Bob Selcoe - cp's OEIS frontend

A330885 Square array T(n,k) read by antidiagonals upwards: T(n,0)=1; T(n,1) = n+1; T(n,2) = 2n+1, T(n,k>2) = T(n,k-1) - T(n,k-2) - T(n,k-3).

1, 1, 1, 1, 2, 1, 1, 3, 3, -1, 1, 4, 5, 0, -3, 1, 5, 7, 1, -5, -3, 1, 6, 9, 2, -7, -8, 1, 1, 7, 11, 3, -9, -13, -3, 7, 1, 8, 13, 4, -11, -18, -7, 10, 9, 1, 9, 15, 5, -13, -23, -11, 13, 21, 1, 1, 10, 17, 6, -15, -28, -15, 16, 33, 14, -15

Offset: 0

Bob Selcoe, May 05 2020

			Array starts:
1  1   1  -1   -3   -3    1   7   9   1  -15   -25
1  2   3   0   -5   -8   -3  10  21  14  -17   -52
1  3   5   1   -7  -13   -7  13  33  27  -19   -79
1  4   7   2   -9  -18  -11  16  45  40  -21  -106
1  5   9   3  -11  -23  -15  19  57  53  -23  -133
1  6  11   4  -13  -28  -19  22  69  66  -25  -160
1  7  13   5  -15  -33  -23  25  81  79  -27  -187

Cf. A078016, A180735.

Columns k: A000012 (k=0), A000027 (k=1), A005408 (k=2), A023443 (k=3), A165747 (k=4), -A016885 (k=5), -A004767 (k=6), A016777 (k=7), A017629 (k=8), A190991 (k=9).

T[n_, k_]:= T[n, k]= If[k<3, k*n+1, T[n, k-1] - T[n, k-2] - T[n, k-3]];
Table[T[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 26 2020 *)

T(0,k) = A180735(k-1).

T(n,k) - T(n-1,k) = -A078016(k+1).

A328037 Irregular triangle T(n,k) read by rows: "quotient trajectories" in reduced Collatz sequences; i.e., T(n,k) = q-value(A256598(n,k)) where q-value(z) = (z - A259663(m,j))/2^(m+j) and (m,j) is the unique pair such that z == A259663(m,j) (mod 2^(m+j)). (See Comments for definitions.)

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 1, 0, 2, 0, 0, 0, 1, 5, 0, 0, 2, 6, 20, 15, 5, 17, 3, 9, 29, 2, 8, 24, 74, 27, 5, 15, 47, 17, 53

Offset: 0

Bob Selcoe, Oct 03 2019

Coefficients T(m,j) in the array A259663 are least residues in congruence classes T(m,j) mod 2^(m+j). T"(m,j) denotes all members of that class.
Reduced Collatz sequences (i.e., reduced sequences) are standard Collatz sequences excluding even terms. Row n in triangle A256598 shows the reduced sequence starting with 2n+1.
Let every positive odd number z = T"(m,j)_q, where q is the quotient of z in T"(m,j). For example, T(2,3) = 7 in A259663, so T"(2,3) contains all numbers == 7 (mod 32). So z = T"(2,3)_0 = 7, z = T"(2,3)_1 = 39, z = T"(2,3)_2 = 71, etc.
T(n,k) is the q-value of A256598(n,k) (see Example below). Thus, each row n is defined here as a "quotient trajectory" for the reduced sequence with starting term 2n+1.

			Triangle starts:
  0;
  0, 0, 0;
  0, 0;
  0, 0, 2, 0, 0, 0;
  1, 0, 0, 2, 0, 0, 0;
  0, 2, 0, 0, 0;
  0, 0, 0;
  0, 0, 0, 0, 0, 0;
  2, 0, 0, 0;
  0, 1, 0, 2, 0, 0, 0;
  0, 0;
  0, 0, 0, 0, 0;
  3, 0, 1, 0, 2, 0, 0, 0;
  ...
n=13 starts with 27 = T"(2,2)_1 and takes 41 steps: 1, 5, 0, 0, 2, 6, 20, 15, 5, ..., 0, 0, 0.
Row n=12 maps to the reduced sequence n=12 in A256598: 25 -> 19 -> 29 -> 11 -> 17 -> 13 -> 5 -> 1, which is T"(2,1)_3 -> T"(3,2)_0 -> T"(3,1)_1 -> T"(2,2)_0 -> T"(2,1)_2 -> T"(3,1)_0 -> T"(4,1)_0 -> T"(2,1)_0.

Cf. A256598, A259663.

