A198643 -id:A198643 - cp's OEIS frontend

A171884 Lexicographically earliest injective nonnegative sequence a(n) satisfying |a(n+1) - a(n)| = n for all n.

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 64, 40, 65, 39, 66, 38, 67, 37, 68, 36, 69, 35, 70, 34, 71, 33, 72, 32, 73, 31, 74, 30, 75, 29, 76, 28, 77, 27, 78, 26, 79, 133, 188, 132, 189, 131, 190, 130, 191, 129, 192, 128, 193, 127, 194

Offset: 1

Robert Munafo, Mar 11 2010

nonn

The map n -> a(n) is an injective map to the nonnegative integers, i.e., no two terms are identical.
Appears not to contain numbers from the following sets (grouped intentionally): {4, 5}, {14, 15, 16, 17, 18, 19}, {44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61}, etc. The numbers of terms in these groups appears to be A008776. - Paul Raff, Mar 15 2010 [This is correct: by the formula below, a(2*3^k+1...2*3^(k+1)) take all the values in the range [3^(k+1)-1, 5*3^k-2] U [7*3^k-1, 3^(k+2)-2], so the numbers not appearing are those in the range [5*3^k-1, 7*3^k-2] for some k. - Jianing Song, Oct 07 2022]
The first 23 terms are shared with Recamán's sequence A005132, but from then on they are different. - Philippe Deléham, Mar 01 2013, Omar E. Pol, Jul 01 2013
From M. F. Hasler, May 09 2013: (Start)
It appears that the starting points of the gaps (4, 14, 44, 134, 404, 1214, ...) are given by A181655(2n) = A198643(n-1), and thus the ending points (5, 19, 61, ...) by A181655(2n) + A048473(n-1).
The first differences have signs (grouped intentionally): +++, -, +++, -+-+-+-+- (5 times "-"), +++, -+...+- (17 times "-"), +++, ... where the number of minus signs is again given by A048473 = A008776-1. (End)
A correspondent, Dennis Reichard, conjectures that (i) a(n) <= 3.5*n for all n and (ii) the sequence covers 2/3 of all natural numbers. - N. J. A. Sloane, Jun 30 2018 [(i) is true: the indices of records for a(n)/n are n = 1, 2, 3, 4, 6, 7, and 2*3^k+2 for k >= 1, with record values 0, 1/2, 1, 1, 3/2, 7/6, 13/7, and (7*3^k-1)/(2*3^k+2) for k >= 1, so a(n) <= 3.5*n. (ii) needs further justification: the lower natural density is lim_{k->+oo} #{terms <= 7*3^k-2}/(7*3^k-2) = lim_{k->+oo} (4*3^k-1)/(7*3^k-2) = 4/7, and the upper natural density is lim_{k->+oo} #{terms <= 5*3^k-2}/(5*3^k-2) = lim_{k->+oo} (4*3^k-1)/(5*3^k-2) = 4/5. - Jianing Song, Oct 07 2022]

			We begin with 0, 0+1=1, 1+2=3. 3-3=0 cannot be the next term because 0 is already in the sequence so we go to 3+3=6. The next could be 6-4=2 or 6+4=10 but we choose 2 because it is smaller.

Jianing Song, Table of n, a(n) for n = 1..13122
Robert Munafo, Lexicographically earliest injective and unbounded sequence A(n) satisfying |A(n+1)-A(n)|=n for all n
Robert Munafo, main-A171884.c(C source code to generate the sequence)

Cf. A005132, which allows duplicate values.

Cf. also A118201, in which every value of a(n) and of |a(n+1)-a(n)| occurs exactly once, but does not ensure that the latter is strictly increasing.

