A120880 -id:A120880 - cp's OEIS frontend

A374361 Irregular table T(n, k), n >= 0, 0 <= k < A120880(n), read by rows; the n-th row contains the terms t of A005836 such that n - t also belongs to A005836.

0, 0, 1, 1, 0, 3, 0, 1, 3, 4, 1, 4, 3, 3, 4, 4, 0, 9, 0, 1, 9, 10, 1, 10, 0, 3, 9, 12, 0, 1, 3, 4, 9, 10, 12, 13, 1, 4, 10, 13, 3, 12, 3, 4, 12, 13, 4, 13, 9, 9, 10, 10, 9, 12, 9, 10, 12, 13, 10, 13, 12, 12, 13, 13, 0, 27, 0, 1, 27, 28, 1, 28, 0, 3, 27, 30, 0, 1, 3, 4, 27, 28, 30, 31

Offset: 0

Rémy Sigrist, Jul 06 2024

In other words, we partition n into pairs of terms of A005836 and list the corresponding terms to get the n-th row.

			Triangle T(n, k) begins:
  n   n-th row
  --  -----------
   0  0
   1  0, 1
   2  1
   3  0, 3
   4  0, 1, 3, 4
   5  1, 4
   6  3
   7  3, 4
   8  4
   9  0, 9
  10  0, 1, 9, 10
  11  1, 10
  12  0, 3, 9, 12

Rémy Sigrist, Table of n, a(n) for n = 0..4095 (rows for n = 0..3^6-1 flattened)

See A374354 for a similar sequence.

Cf. A005836, A120880, A374362, A374363.

row(n) = { my (r = [0], t = 1, d); while (n, d = n % 3; n \= 3; if (d==1, r = concat(r, [v + t | v <- r]), d==2, r = [v + t | v <- r]); t *= 3;); return (r); }

T(n, 0) = 0 iff n belongs to A005836.
T(n, k) + T(n, A120880(k)-1-k) = n.
T(n, 0) = A374362(n).
T(n, A120880(k)-1) = A374363(n).

A374791 Irregular triangle T(n, k), n >= 0, k = 1..A120880(n), read by rows; the n-th lists the integers m such that A374560(m) = n.

Original entry on oeis.org

0, 1, 2, 4, 3, 5, 6, 7, 8, 9, 11, 13, 12, 17, 18, 24, 10, 14, 15, 16, 19, 20, 22, 26, 21, 23, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 39, 41, 43, 38, 42, 47, 48, 51, 52, 58, 62, 40, 49, 50, 60, 59, 61, 70, 71, 72, 73, 83, 85, 84, 97, 98, 112, 36, 44, 45, 46, 53, 54

Offset: 0

Rémy Sigrist, Jul 20 2024

A374560 corresponds to a square array, but we consider it here as a flat sequence (when its values are read according along its antidiagonals).

As a flat sequence, this is a permutation of the nonnegative integers with inverse A374792.

			Triangle T(n, k) begins:
  n   n-th row
  --  ------------------------------
   0  0
   1  1, 2
   2  4
   3  3, 5
   4  6, 7, 8, 9
   5  11, 13
   6  12
   7  17, 18
   8  24
   9  10, 14
  10  15, 16, 19, 20
  11  22, 26
  12  21, 23, 25, 27
  13  28, 29, 30, 31, 32, 33, 34, 35
  14  37, 39, 41, 43
  15  38, 42

Rémy Sigrist, Table of n, a(n) for n = 0..8255
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A120880, A374560, A374792 (inverse).

PARI
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A309677 G.f. A(x) satisfies: A(x) = A(x^3) / (1 - x)^2.

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 18, 24, 30, 42, 54, 66, 87, 108, 129, 162, 195, 228, 279, 330, 381, 456, 531, 606, 711, 816, 921, 1068, 1215, 1362, 1563, 1764, 1965, 2232, 2499, 2766, 3120, 3474, 3828, 4290, 4752, 5214, 5805, 6396, 6987, 7740, 8493, 9246, 10194, 11142

Offset: 0

Ilya Gutkovskiy, Aug 12 2019

nonn

Self-convolution of A062051.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A005704, A062051, A120880, A171238, A309678, A309679.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      b(n, i-1)+(p-> `if`(p>n, 0, b(n-p, i)))(3^i)))
    end:
a:= n-> add(b(j, ilog[3](j))*b(n-j, ilog[3](n-j)), j=0..n):
seq(a(n), n=0..52);  # Alois P. Heinz, Aug 12 2019

nmax = 52; A[] = 1; Do[A[x] = A[x^3]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 52; CoefficientList[Series[Product[1/(1 - x^(3^k))^2, {k, 0, Floor[Log[3, nmax]] + 1}], {x, 0, nmax}], x]

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

A374363 a(n) is the greatest term t <= n of A005836 such that n - t also belongs to A005836.

