A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.
1, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 3, 4, 7, 2, 1, 1, 4, 5, 13, 5, 5, 1, 1, 5, 6, 21, 10, 31, 1, 1, 1, 6, 7, 31, 17, 121, 3, 7, 1, 1, 7, 8, 43, 26, 341, 7, 127, 2, 1, 1, 8, 9, 57, 37, 781, 13, 1093, 17, 3, 1, 1, 9, 10, 73, 50, 1555, 21, 5461, 82, 73, 1, 1, 1, 10, 11, 91, 65, 2801, 31, 19531, 257, 757, 11, 11, 1, 1, 11, 12, 111, 82, 4681, 43, 55987, 626, 4161, 61, 2047, 1, 1
Offset: 0
Examples
Read by antidiagonals: m\n 0 1 2 3 4 5 6 7 8 9 10 11 12 ------------------------------------------------------ 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 0 1 2 3 4 5 6 7 8 9 10 11 2 1 2 3 4 5 6 7 8 9 10 11 12 13 3 1 3 7 13 21 31 43 57 73 91 111 133 157 4 1 2 5 10 17 26 37 50 65 82 101 122 145 5 1 5 31 121 341 781 ... ... ... ... ... ... ... 6 1 1 3 7 13 21 31 43 57 73 91 111 133 etc. The cyclotomic polynomials are: n n-th cyclotomic polynomial 0 1 1 x-1 2 x+1 3 x^2+x+1 4 x^2+1 5 x^4+x^3+x^2+x+1 6 x^2-x+1 ...
Links
- Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)
- Eric Weisstein's World of Mathematics, Cyclotomic polynomial
- Wikipedia, Cyclotomic polynomial
- Index entries for Cyclotomic polynomials, values at X
Crossrefs
Rows 0-16 are A000012, A023443, A000027, A002061, A002522, A053699, A002061, A053716, A002523, A060883, A060884, A060885, A060886, A060887, A060888, A060889, A060890.
Columns 0-13 are A158388, A020500, A019320, A019321, A019322, A019323, A019324, A019325, A019326, A019327, A019328, A019329, A019330, A019331.
Main diagonal is A070518.
Indices of primes in n-th row for n = 1-20 are A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A250174, A250175, A006314, A217071, A164989, A217072, A250176.
Indices of primes in n-th column for n = 1-10 are A246655, A072226, A138933, A138934, A138935, A138936, A138937, A138938, A138939, A138940.
Indices of primes in main diagonal is A070519.
Cf. A117544 (indices of first prime in n-th row), A085398 (indices of first prime in n-th row apart from column 1), A117545 (indices of first prime in n-th column).
Cf. A206942 (all terms (sorted) for rows>2 and columns>1).
Cf. A206864 (all primes (sorted) for rows>2 and columns>1).
Programs
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Mathematica
Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}] -
PARI
t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2) t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1) T(m, n) = if(m==0, 1, polcyclo(m, n)) a(n) = T(t1(n), t2(n))
Formula
T(m, n) = Phi_m(n)
Comments