A171884[{}, , ] := {};
A171884[L_List, max_Integer, True] := If[Length[L] == max, L, With[{n = Length[L]},
  If[Last[L] - n < 1 || MemberQ[L, Last[L] - n],
    If[MemberQ[L, Last[L] + n],
       A171884[Drop[L, -1], max, False],
       A171884[Append[L, Last[L] + n], max, True]],
    A171884[Append[L, Last[L] - n], max, True]]]]
A171884[L_List, max_Integer, False] := With[{n = Length[L]},
  If[MemberQ[L, Last[L] + n],
     A171884[Drop[L, -1], max, False],
     A171884[Append[L, Last[L] + n], max, True]]]
A171884[{0}, 200, True] (* Paul Raff, Mar 15 2010 *)

A171884_upto(N,a=0,t=2)=vector(N,k, a+=if(!bitand(k,1), k-1, t-=1, 1-k, t=k-1)) \\ or:
A171884_upto(N,a)=vector(N,k,a+=if(bitand(k,1)&&k\2!=3^valuation(k-(k>1),3),1-k,k-1)) \\ M. F. Hasler, Apr 05 2019
a(n) = if(n<=2, n-1, my(k=logint((n-1)\2, 3), r=n-2*3^k); if(r%2, 5*3^k-1-(r+1)/2, 7*3^k-2+r/2)) \\ Jianing Song, Oct 07 2022

a(n+1) = a(n) +- n with - iff n is even but not n = 2 + 2*3^k. (Cf. comment from May 09 2013.) - M. F. Hasler, Apr 05 2019

a(2*3^k + 2*r - 1) = 5*3^k - 1 - r, a(2*3^k + 2*r) = 7*3^k - 2 + r, for k >= 0 and 1 <= r <= 2*3^k. - Jianing Song, Oct 07 2022

Definition edited by M. F. Hasler, Apr 01 2019

A240917 a(n) = 23^(2n) - 3*3^n + 1.

Original entry on oeis.org

0, 10, 136, 1378, 12880, 117370, 1060696, 9559378, 86073760, 774781930, 6973391656, 62761587778, 564857478640, 5083726873690, 45753570561016, 411782221142578, 3706040248563520, 33354363011912650, 300189269431736776, 2701703431859199778

Offset: 0

Kival Ngaokrajang, Apr 14 2014

a(n) is the total number of holes of a triflake-like fractal (fan pattern) after n iterations. The scale factor for this case is 1/3, but for the actual triflake case, it is 1/2, i.e., Sierpiński triangle. The total number of sides is 3*(A198643-1). The perimeter seems to converge to 10/6.

Kival Ngaokrajang, Illustration of triflake like fractal (fan pattern) for n = 0..3
Wikipedia, n-flake,
Index entries for linear recurrences with constant coefficients, signature (13,-39, 27).

Cf. A198643, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240840 (hendecaflake), A240735 (dodecaflake), A240841 (tridecaflake).

A240917:=n->2*3^(2*n) - 3*3^n + 1; seq(A240917(n), n=0..30); # Wesley Ivan Hurt, Apr 15 2014

Table[2*3^(2 n) - 3*3^n + 1, {n, 0, 30}] (* Wesley Ivan Hurt, Apr 15 2014 *)

a(n)= 2*3^(2*n) - 3*3^n + 1
       for(n=0,100,print1(a(n),", "))

concat(0, Vec(-2*x*(3*x+5)/((x-1)*(3*x-1)*(9*x-1)) + O(x^100))) \\ Colin Barker, Apr 15 2014

a(n) = 2*A007742(A003462(n)).
a(n) = 9*(a(n-1) + 2*A048473(n-1)) + 1.
From Colin Barker, Apr 15 2014: (Start)
a(n) = 1-3^(1+n)+2*9^n.
a(n) = 13*a(n-1)-39*a(n-2)+27*a(n-3).
G.f.: -2*x*(3*x+5) / ((x-1)*(3*x-1)*(9*x-1)). (End).

A181655 Expansion of (1+2x-x^3+x^4)/(1-4x^2+3x^4).

Original entry on oeis.org

1, 2, 4, 7, 14, 22, 44, 67, 134, 202, 404, 607, 1214, 1822, 3644, 5467, 10934, 16402, 32804, 49207, 98414, 147622, 295244, 442867, 885734, 1328602, 2657204, 3985807, 7971614, 11957422, 23914844, 35872267, 71744534, 107616802, 215233604

Offset: 0

Paul Barry, Nov 03 2010

Row sums of A181654.

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-3).

