A141023 -id:A141023 - cp's OEIS frontend

A309597 a(n) is the A325907(n)-th triangular number.

6, 666, 5656566, 555665666566566, 5555555666655656666556566566566, 555555555555555666666665555665666666666555566566666556566566566

Offset: 1

Seiichi Manyama, Sep 14 2019

nonn

a(n) decimal expansion includes A141023(n-1) 5's and A052950(n) 6's in digits.

All terms are elements of A213516.

			a(1) =               6 =               6 +        0 +    0 * 10^1.
a(2) =             666 =             556 +       10 +    1 * 10^2.
a(3) =         5656566 =         5555556 +     1010 +   10 * 10^4.
a(4) = 555665666566566 = 555555555555556 + 11011010 + 1101 * 10^8.
------------------------------------------------------------------
a(2) =                                 6 6 6. (            3 6's)
                                       - -
a(3) =                           5 65 65  66. ( 3 5's and  4 6's)
                                 - -- --
a(4) =                  555 6656 6656   6566. ( 6 5's and  9 6's)
                        --- ---- ----
a(5) = 5555555 66665565 66665565    66566566. (15 5's and 16 6's)
       ------- -------- --------

Seiichi Manyama, Table of n, a(n) for n = 1..9

Cf. A000217, A052950, A093142, A141023, A213516, A325907, A325910, A325493.

def A325907(n)
  a = [3]
  (2..n).each{|i|
    j = 10 ** (2 ** (i - 2))
    a << (j + 3) * (j - 1) / 3 - a[-1]
  }
  a
end
def A309597(n)
  A325907(n).map{|i| i * (i + 1) / 2}
end
p A309597(10)

a(n) = A000217(A325907(n)).

a(n) = A093142(2^n - 1) + A325493(n-1) + A325910(n-1) * 10^(2^(n-1)).

A290258 Triangle read by rows: row n (>=2) contains in increasing order the integers for which the binary representation has length n and all runs of 1's have even length.

Original entry on oeis.org

3, 6, 12, 15, 24, 27, 30, 48, 51, 54, 60, 63, 96, 99, 102, 108, 111, 120, 123, 126, 192, 195, 198, 204, 207, 216, 219, 222, 240, 243, 246, 252, 255, 384, 387, 390, 396, 399, 408, 411, 414, 432, 435, 438, 444, 447, 480, 483, 486, 492, 495, 504, 507, 510

Offset: 2

Emeric Deutsch, Sep 12 2017

The viabin numbers of integer partitions having only even parts. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [6,4,4,2]. The southeast border of its Ferrers board yields 110110011 (length is 9), leading to the viabin number 435 (a term in row 9).
Number of entries in row n is the Fibonacci number F(n-1) = A000045(n-1).
T(n,k) = A290259(n-1,k) + 2^(n-1).
Last entry in row n = A141023(n).
Basically the same as A277335.

			399 is in the sequence because all the runs of 1's of its binary representation, namely 110001111, have even lengths.
Triangle begins:
  3;
  6;
  12,15;
  24,27,30;
  48,51,54,60,63;
  96,99,102,108,111,120,123,126;
  ...

Cf. A000045, A141023, A277335, A290259.

A[2] := {3}; A[3] := {6}; for n from 4 to 10 do A[n] := `union`(map(proc (x) 2*x end proc, A[n-1]), map(proc (x) 4*x+3 end proc, A[n-2])) end do; # yields sequence in triangular form

A[2] = {3}; A[3] = {6};
For[n = 4, n <= 10, n++, A[n] = Union[2 A[n-1], 4 A[n-2] + 3]];
Table[A[n], {n, 2, 10}] // Flatten (* Jean-François Alcover, Aug 19 2024, after Maple program *)

The entries in row n (n>=4) are: (i) 2x, where x is in row n-1 and (ii) 4y + 3, where y is in row n-2. The Maple program is based on this.

A290604 a(0) = 2, a(1) = 2; for n > 1, a(n) = a(n-1) + 2*a(n-2) + 3.

Original entry on oeis.org

2, 2, 9, 16, 37, 72, 149, 296, 597, 1192, 2389, 4776, 9557, 19112, 38229, 76456, 152917, 305832, 611669, 1223336, 2446677, 4893352, 9786709, 19573416, 39146837, 78293672, 156587349, 313174696, 626349397, 1252698792, 2505397589, 5010795176, 10021590357

Offset: 0

Iain Fox, Oct 11 2017

Ratio of successive terms approaches 2.

			a(0) = 2.
a(1) = 2.
a(2) = 2 + 2*2 + 3 = 9.
a(3) = 9 + 2*2 + 3 = 16.
a(4) = 16 + 9*2 + 3 = 37.
...

Iain Fox, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2)

[(2^(n+2)+2*(-1)^n)/3+2^n-(3-(-1)^n)/2: n in [0..40]]; // Vincenzo Librandi, Oct 20 2017

Table[(2^(n + 2) + 2 (-1)^n) / 3 + 2^n - (3 - (-1)^n) / 2, {n, 0, 40}] (* Vincenzo Librandi, Oct 20 2017 *)
LinearRecurrence[{2,1,-2},{2,2,9},50] (* Harvey P. Dale, Mar 06 2025 *)

Vec((2/(1-x-2*x^2)) + (3*x^2/((1-x)*(1-x-2*x^2))) + O(x^50)) \\ Michel Marcus, Oct 12 2017

first(n) = Vec((2 - 2*x + 3*x^2)/(1 - 2*x - x^2 + 2*x^3) + O(x^n)) \\ Iain Fox, Dec 18 2017

a(n) = (2^(n+2) + 2*(-1)^n)/3 + 2^n - (3-(-1)^n)/2.
a(n) = A014113(n+1) + A141023(n).
G.f.: (2 - 2*x + 3*x^2)/(1 - 2*x - x^2 + 2*x^3).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), n > 2. - Iain Fox, Dec 18 2017

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A309597 a(n) is the A325907(n)-th triangular number.

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A290258 Triangle read by rows: row n (>=2) contains in increasing order the integers for which the binary representation has length n and all runs of 1's have even length.

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A290604 a(0) = 2, a(1) = 2; for n > 1, a(n) = a(n-1) + 2*a(n-2) + 3.

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