A164094 -id:A164094 - cp's OEIS frontend

A195744 a(n) = 15*2^(n+1) + 1.

31, 61, 121, 241, 481, 961, 1921, 3841, 7681, 15361, 30721, 61441, 122881, 245761, 491521, 983041, 1966081, 3932161, 7864321, 15728641, 31457281, 62914561, 125829121, 251658241, 503316481, 1006632961, 2013265921, 4026531841, 8053063681, 16106127361

Offset: 0

Brad Clardy, Sep 23 2011

Binary numbers of form 1111(0^n)1 where n is the index and number of 0's.
Base 10 numbers of this sequence always end in 1.
An Engel expansion of 1/15 to the base 2 as defined in A181565, with the associated series expansion 1/15 = 2/31 + 2^2/(31*61) + 2^3/(31*61*121) + 2^4/(31*61*121*241) + ... . - Peter Bala, Oct 29 2013
The only squares in this sequence are 121 = 11^2 and 961 = 31^2. - Antti Karttunen, Sep 24 2023

			First few terms in binary are 11111, 111101, 1111001, 11110001, 111100001.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A052996, A164094.
Cf. A020737, A083575, A083683, A083686, A083705, A168596, A181565.
Cf. A110286, A128470.
Cf. A005940, A030514.

[15*2^(n+1) + 1: n in [0..30]]; // Vincenzo Librandi, Sep 24 2011

15*2^Range[50] + 1 (* Paolo Xausa, Apr 02 2024 *)

a(n)=30*2^n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A052996(n+3) + A164094(n+2).
From Bruno Berselli, Sep 23 2011: (Start)
G.f.: (31-32*x)/(1-3*x+2*x^2).
a(n) = 2*a(n-1)-1.
a(n) = A110286(n+1)+1 = A128470(2^n). (End)
E.g.f.: exp(x)*(1 + 30*exp(x)). - Stefano Spezia, Oct 08 2022
For n >= 0, A005940(a(n)) = A030514(2+n). - Antti Karttunen, Sep 24 2023

Corrected by Arkadiusz Wesolowski, Sep 23 2011

A240192 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or one plus the sum of the elements above it or three plus the sum of the elements diagonally to its northwest, modulo 4.

Original entry on oeis.org

1, 1, 2, 1, 5, 3, 1, 8, 12, 4, 1, 14, 37, 27, 7, 1, 26, 129, 138, 73, 10, 1, 50, 478, 771, 680, 154, 15, 1, 98, 1908, 5240, 7170, 2413, 358, 24, 1, 194, 7868, 40765, 91879, 44594, 10017, 872, 35, 1, 386, 32888, 336257, 1399773, 1005029, 333607, 43956, 1871, 54, 1

Offset: 1

R. H. Hardin, Apr 02 2014

Table starts
..1....1......1.........1...........1............1.............1.............1
..2....5......8........14..........26...........50............98...........194
..3...12.....37.......129.........478.........1908..........7868.........32888
..4...27....138.......771........5240........40765........336257.......2843914
..7...73....680......7170.......91879......1399773......22849697.....385366572
.10..154...2413.....44594.....1005029.....28061567.....865984451...28244997476
.15..358..10017....333607....14022582....733907809...43398047802.2752449791995
.24..872..43956...2715035...206345434..19388521135.2070573220929
.35.1871.159668..17332017..2336659626.394134037392
.54.4438.681760.134735700.33576330306

			Some solutions for n=4 k=4
..2..0..0..0....2..0..0..0....2..0..0..0....2..0..0..0....2..0..0..0
..1..2..0..0....2..3..0..2....2..0..0..0....2..0..0..0....2..0..0..0
..2..1..0..2....2..3..3..1....2..0..3..0....2..3..0..2....2..0..3..2
..1..3..2..0....1..2..1..1....2..0..3..3....1..2..2..0....1..0..2..1

R. H. Hardin, Table of n, a(n) for n = 1..112

Column 1 is A159288

Row 2 is A164094(n-2)

Empirical for column k:
k=1: a(n) = a(n-2) +2*a(n-3)
k=2: [order 13]
Empirical for row n:
n=1: a(n) = a(n-1)
n=2: a(n) = 3*a(n-1) -2*a(n-2) for n>3
n=3: a(n) = 9*a(n-1) -27*a(n-2) +29*a(n-3) +6*a(n-4) -32*a(n-5) +16*a(n-6) for n>9
n=4: [order 29] for n>34