Cf. A060816, A198643 (bisections).

CoefficientList[Series[(1+2x-x^3+x^4)/(1-4x^2+3x^4),{x,0,40}],x] (* or *) Join[{1},LinearRecurrence[{0,4,0,-3},{2,4,7,14},40]] (* Harvey P. Dale, Jan 11 2012 *)

A181655(n)=if(bitand(n,1), 3^(n\2)*5\2, n, 3^(n\2-1)*5-1, 1) \\ M. F. Hasler, Apr 06 2019

G.f.: (1+2*x-x^3+x^4)/((1-x^2)*(1-3*x^2)).
a(n) = 5*A038754(n+1)/6 - A040001(n)/2. - R. J. Mathar, May 14 2016
a(2n-1) = A060816(n-1), a(2n) = A198643(n-1); n >= 1. a(n+1) = 2*a(n) if n is odd. - M. F. Hasler, Apr 06 2019

A355492 a(n) = 7*3^n - 2.

Original entry on oeis.org

5, 19, 61, 187, 565, 1699, 5101, 15307, 45925, 137779, 413341, 1240027, 3720085, 11160259, 33480781, 100442347, 301327045, 903981139, 2711943421, 8135830267, 24407490805, 73222472419, 219667417261, 659002251787, 1977006755365, 5931020266099, 17793060798301, 53379182394907

Offset: 0

Jianing Song, Oct 07 2022

Right ending points of the gaps in A171884. The left ending points are given in A198643.

			The numbers not appearing in A171884 are those in the range [5*3^k-1, 7*3^k-2] for some k; that is, [4, 5] U [14, 19] U [44, 61] U ...

Jianing Song, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A171884, A198643.

7*3^Range[0, 50] - 2 (* Paolo Xausa, Jun 10 2024 *)

PARI
```
a(n)=7*3^n-2
```

G.f.: (5-x)/((1-x)*(1-3*x)).

E.g.f.: 7*exp(3*x) - 2*exp(x).

A238207 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) is A032766(k) and T(n,k) = 3*T(n-1,k) + 2 for n>0.

Original entry on oeis.org

0, 1, 2, 3, 5, 8, 4, 11, 17, 26, 6, 14, 35, 53, 80, 7, 20, 44, 107, 161, 242, 9, 23, 62, 134, 323, 485, 728, 10, 29, 71, 188, 404, 971, 1457, 2186, 12, 32, 89, 215, 566, 1214, 2915, 4373, 6560, 13, 28, 98, 269, 647, 1700, 3644, 8747, 13121, 19682, 15, 41, 116

Offset: 0

Philippe Deléham, Feb 20 2014

Permutation of nonnegative integers.

			Square array begins:
0, 1, 3, 4, 6, 7, 9, 10, ...
2, 5, 11, 14, 20, 23, 29, 32, ...
8, 17, 35, 44, 62, 71, 89, 98, ...
26, 53, 107, 134, 188, 215, 269, 296, ...
80, 161, 323, 404, 566, 647, 809, 890, ...
242, 485, 971, 1214, 1700, 1943, 2429, 2672, ...
728, 1457, 2915, 3644, 5102, 5831, 7289, 8018, ...
2186, 4373, 8747, 10934, 15308, 17495, 21869, 24056, ...
...

Cf. A024023, A027107, A032766, A048473, A171498, A198643, A198644, A198645, A198646.

T(n,k) = T(0,k)*3^n + T(n,0) where T(0,k) = A032766(k) and T(n,0) = 3^n - 1 = A024023(n).

cp's OEIS Frontend

A171884 Lexicographically earliest injective nonnegative sequence a(n) satisfying |a(n+1) - a(n)| = n for all n.

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A240917 a(n) = 2*3^(2*n) - 3*3^n + 1.

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A181655 Expansion of (1+2x-x^3+x^4)/(1-4x^2+3x^4).

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A355492 a(n) = 7*3^n - 2.

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A238207 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) is A032766(k) and T(n,k) = 3*T(n-1,k) + 2 for n>0.

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A240917 a(n) = 23^(2n) - 3*3^n + 1.