A342637 -id:A342637 - cp's OEIS frontend

A342603 a(0) = 0, a(1) = 1; a(2n) = a(n), a(2n+1) = 6*a(n) + a(n+1).

0, 1, 1, 7, 1, 13, 7, 43, 1, 19, 13, 85, 7, 85, 43, 259, 1, 25, 19, 127, 13, 163, 85, 517, 7, 127, 85, 553, 43, 517, 259, 1555, 1, 31, 25, 169, 19, 241, 127, 775, 13, 241, 163, 1063, 85, 1027, 517, 3109, 7, 169, 127, 847, 85, 1063, 553, 3361, 43, 775, 517, 3361, 259, 3109, 1555, 9331, 1

Offset: 0

Ilya Gutkovskiy, Mar 17 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..16384

Cf. A002487, A003464, A016921, A116528, A178243, A342633, A342634, A342635, A342636, A342637, A342638.

a:= proc(n) option remember; `if`(n<2, n, (q->
     `if`(d=1, 6*a(q)+a(q+1), a(q)))(iquo(n, 2, 'd')))
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Mar 17 2021

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], 6 a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 64}]
nmax = 64; CoefficientList[Series[x Product[(1 + x^(2^k) + 6 x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

G.f.: x * Product_{k>=0} (1 + x^(2^k) + 6*x^(2^(k+1))).

a(2^n-1) = (6^n - 1)/5 = A003464(n); a(2^n) = 1; a(2^n+1) = 6*n + 1 = A016921(n). - Alois P. Heinz, Mar 17 2021

A342633 a(0) = 0, a(1) = 1; a(2n) = a(n), a(2n+1) = 3*a(n) + a(n+1).

Original entry on oeis.org

0, 1, 1, 4, 1, 7, 4, 13, 1, 10, 7, 25, 4, 25, 13, 40, 1, 13, 10, 37, 7, 46, 25, 79, 4, 37, 25, 88, 13, 79, 40, 121, 1, 16, 13, 49, 10, 67, 37, 118, 7, 67, 46, 163, 25, 154, 79, 241, 4, 49, 37, 136, 25, 163, 88, 277, 13, 118, 79, 277, 40, 241, 121, 364, 1, 19, 16, 61, 13, 88, 49, 157

Offset: 0

Ilya Gutkovskiy, Mar 17 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..16384

Cf. A002487, A116528, A178243, A342603, A342634, A342635, A342636, A342637, A342638.

a:= proc(n) option remember; `if`(n<2, n, (q->
     `if`(d=1, 3*a(q)+a(q+1), a(q)))(iquo(n, 2, 'd')))
    end:
seq(a(n), n=0..71);  # Alois P. Heinz, Mar 17 2021

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], 3 a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 71}]
nmax = 71; CoefficientList[Series[x Product[(1 + x^(2^k) + 3 x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

G.f.: x * Product_{k>=0} (1 + x^(2^k) + 3*x^(2^(k+1))).

A342634 a(0) = 0, a(1) = 1; a(2n) = a(n), a(2n+1) = 4*a(n) + a(n+1).

Original entry on oeis.org

0, 1, 1, 5, 1, 9, 5, 21, 1, 13, 9, 41, 5, 41, 21, 85, 1, 17, 13, 61, 9, 77, 41, 169, 5, 61, 41, 185, 21, 169, 85, 341, 1, 21, 17, 81, 13, 113, 61, 253, 9, 113, 77, 349, 41, 333, 169, 681, 5, 81, 61, 285, 41, 349, 185, 761, 21, 253, 169, 761, 85, 681, 341, 1365, 1, 25, 21, 101, 17

Offset: 0

Ilya Gutkovskiy, Mar 17 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..16384

Cf. A002487, A116528, A178243, A342603, A342633, A342635, A342636, A342637, A342638.

a:= proc(n) option remember; `if`(n<2, n, (q->
     `if`(d=1, 4*a(q)+a(q+1), a(q)))(iquo(n, 2, 'd')))
    end:
seq(a(n), n=0..68);  # Alois P. Heinz, Mar 17 2021

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], 4 a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 68}]
nmax = 68; CoefficientList[Series[x Product[(1 + x^(2^k) + 4 x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

G.f.: x * Product_{k>=0} (1 + x^(2^k) + 4*x^(2^(k+1))).

a(n) == 1 (mod 4) for n >= 1. - Hugo Pfoertner, Mar 17 2021

A342635 a(0) = 0, a(1) = 1; a(2n) = a(n), a(2n+1) = 5*a(n) + a(n+1).