A334164 a(n) is the number of ON-cells in the completed n-th level of a triangular wedge in the hexagonal grid of A151723 (i.e., after 2^k >= n generations of the automaton in A151723 have been computed).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 4, 8, 2, 10, 6, 10, 4, 14, 8, 16, 2, 18, 10, 16, 8, 20, 12, 22, 6, 26, 14, 22, 8, 30, 16, 32, 2, 34, 18, 28, 16, 34, 18, 32, 14, 40, 22, 34, 16, 42, 24, 44, 10, 50, 26, 40, 20, 48, 28, 50, 14, 58, 30, 46, 16, 62, 32, 64

Offset: 1

Hartmut F. W. Hoft, Apr 17 2020

nonn

Conjecture 1: Except for a(2^n + 1) = 2, n >= 1, for odd-numbered completed levels a lower bound of the ratio of ON-cells to the length of the level is (2^n + 2)/(3*2^(n+1) + 1) with limit 1/6, determined by the subsequence of levels starting with: 13, 25, 49, 97, 193, 385, 769, 1537, 3073, ..., and the associated ON-cell counts: 4, 6, 10, 18, 34, 66, 130, 258, 514, ..., as listed in the second column of each of the two triangles below.
The ON-cell counts for the indices in each row define a line of slope 1/2.  The formula for the indices of levels in row k >= 2 is  L(k, i) = 1 + Sum_{j = 0, ..., i} 2^(k-j), 0 <= i <= k - 2, and the formula for the associated numbers of ON-cells is C(k, i) = 2 + Sum_{j = 1..i} 2^(k-1-j), 0 <= i <= k - 2:
Index of the level: L(k, i) number of ON-cells: C(k, i)
k\i  0   1   2   3   4   5   6    k/i  0   1   2   3   4   5   6
2:   5                             2:  2
3:   9  13                         3:  2   4
4:  17  25  29                     4:  2   6   8
5:  33  49  57  61                 5:  2  10  14  16
6:  65  97 113 121 125             6:  2  18  26  30  32
7: 129 193 225 241 249 253         7:  2  34  50  58  62  64
8: 257 385 449 481 497 505 509     8:  2  66  98 114 122 126 128
...
The pairs ( L(k, i), C(k, i) ), for 0 <= i <= k-2, define a line of slope 1/2 for each k >= 3.
For triangle L(k, i): column 0 is A000051(n), n >= 2; column 1 is A181565(n), n >= 3; column 2 is A083686(n), n >= 2; columns 3 is A195744(n), n >= 1; column 4 is A206371(n), n >= 2; column 5 is A196657(n), n >= 1; the bounding diagonal is A036563(n), n >= 3.
For triangle C(k, i): column 1 is A052548(n), n >= 1; column 2 is A164094(n), n >= 1.
Conjecture 2: In an even-numbered completed level 2*n the fraction of ON-cells is bounded below by (23 * 2^n - 24)/(2^(n+5) - 36) with limit 23/32, determined by the subsequence of levels starting with: 28, 92, 220, 476, 988, 2012, 4060, ... .
There are 16 numbers less than 1000 that do not occur as the number of ON-cells in a completed level through level 16384: 136, 164, 330, 334, 402, 444, 526, 570, 598, 604, 614, 714, 740, 822, 832, 878.
Sequence A334169 of even-numbered completed levels in which all cells are ON-cells is a subsequence of this sequence.

Hartmut F. W. Hoft, Diagram of ON-cell counts

Cf. A000051, A036563, A052548, A083686, A151723, A164094, A181565, A195744, A196657, A206371, A334169.

(* a169781[] and support functions are defined in A169781 and create the list nTriangle *)
a334164[n_] := Module[{k, levels={}}, a169781[n]; For[k=1, k<=n, k++, AppendTo[levels, Count[nTriangle[[k]], 1] - 2]]; levels]/;(n>=3 && IntegerQ[Log[2,n]])
a334164[64] (* sequence data *)

cp's OEIS Frontend

A195744 a(n) = 15*2^(n+1) + 1.

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A240192 T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or one plus the sum of the elements above it or three plus the sum of the elements diagonally to its northwest, modulo 4.

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A334164 a(n) is the number of ON-cells in the completed n-th level of a triangular wedge in the hexagonal grid of A151723 (i.e., after 2^k >= n generations of the automaton in A151723 have been computed).

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