Tdt(n, k) = if (n==2, if (k%2, 2^k-1, 3*2^k-1), if (n==3, if (k%2, 7*2^k-1, 5*2^k-1), mj = 2^(n-3) % 2^(n-2); mk = k % 2^(n-2); (2^k*3^(mj-mk) - 1) % 2^(n+k))); \\ A259663
qvalue(m) = {my(line = 2, i, md); while (1, i = line; for (j=1, line-1, md = Tdt(i, j); if (m % (2^(i+j)) == md % (2^(i+j)), return((m-md)/2^(i+j))); i--;); line ++;);}
row(n) = {my(oddn = 2*n+1, vl = List(oddn), x); while (oddn != 1, x = 3*oddn+1; oddn = x >> valuation(x, 2); listput(vl, oddn)); my(v = Vec(vl)); for (i=1, #v, v[i] = qvalue(v[i]);); v;} \\ A256598
tabf(nn) = {for (n=0, nn, my(rown = row(n)); for (k=1, #rown, print1(rown[k], ", ")); print;);} \\ Michel Marcus, Oct 04 2019

A327283 Irregular triangle T(n,k) read by rows: "residual summands" in reduced Collatz sequences (see Comments for definition and explanation).

Original entry on oeis.org

1, 1, 5, 1, 5, 19, 73, 347, 1, 7, 29, 103, 373, 1631, 1, 5, 23, 133, 1, 11, 1, 5, 19, 65, 451, 1, 7, 53, 1, 5, 31, 125, 503, 2533, 1, 1, 5, 19, 185, 1, 7, 29, 151, 581, 2255, 10861, 1, 5, 23, 85, 287, 925

Offset: 1

Bob Selcoe, Sep 15 2019

Let R_s be the reduced Collatz sequence (cf. A259663) starting with s and let R_s(k), k >= 0 be the k-th term in R_s. Then R_(2n-1)(k) = (3^k*(2n-1) + T(n,k))/2^j, where j is the total number of halving steps from R_(2n-1)(0) to R_(2n-1)(k). T(n,k) is defined here as the "residual summand".

The sequence without duplicates is a permutation of A116641.

			Triangle starts:
  1;
  1, 5;
  1;
  1, 5, 19, 73,  347;
  1, 7, 29, 103, 373, 1631;
  1, 5, 23, 133;
  1, 11;
  1, 5, 19, 65,  451;
  1, 7, 53;
  1, 5, 31, 125, 503, 2533;
  1;
  1, 5, 19, 185;
  1, 7, 29, 151, 581, 2255, 10861;
  ...
T(5,4)=103 because R_9(4) = 13; the number of halving steps from R_9(0) to R_9(4) is 6, and 13 = (81*9 + 103)/64.

Cf. A116623, A116641, A259663.

T(n,k) = 2^j*R_(2n-1)(k) - 3^k*(2n-1), as defined in Comments.

T(n,1) = 1; for k>1: T(n,k) = 3*T(n,k-1) + 2^i, where i is the total number of halving steps from R_(2n-1)(0) to R_(2n-1)(k-1).

A308999 Irregular triangle T(n,k) read by rows: Lexicographically smallest marks on "perfect rulers" (as defined in A103294) of length n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 3, 0, 1, 2, 4, 0, 1, 2, 5, 0, 1, 4, 6, 0, 1, 2, 3, 7, 0, 1, 2, 5, 8, 0, 1, 2, 6, 9, 0, 1, 2, 3, 6, 10, 0, 1, 2, 3, 7, 11, 0, 1, 2, 3, 8, 12, 0, 1, 2, 6, 10, 13, 0, 1, 2, 3, 4, 9, 14, 0, 1, 2, 3, 4, 10, 15, 0, 1, 2, 3, 8, 12, 16

Offset: 0

Bob Selcoe, Jul 04 2019

Refer to A103294 for additional definitions, references and links.
All rulers (rows) start with mark 0 and end with mark n.
Row lengths are A103298(n) + 1.

			Triangle starts:
  0;
  0,  1;
  0,  1,  2;
  0,  1,  3;
  0,  1,  2,  4;
  0,  1,  2,  5;
  0,  1,  4,  6;
  0,  1,  2,  3,  7;
  0,  1,  2,  5,  8;
  0,  1,  2,  6,  9;
  0,  1,  2,  3,  6, 10;
  0,  1,  2,  3,  7, 11;
  0,  1,  2,  3,  8, 12;
  0,  1,  2,  6, 10, 13;
  0,  1,  2,  3,  4,  9, 14;
  0,  1,  2,  3,  4, 10, 15;
  0,  1,  2,  3,  8, 12, 16;

Peter Luschny, The perfect ruler pyramid
Peter Luschny, Perfect rulers

Cf. A103294, A103298.