Original entry on oeis.org

0, 1, 1, 6, 1, 11, 6, 31, 1, 16, 11, 61, 6, 61, 31, 156, 1, 21, 16, 91, 11, 116, 61, 311, 6, 91, 61, 336, 31, 311, 156, 781, 1, 26, 21, 121, 16, 171, 91, 466, 11, 171, 116, 641, 61, 616, 311, 1561, 6, 121, 91, 516, 61, 641, 336, 1711, 31, 466, 311, 1711, 156, 1561, 781, 3906, 1, 31, 26

Offset: 0

Ilya Gutkovskiy, Mar 17 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..16384

Cf. A002487, A116528, A178243, A342603, A342633, A342634, A342636, A342637, A342638.

a:= proc(n) option remember; `if`(n<2, n, (q->
     `if`(d=1, 5*a(q)+a(q+1), a(q)))(iquo(n, 2, 'd')))
    end:
seq(a(n), n=0..66);  # Alois P. Heinz, Mar 17 2021

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], 5 a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 66}]
nmax = 66; CoefficientList[Series[x Product[(1 + x^(2^k) + 5 x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

G.f.: x * Product_{k>=0} (1 + x^(2^k) + 5*x^(2^(k+1))).

a(n) == 1 (mod 5) for n >= 1. - Hugo Pfoertner, Mar 17 2021

A342636 a(0) = 0, a(1) = 1; a(2n) = a(n), a(2n+1) = 7*a(n) + a(n+1).

Original entry on oeis.org

0, 1, 1, 8, 1, 15, 8, 57, 1, 22, 15, 113, 8, 113, 57, 400, 1, 29, 22, 169, 15, 218, 113, 799, 8, 169, 113, 848, 57, 799, 400, 2801, 1, 36, 29, 225, 22, 323, 169, 1198, 15, 323, 218, 1639, 113, 1590, 799, 5601, 8, 225, 169, 1296, 113, 1639, 848, 5993, 57, 1198, 799, 5993, 400, 5601, 2801

Offset: 0

Ilya Gutkovskiy, Mar 17 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..16384

Cf. A002487, A116528, A178243, A342603, A342633, A342634, A342635, A342637, A342638.

a:= proc(n) option remember; `if`(n<2, n, (q->
     `if`(d=1, 7*a(q)+a(q+1), a(q)))(iquo(n, 2, 'd')))
    end:
seq(a(n), n=0..62);  # Alois P. Heinz, Mar 17 2021

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], 7 a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 62}]
nmax = 62; CoefficientList[Series[x Product[(1 + x^(2^k) + 7 x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

G.f.: x * Product_{k>=0} (1 + x^(2^k) + 7*x^(2^(k+1))).

A342638 a(0) = 0, a(1) = 1; a(2n) = a(n), a(2n+1) = 9*a(n) + a(n+1).

Original entry on oeis.org

0, 1, 1, 10, 1, 19, 10, 91, 1, 28, 19, 181, 10, 181, 91, 820, 1, 37, 28, 271, 19, 352, 181, 1639, 10, 271, 181, 1720, 91, 1639, 820, 7381, 1, 46, 37, 361, 28, 523, 271, 2458, 19, 523, 352, 3349, 181, 3268, 1639, 14761, 10, 361, 271, 2620, 181, 3349, 1720, 15571, 91, 2458, 1639, 15571, 820

Offset: 0

Ilya Gutkovskiy, Mar 17 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..16384

Cf. A002487, A116528, A178243, A342603, A342633, A342634, A342635, A342636, A342637.

a:= proc(n) option remember; `if`(n<2, n, (q->
     `if`(d=1, 9*a(q)+a(q+1), a(q)))(iquo(n, 2, 'd')))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 17 2021

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], 9 a[(n - 1)/2] + a[(n + 1)/2]]; Table[a[n], {n, 0, 60}]
nmax = 60; CoefficientList[Series[x Product[(1 + x^(2^k) + 9 x^(2^(k + 1))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]

G.f.: x * Product_{k>=0} (1 + x^(2^k) + 9*x^(2^(k+1))).

