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 A099884 XOR difference triangle of the powers of 2, read by rows; Square array A(row,col): A(0,col) = 2^col, A(row,col) = A048724(A(row-1, col)) for row > 0, read by descending antidiagonals. 33
 1, 2, 3, 4, 6, 5, 8, 12, 10, 15, 16, 24, 20, 30, 17, 32, 48, 40, 60, 34, 51, 64, 96, 80, 120, 68, 102, 85, 128, 192, 160, 240, 136, 204, 170, 255, 256, 384, 320, 480, 272, 408, 340, 510, 257, 512, 768, 640, 960, 544, 816, 680, 1020, 514, 771, 1024, 1536, 1280, 1920 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Define an "XOR difference triangle" for a sequence A by the following process. Start with A in the leftmost column. Generate the next column by performing the XOR operation between adjacent terms of the prior column. Repeat this process to generate the XOR difference triangle for A. Further, we define the "XOR BINOMIAL transform" of A as the main diagonal in the XOR difference triangle for A. The XOR BINOMIAL transform is its self-inverse. Let a sequence B be the XOR BINOMIAL transform of A, then we may express B by: B(n) = SumXOR_{k=0..n} A047999(n,k)*A(k), which is equivalent to: B(n) = (C(n,0)mod 2)*A(0) XOR (C(n,1)mod 2)*A(1) XOR (C(n,2)mod 2)*A(2) XOR ... XOR (X(n,n)mod 2)*A(n), where the coefficients are C(n,k)(mod 2) = A047999(n,k). This sequence is a rearrangement of the numbers which are 2^k times distinct Fermat numbers (numbers of the form 2^(2^m) + 1). This matches the sizes of polygons constructible with compass and straightedge (A003401) up to 2^32+1, which is the first nonprime Fermat number. - Franklin T. Adams-Watters, Jun 16 2006 LINKS Paul D. Hanna, First 45 Rows of Triangle, in flattened form. SeqFan-mailing list, Discussion of about related array A255483 FORMULA T(n, k) = 2^(n-k)*A001317(k). T(n, n) = A001317(n) = SumXOR_{k=0..n} A047999(n, k)*2^k, where SumXOR is the analog of summation under the binary XOR operation. From Antti Karttunen, Sep 19 2016: (Start) When viewed as a square array A(row,col), with row >= 0, col >= 0, the following recurrences and formulas are valid: A(0,col) = A000079(col), for row > 0, A(row,col) = A048724(A(row-1, col)). A(row,0) = A001317(row), for col > 0, A(row,col) = 2*A(row,col-1). A(row,col) = A248663(A066117(row+1,col+1)) = A048675(A255483(row,col+1)). (End) With the definitions from Antti Karttunen above, A(row+1, col) = A048720(3, A(row, col)). - Peter Munn, Jan 13 2020 A(n,k) = A193231(A(k,n)) = A091202(A036561(n,k)). - Antti Karttunen, Jan 18 2020 EXAMPLE The main diagonal equals A001317 (Pascal's triangle mod 2 in decimal): {1,3,5,15,17,51,85,255,257,771,1285,3855,...}, and defines the XOR BINOMIAL transform of the powers of 2. Rows begin: 1; 2, 3; 4, 6, 5; 8, 12, 10, 15; 16, 24, 20, 30, 17; 32, 48, 40, 60, 34, 51; 64, 96, 80, 120, 68, 102, 85; 128, 192, 160, 240, 136, 204, 170, 255; 256, 384, 320, 480, 272, 408, 340, 510, 257; 512, 768, 640, 960, 544, 816, 680, 1020, 514, 771; 1024, 1536, 1280, 1920, 1088, 1632, 1360, 2040, 1028, 1542, 1285; 2048, 3072, 2560, 3840, 2176, 3264, 2720, 4080, 2056, 3084, 2570, 3855; ... From Antti Karttunen, Sep 19 2016: (Start) Viewed as a square array, the top left corner looks like this: 1, 2, 4, 8, 16, 32, 64, 128 3, 6, 12, 24, 48, 96, 192, 384 5, 10, 20, 40, 80, 160, 320, 640 15, 30, 60, 120, 240, 480, 960, 1920 17, 34, 68, 136, 272, 544, 1088, 2176 51, 102, 204, 408, 816, 1632, 3264, 6528 85, 170, 340, 680, 1360, 2720, 5440, 10880 255, 510, 1020, 2040, 4080, 8160, 16320, 32640 257, 514, 1028, 2056, 4112, 8224, 16448, 32896 771, 1542, 3084, 6168, 12336, 24672, 49344, 98688 1285, 2570, 5140, 10280, 20560, 41120, 82240, 164480 3855, 7710, 15420, 30840, 61680, 123360, 246720, 493440 4369, 8738, 17476, 34952, 69904, 139808, 279616, 559232 ... (End) The square array shown above can be viewed as a subtable of a multiplication table with particular relevance to the carryless multiplication defined by A048720, as the first column gives the A048720 powers of 3 (and the first row gives powers of 2, which are the same as in standard arithmetic). - Peter Munn, Jan 13 2020 MATHEMATICA a[n_]:= Sum[Mod[Binomial[n, i], 2]*2^i, {i, 0, n}]; T[n_, k_]:=2^(n - k)a[k]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* Indranil Ghosh, Apr 11 2017 *) PROG (PARI) {T(n, k)=local(B); B=0; for(i=0, k, B=bitxor(B, binomial(k, i)%2*2^(n-i))); B} for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) (Scheme) (define (A099884 n) (A099884bi (A002262 n) (A025581 n))) ;; Then use either this recurrence: (define (A099884bi row col) (if (zero? row) (A000079 col) (A048724 (A099884bi (- row 1) col)))) ;; or this one: (define (A099884bi row col) (if (zero? col) (A001317 row) (* 2 (A099884bi row (- col 1))))) ;; Antti Karttunen, Sep 19 2016 (Python) from sympy import binomial def a(n): return sum((binomial(n, i)%2)*2**i for i in range(n + 1)) def T(n, k): return 2**(n - k)*a(k) for n in range(21): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 11 2017 CROSSREFS Essentially GF(2)[X] analog of table A036561. - Antti Karttunen, Jan 18 2020 Cf. A047999, A158875 (row sums). Cf. A000215, A003401, A048675, A048720, A048724, A066117, A248663, A255483, A276586. Cf. A000079 (first column of triangular table, the topmost row of square array). Cf. A001317 (the rightmost diagonal of triangular table, the leftmost column of square array). Cf. A099885, A117998 (central diagonals). Cf. A276618 (transpose), A091202, A193231. Sequence in context: A054582 A257797 A220347 * A191446 A230764 A276685 Adjacent sequences: A099881 A099882 A099883 * A099885 A099886 A099887 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Oct 28 2004 EXTENSIONS Square array interpretation added as a second, alternative description by Antti Karttunen, Sep 19 2016 STATUS approved

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Last modified February 1 12:28 EST 2023. Contains 359993 sequences. (Running on oeis4.)