Original entry on oeis.org

0, 1, 1, 3, 4, 4, 3, 4, 4, 9, 10, 10, 12, 13, 13, 12, 13, 13, 9, 10, 10, 12, 13, 13, 12, 13, 13, 27, 28, 28, 30, 31, 31, 30, 31, 31, 36, 37, 37, 39, 40, 40, 39, 40, 40, 36, 37, 37, 39, 40, 40, 39, 40, 40, 27, 28, 28, 30, 31, 31, 30, 31, 31, 36, 37, 37, 39, 40

Offset: 0

Rémy Sigrist, Jul 06 2024

To compute a(n): in the ternary expansion of n, 2's by 1's.

			The first terms, in decimal and in ternary, are:
  n   a(n)  ter(n)  ter(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     1       2          1
   3     3      10         10
   4     4      11         11
   5     4      12         11
   6     3      20         10
   7     4      21         11
   8     4      22         11
   9     9     100        100
  10    10     101        101
  11    10     102        101
  12    12     110        110
  13    13     111        111
  14    13     112        111
  15    12     120        110

Rémy Sigrist, Table of n, a(n) for n = 0..6560

Cf. A005836, A120880, A289831, A374361, A374362.

a(n) = fromdigits(apply(d -> [0, 1, 1][1+d], digits(n, 3)), 3)

a(n) = T(n, A120880(k)-1).
a(n) = n - A374362(n).
a(n) <= n with equality iff n belongs to A005836.
a(n) = A005836(1+A289831(n)).

A374560 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = A005836(n+1) + A005836(k+1).

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 4, 4, 4, 4, 9, 5, 6, 5, 9, 10, 10, 7, 7, 10, 10, 12, 11, 12, 8, 12, 11, 12, 13, 13, 13, 13, 13, 13, 13, 13, 27, 14, 15, 14, 18, 14, 15, 14, 27, 28, 28, 16, 16, 19, 19, 16, 16, 28, 28, 30, 29, 30, 17, 21, 20, 21, 17, 30, 29, 30, 31, 31, 31, 31, 22, 22, 22, 22, 31, 31, 31, 31

Offset: 0

Rémy Sigrist, Jul 20 2024

For any v >= 0, the value v appears A120880(v) times.

			Array A(n, k) begins:
  n\k |  0   1   2   3   4   5   6   7   8   9  10  11  12
  ----+---------------------------------------------------
    0 |  0   1   3   4   9  10  12  13  27  28  30  31  36
    1 |  1   2   4   5  10  11  13  14  28  29  31  32  37
    2 |  3   4   6   7  12  13  15  16  30  31  33  34  39
    3 |  4   5   7   8  13  14  16  17  31  32  34  35  40
    4 |  9  10  12  13  18  19  21  22  36  37  39  40  45
    5 | 10  11  13  14  19  20  22  23  37  38  40  41  46
    6 | 12  13  15  16  21  22  24  25  39  40  42  43  48
    7 | 13  14  16  17  22  23  25  26  40  41  43  44  49
    8 | 27  28  30  31  36  37  39  40  54  55  57  58  63
    9 | 28  29  31  32  37  38  40  41  55  56  58  59  64
   10 | 30  31  33  34  39  40  42  43  57  58  60  61  66
   11 | 31  32  34  35  40  41  43  44  58  59  61  62  67
   12 | 36  37  39  40  45  46  48  49  63  64  66  67  72

Rémy Sigrist, Table of n, a(n) for n = 0..8255 (antidiagonals for n+k = 0..127 flattened)

Cf. A005836, A120880, A374789, A374790, A374791, A374792.

A(n, k) = fromdigits(binary(n), 3) + fromdigits(binary(k), 3)

A(n, k) = A(k, n).

A(2*n, 2*k) = 3*A(n, k).

A205565 Number of ways of writing n = u + v with u <= v, and u,v having in ternary representation no 3.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 4, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 4, 2, 1, 2, 1, 2, 4, 2, 4, 8, 4, 2, 4, 2, 1, 2, 1, 2, 4, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 4, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 4

Offset: 0

Reinhard Zumkeller, Jan 28 2012

nonn

			a(30) = #{0+30, 3+27} = 2;
a(31) = #{0+31, 1+30, 3+28, 4+27} = 4;
a(32) = #{1+31, 4+28} = 2;
a(33) = #{3+30} = 1;
a(34) = #{3+31, 4+30} = 2;
a(35) = #{4+31} = 1;
a(36) = #{0+36, 9+27} = 2;
a(37) = #{0+37, 1+36, 9+28, 10+27} = 4;
a(38) = #{1+37, 10+28} = 2;
a(39) = #{0+39, 3+36, 9+30, 12+27} = 4;
a(40) = #{0+40, 1+39, 3+37, 4+36, 9+31, 10+30, 12+28, 13+27} = 8.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A039966, A005836, A120880.

a205565 n = sum $ map (a039966 . (n -)) $
                  takeWhile (<= n `div` 2) a005836_list

A120879 G.f. satisfies: A(x) = A(x^3)(1 + 3x + 2*x^2).