A178239 Triangle read by rows, antidiagonals of an array generated from a(n) = a(2n), a(2n+1) = r*a(n) + a(n+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 5, 1, 5, 2, 1, 1, 1, 6, 1, 7, 3, 3, 1, 1, 1, 7, 1, 9, 4, 7, 1, 1, 1, 1, 8, 1, 11, 5, 13, 1, 4, 1, 1, 1, 9, 1, 13, 6, 21, 1, 7, 3, 1, 1, 1, 10, 1, 15, 7, 31, 1, 10, 5, 5, 1

Offset: 0

Gary W. Adamson, May 23 2010

Partial sums of array terms in groups of 1, next 2, next 4, ... 8 = powers of (r+2).
Row sums = A178240: (1, 2, 3, 5, 7, 11, 16, 23, ...).
Row 1 of the array = A002487.
Row 2 = .............A116528.
Row 3 = .............A342633.
Row 4 = .............A342634.
...
Row 10 = ............A178243.
Polcoeff row r of the array as f(x) satisfies f(x)/f(x^2) = (1 + x + r*x^2).
Let q(x) = (1 + x + r*x^2). Then polcoeff row 4 = q(x) * q(x^2) * q(x^4) * q(x^8) * ...

			First few rows of the array =
      n=1  n=2  n=3  n=4  n=5  n=6  n=7  n=8  n=9 n=10 n=11 n=12 n=13 n=14 n=15
  r=0:  1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
  r=1:  1,   1,   2,   1,   3,   2,   3,   1,   4,   3,   5,   2,   5,   3,   4, ...
  r=2:  1,   1,   3,   1,   5,   3,   7,   1,   7,   5,  13,   3,  13,   7,  15, ...
  r=3:  1,   1,   4,   1,   7,   4,  13,   1,  10,   7,  25,   4,  25,  13,  40, ...
  r=4:  1,   1,   5,   1,   9,   5,  21,   1,  13,   9,  41,   5,  41,  21,  85, ...
  r=5:  1,   1,   6,   1,  11,   6,  31,   1,  16,  11,  61,   6,  61,  31, 156, ...
  ...
Example: In row 3: (1, 1, 4, 1, 7, 4, 13, ...) = A342633, r = 3.
A342633(7) = 13 = 3*4 + 1. In blocks of 1, 2, 4, 8, ... terms, partial sums are powers of (r+2) = 5: (1, 5, 25, ...).
First few rows of the triangle =
  1;
  1, 1;
  1, 1,  1;
  1, 1,  2, 1;
  1, 1,  3, 1,  1;
  1, 1,  4, 1,  3,  1;
  1, 1,  5, 1,  5,  2,  1;
  1, 1,  6, 1,  7,  3,  3, 1;
  1, 1,  7, 1,  9,  4,  7, 1,  1;
  1, 1,  8, 1, 11,  5, 13, 1,  4,  1;
  1, 1,  9, 1, 13,  6, 21, 1,  7,  3,  1;
  1, 1, 10, 1, 15,  7, 31, 1, 10,  5,  5, 1;
  1, 1, 11, 1, 17,  8, 43, 1, 13,  7, 13, 2,  1;
  1, 1, 12, 1, 19,  9, 57, 1, 16,  9, 21, 3,  5, 1;
  1, 1, 13, 1, 21, 11, 73, 1, 19, 11, 31, 4, 13, 2, 1;
  ...

Cf. A178240, A359250 (column polynomials).

Array rows r=1..10: A002487, A116528, A342633, A342634, A342635, A342603, A342636, A342637, A342638, A178243.

Antidiagonals of an array generated from a(n) = a(2n); a(2n+1) = r*a(n) + a(n+1).

Given a triangle M with columns stepped down twice from the previous column, for columns > 0, with (1, 1, r, 0, 0, 0, ...) in each column, r-th row of the array = lim_{n->oo} M^n.