def Partsum(T) :
    return [add([T[j] for j in range(i)]) for i in (0..len(T))]
def Ruler(L, S) :
    return map(Partsum, Compositions(L, length=S))
def isComplete(R) :
    S = Set([])
    L = len(R)-1
    for i in range(L,0,-1) :
        for j in (1..i) :
            S = S.union(Set([R[i]-R[i-j]]))
    return len(S) == R[L]
def CompleteRuler(L, S) :
    return list(filter(isComplete, Ruler(L, S)))
def PerfectRulers(L) :
    for i in (0..L) :
        R = CompleteRuler(L, i)
        if R: return R
    return []
def A308999list(L):
    for n in (0..L):
        print(PerfectRulers(n)[-1])
A308999list(16) # Peter Luschny, Aug 21 2019

A316625 Terms in A259663, in ascending order.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 19, 21, 23, 31, 35, 47, 53, 55, 63, 79, 85, 87, 95, 99, 127, 143, 151, 191, 213, 223, 227, 255, 271, 319, 341, 351, 383, 407, 483, 511, 575, 663, 739, 767, 783, 853, 863, 895, 1023, 1175, 1251, 1279, 1365, 1407, 1535, 1599, 1807, 1887, 2047

Offset: 1

Bob Selcoe, Jul 08 2018

nonn

See A259663 for discussion of these terms in relation to Collatz sequences.
There are k terms in the interval [2^k, 2^(k+1)], k >= 1; terms in each interval are of the form 2^k + a(n) for some n.
The sequence is a permutation (without repeating terms) of the following numbers:
2^i-1 and 7*2^i-1 when i is odd, i >= 1;
3^2^i-1 and 5^2^i-1 when i is even, i >= 2;
For fixed k >= 4: least residues of 3^j*(2^(2^(k-3) + i*2^(k-2) - j)) - 1 mod 2^(2^(k-3) + i*2^(k-2) + k-j), i >= 0, 0 <= j < 2^(k-3) + i*2^(k-2) .  (See example).

			k=5, i=1 -- terms are least residues of 3^j*2^(12-j)-1 mod 2^(17-j), 0 <= j < 12:
j=0: 4096-1 mod 131072 = 4095;
j=1: 3*2048-1 mod 65536 = 6143;
j=2: 9*1024-1 mod 32768 = 9215;
j=3: 27*512-1 mod 16384 = 13823;
j=4: 81*256-1 mod 8192 = 20735 mod 8192 == 4351;
j=5: 243*128-1 mod 4096 = 31103 mod 4096 == 2431;
j=6: 729*64-1 mod 2048 = 46655 mod 2048 == 1599;
j=7: 2187*32-1 mod 1024 = 69983 mod 1024 == 351;
j=8: 6561*16-1 mod 512 = 104975 mod 512 == 15;
j=9: 19683*8-1 mod 256 = 157463 mod 256 == 23;
j=10: 59049*4-1 mod 128 = 236195 mod 128 == 35;
j=11: 177147*2-1 mod 64 = 354293 mod 64 == 53.
Note: k=5, i=0 is equivalent to starting with j=0: 15 mod 512.

Cf. A259663.

More terms from Michel Marcus, Jul 10 2018

A299109 Probable primes in A030221.

Original entry on oeis.org

29, 139, 3191, 15289, 350981, 1681651, 20344613659, 2237722536169, 5650248517599839, 1464318118372209903213451940281613111, 471219735266432821374794400248484805597413615642086220363989152195627985749609

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Feb 02 2018

From a problem in A269254. For detailed theory, see [Hone].

Subsequent terms have too many digits to display.

Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Positions of primes in A030221, A294099 (row 5).

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269253, A269254, A298675, A298677, A298878, A299045, A299071.

a(n) = A030221(A299101(n)).

A299107 Probable primes in sequence {s_k(4)}, where s_k(4) = 4*s_{k-1}(4) - s_{k-2}(4), k >= 2, s_0(4) = 1, s_1(4) = 5.

Original entry on oeis.org

5, 19, 71, 3691, 191861, 138907099, 26947261171, 436315574686414344004975231616076636245689199862837798457639364993981991744926792179

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Feb 02 2018

From a problem in A269254. For detailed theory, see [Hone].

Subsequent terms have too many digits to display.

Hugo Pfoertner, Table of n, a(n) for n = 1..14
Gary Greaves and Jose Yip, The coherent rank of a graph with three eigenvalues, arXiv:2406.17395 [math.CO], 2024. See p. 31.
Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

A269254, A001834, A294099 (row 4),

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269253, A298675, A298677, A298878, A299045, A299071, A086386.

a(n) = s_{A299100(n)}(4) = A001834(A299100(n)).

A299101 Indices of (probable) primes in A030221.

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 15, 18, 23, 53, 114, 194, 564, 575, 585, 2594, 3143, 4578, 4970, 9261, 11508, 13298, 30018, 54993, 198476

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Feb 02 2018

From a problem in A269254. For detailed theory, see [Hone].

a(25) > 2*10^5. - Robert Price, Jul 03 2020

Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Cf. A269254, A030221, A294099 (row 5).