Original entry on oeis.org

1, 3, 2, 3, 9, 6, 2, 6, 4, 3, 9, 6, 9, 27, 18, 6, 18, 12, 2, 6, 4, 6, 18, 12, 4, 12, 8, 3, 9, 6, 9, 27, 18, 6, 18, 12, 9, 27, 18, 27, 81, 54, 18, 54, 36, 6, 18, 12, 18, 54, 36, 12, 36, 24, 2, 6, 4, 6, 18, 12, 4, 12, 8, 6, 18, 12, 18, 54, 36, 12, 36, 24, 4, 12, 8, 12, 36, 24, 8, 24, 16, 3, 9

Offset: 0

Paul D. Hanna, Jul 11 2006

a(n) = 3^A062756(n) * 2^A081603(n), where A062756(n) is the number of 1's and A081603(n) is the number of 2's, in the ternary expansion of n.

More generally, if g.f. of {a(n)} satisfies: A(x) = A(x^d)*(1+Sum_{k=1..d-1} c(k)*x^k), then a(n) = Product_{k=1..d-1} c(k)^digits(n,k,d), where digits(n,k,d) is the number of k's in the d-ary expansion of n and d is any integer > 1. This sequence is a simple example for d=3 with c(1)=3 and c(2)=2.

Paul D. Hanna, Table of n, a(n) for n = 0..10000

Cf. A120880, A062756, A081603.

{a(n)=local(A=1+x+x*O(x^n));for(i=1,floor(log(n+1)/log(3))+1, A=subst(A,x,x^3+x*O(x^n))*(1+3*x+2*x^2));polcoeff(A,n,x)}

/* Recurrence: */ {a(n)=if(n==0,1,a(n\3)*3^((n%3)%2)*2^((n%3)\2))}

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

a(n) = a(floor(n/3)) * 3^((n mod 3) mod 2) * 2^floor((n mod 3)/2) with a(0)=1.

A374628 Expansion of Product_{k>=0} (1 + x^(3^k))^3.

Original entry on oeis.org

1, 3, 3, 4, 9, 9, 6, 9, 9, 7, 12, 12, 13, 27, 27, 18, 27, 27, 15, 18, 18, 15, 27, 27, 18, 27, 27, 16, 21, 21, 19, 36, 36, 24, 36, 36, 25, 39, 39, 40, 81, 81, 54, 81, 81, 45, 54, 54, 45, 81, 81, 54, 81, 81, 42, 45, 45, 33, 54, 54, 36, 54, 54, 33, 45, 45, 42, 81, 81, 54, 81, 81, 45, 54, 54, 45, 81, 81, 54, 81, 81

Offset: 0

Seiichi Manyama, Jul 14 2024

nonn

Cf. A039966, A120880.

my(N=90, x='x+O('x^N)); Vec(prod(k=0, logint(N, 3), 1+x^3^k)^3)

G.f. A(x) satisfies A(x) = (1 + x)^3 * A(x^3).

cp's OEIS Frontend

A374361 Irregular table T(n, k), n >= 0, 0 <= k < A120880(n), read by rows; the n-th row contains the terms t of A005836 such that n - t also belongs to A005836.

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A374791 Irregular triangle T(n, k), n >= 0, k = 1..A120880(n), read by rows; the n-th lists the integers m such that A374560(m) = n.

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PARI

A309677 G.f. A(x) satisfies: A(x) = A(x^3) / (1 - x)^2.

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Maple

Mathematica

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A374363 a(n) is the greatest term t <= n of A005836 such that n - t also belongs to A005836.

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PARI

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A374560 Square array A(n, k), n, k >= 0, read by antidiagonals; A(n, k) = A005836(n+1) + A005836(k+1).

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PARI

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A205565 Number of ways of writing n = u + v with u <= v, and u,v having in ternary representation no 3.

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Haskell

A120879 G.f. satisfies: A(x) = A(x^3)*(1 + 3*x + 2*x^2).

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PARI

PARI

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A374628 Expansion of Product_{k>=0} (1 + x^(3^k))^3.

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A120879 G.f. satisfies: A(x) = A(x^3)(1 + 3x + 2*x^2).