A359250 Irregular triangle read by rows where T(n,k) is the coefficient of y^k in polynomial P(n) defined by P(2n) = P(n) and P(2n+1) = y*P(n) + P(n+1) starting P(0) = 0, P(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 3, 1, 3, 3, 1, 2, 1, 3, 4, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 3, 3, 1, 2, 2, 1, 2, 3, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 4, 1, 4, 4

Offset: 0

Kevin Ryde, Dec 28 2022

Row n length is wt(n) = A000120(n), the binary weight of n, so that k ranges 0 <= k < wt(n).
Evaluated at y=1, the recurrence for P is per Stern's diatomic sequence so that row n has sum A002487(n).
Evaluated at y=2, the recurrence for P is per A116528 so that Sum T(n,k)*2^k = A116528(n), and similarly variations such as y=10 for A178243.
Array A178239(r,n) is P(n) evaluated at y=r, and in particular P(n) is the polynomial for the values in column n there.
Column k=1, when that value exists, is T(n,1) = A080791(n-1), the number of 0 bits in n-1, since expressing the recurrence in Q(m) = P(m+1) adds y*Q(m-1) at each 0 bit in m.
Reversing the bits of n is no change to P(n), so that P(n) = P(A030101(n)).

			Triangle begins:
      k=0  1  2
  n=0:  (empty)
  n=1:  1,
  n=2:  1,
  n=3:  1, 1,
  n=4:  1,
  n=5:  1, 2,
  n=6:  1, 1,
  n=7:  1, 1, 1,
  n=8:  1,
  n=9:  1, 3,

Kevin Ryde, Table of n, a(n) for rows 0..2048, flattened
Kevin Ryde, PARI/GP Code and Notes
Thomas Scheuerle, Scatter plot a(n) for n = 1 to 114689. This is in T(n,k) the range of n = 1 to 2^14. It looks a bit like narrow and tall fir trees like you could see in the Schwarzwald region.

Cf. A000120 (row lengths), A002487 (row sums).
Cf. A178239 (array), A030101 (bit reversal).
Evaluated at y=2..10: A116528, A342633, A342634, A342635, A342603, A342636, A342637, A342638, A178243.
Cf. A125184.

function a = A359250( max_n )
    ac = cell(1,1); ac{1} = [1];
    for n = 2:max_n
        m = floor(n/2);
        if 2*m == n
            ac{n} = ac{m};
        else
            ac{n} = [ac{m+1} zeros(1,(length(ac{m})+1)-length(ac{m+1}))] ...
            + [0 ac{m}];
        end
    end
    a = cell2mat(ac);
end % Thomas Scheuerle, Dec 28 2022

PARI
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G.f.: Sum_{n>=0} P(n)*x^n = x * Product_{e>=0} 1 + x^(2^e) + y*x^(2^(e+1)).

cp's OEIS Frontend

A342603 a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = 6*a(n) + a(n+1).

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A342633 a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = 3*a(n) + a(n+1).

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A342634 a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = 4*a(n) + a(n+1).

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A342635 a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = 5*a(n) + a(n+1).

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A342636 a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = 7*a(n) + a(n+1).

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A342638 a(0) = 0, a(1) = 1; a(2*n) = a(n), a(2*n+1) = 9*a(n) + a(n+1).

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A178239 Triangle read by rows, antidiagonals of an array generated from a(n) = a(2n), a(2n+1) = r*a(n) + a(n+1).

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A359250 Irregular triangle read by rows where T(n,k) is the coefficient of y^k in polynomial P(n) defined by P(2n) = P(n) and P(2n+1) = y*P(n) + P(n+1) starting P(0) = 0, P(1) = 1.

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A342603 a(0) = 0, a(1) = 1; a(2n) = a(n), a(2n+1) = 6*a(n) + a(n+1).

A342633 a(0) = 0, a(1) = 1; a(2n) = a(n), a(2n+1) = 3*a(n) + a(n+1).

A342634 a(0) = 0, a(1) = 1; a(2n) = a(n), a(2n+1) = 4*a(n) + a(n+1).

A342635 a(0) = 0, a(1) = 1; a(2n) = a(n), a(2n+1) = 5*a(n) + a(n+1).

A342636 a(0) = 0, a(1) = 1; a(2n) = a(n), a(2n+1) = 7*a(n) + a(n+1).

A342638 a(0) = 0, a(1) = 1; a(2n) = a(n), a(2n+1) = 9*a(n) + a(n+1).