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269253, A298675, A298677, A298878, A299045, A299071.

-1 + Position[Nest[Append[#, 5 #[[-1]] - #[[-2]] ] &, {1, 6}, 2600], ?PrimeQ][[All, 1]] (* _Michael De Vlieger, Jul 03 2020 *)

A299109(n) = A030221(a(n)). - R. J. Mathar, Jul 22 2022

a(24) from Robert Price, Jul 03 2020

A299100 Indices k such that s_k(4) is a (probable) prime, where s_k(4) = 4*s_{k-1}(4) - s_{k-2}(4), k >= 2, s_0(4) = 1, s_1(4) = 5.

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 18, 146, 216, 293, 704, 1143, 1530, 1593, 2924, 7163, 9176, 9489, 11531, 39543, 50423, 60720, 62868, 69993, 69995, 88103, 88163, 104606, 164441, 178551

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Feb 02 2018

From a problem in A269254. For detailed theory, see [Hone].

a(31) > 2*10^5. - Robert Price, May 29 2020

Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Cf. A269254. Indices of primes in A001834, A294099 (row 4).

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269253, A298675, A298677, A298878, A299045, A299071.

s[k_, m_] := s[k, m] = Which[k == 0, 1, k == 1, 1 + m, True, m s[k - 1, m] - s[k - 2, m]]; Select[Range@ 2000, PrimeQ@ Abs@ s[#, 4] &] (* Michael De Vlieger, Feb 03 2018 *)

a(24)-a(30) from Robert Price, May 29 2020

A299071 Union_{odd primes p, n >= 3} {T_p(n)}, where T_m(x) = x*T_{m-1}(x) - T_{m-2}(x), m >= 2, T_0(x) = 2, T_1(x) = x (dilated Chebyshev polynomials of the first kind).

Original entry on oeis.org

18, 52, 110, 123, 198, 488, 702, 724, 843, 970, 1298, 1692, 2158, 2525, 3330, 4048, 4862, 5778, 6726, 6802, 7940, 9198, 10084, 10582, 13752, 15550, 17498, 19602, 21868, 24302, 26910, 29698, 30248, 32672, 35838, 39603, 42770, 46548, 50542

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Feb 01 2018

nonn

From a problem in A269254. For detailed theory, see [Hone].

Sequence avoids numbers of the form T_p(T_2(j)).

Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Cf. A008865 (T_2(n)), A298878 (T_p(n), p prime).

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269253, A269254, A294099, A298675, A298677, A299045, A299071.

maxT = 55000; maxn = 12;
T[0][] = 2; T[1][x] = x;
T[m_][x_] := T[m][x] = x T[m-1][x] - T[m-2][x];
TT = Table[T[p][n], {p, Prime[Range[2, maxn]]}, {n, 3, Prime[maxn]}] // Flatten // Union // Select[#, # <= maxT&]&;
avoid = Table[T[p][T[2][n]], {p, Prime[Range[2, maxn]]}, {n, 3, Prime[maxn] }] // Flatten // Union // Select[#, # <= maxT&]&;
Complement[TT, avoid] (* Jean-François Alcover, Nov 03 2018 *)

cp's OEIS Frontend

User: Bob Selcoe

A330885 Square array T(n,k) read by antidiagonals upwards: T(n,0)=1; T(n,1) = n+1; T(n,2) = 2n+1, T(n,k>2) = T(n,k-1) - T(n,k-2) - T(n,k-3).

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A328037 Irregular triangle T(n,k) read by rows: "quotient trajectories" in reduced Collatz sequences; i.e., T(n,k) = q-value(A256598(n,k)) where q-value(z) = (z - A259663(m,j))/2^(m+j) and (m,j) is the unique pair such that z == A259663(m,j) (mod 2^(m+j)). (See Comments for definitions.)

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PARI

A327283 Irregular triangle T(n,k) read by rows: "residual summands" in reduced Collatz sequences (see Comments for definition and explanation).

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A308999 Irregular triangle T(n,k) read by rows: Lexicographically smallest marks on "perfect rulers" (as defined in A103294) of length n.

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A316625 Terms in A259663, in ascending order.

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A299109 Probable primes in A030221.

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A299107 Probable primes in sequence {s_k(4)}, where s_k(4) = 4*s_{k-1}(4) - s_{k-2}(4), k >= 2, s_0(4) = 1, s_1(4) = 5.

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A299101 Indices of (probable) primes in A030221.

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Extensions

A299100 Indices k such that s_k(4) is a (probable) prime, where s_k(4) = 4*s_{k-1}(4) - s_{k-2}(4), k >= 2, s_0(4) = 1, s_1(4) = 5.

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