding of parenthesizations given in a "global arithmetic order", using A001477 as the packing bijection N X N -> N.

Original entry on oeis.org

10, 1100, 1010, 111000, 110010, 101100, 110100, 11100010, 11001100, 101010, 11110000, 11010010, 1110001100, 11001010, 10111000, 11100100, 1111000010, 1101001100, 1110001010, 1100111000, 10110010, 11011000, 1110010010

Offset: 1

Antti Karttunen, May 30 2002

A. Karttunen, Gatomorphisms (Includes the complete Scheme source for computing this sequence)

Cf. A071672(n) = A063171(A071654(n)). Permutation of A063171 and A071671. Particularly, applying the automorphism ReflectBinTree (A057163) to A071672(n) yields A071671(n). The length of each term / 2: A071673.

cp's OEIS Frontend

A286145 Square array read by antidiagonals: A(n,k) = T(n XOR k, k), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

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A286147 Square array read by antidiagonals: A(n,k) = T(n XOR k, n), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

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A286150 Square array read by antidiagonals: A(n,k) = T(n XOR k, min(n,k)), where T(n,k) is sequence A001477 considered as a two-dimensional table, and XOR is bitwise-xor (A003987).

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A182439 Table a(k,i), read by antidiagonals, in which the n-th row comprises A214206(n) in position 0 followed by a second order recursive series G in which each product G(i)*G(i+1) lies in the same row of A001477 (interpreted as a square array - see below).

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A182441 Table, read by antidiagonals, in which the n-th row comprises A214206(n) in position 0 followed by a second order recursive series G in which each product G(i)*G(i+1) lies in the same row of A001477 (interpreted as a square array - see below).

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A286098 Square array read by antidiagonals: A(n,k) = T(n AND k, n OR k), where T(n,k) is sequence A001477 considered as a two-dimensional table, AND is bitwise-and (A004198) and OR is bitwise-or (A003986).

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A286099 Square array read by antidiagonals: A(n,k) = T(n OR k, n AND k), where T(n,k) is sequence A001477 considered as a two-dimensional table, AND is bitwise-and (A004198) and OR is bitwise-or (A003986).

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