A158953 -id:A158953 - cp's OEIS frontend

A098502 a(n) = 16*n - 4.

12, 28, 44, 60, 76, 92, 108, 124, 140, 156, 172, 188, 204, 220, 236, 252, 268, 284, 300, 316, 332, 348, 364, 380, 396, 412, 428, 444, 460, 476, 492, 508, 524, 540, 556, 572, 588, 604, 620, 636, 652, 668, 684, 700, 716, 732, 748, 764, 780, 796, 812, 828, 844

Offset: 1

Ralf Stephan, Sep 15 2004

For n > 3, the number of squares on the infinite 4-column chessboard at <= n knight moves from any fixed start point.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).
Index entries for sequences which agree for a long time but are different.

Cf. A004767, A008590, A017113, A017137, A017629, A098498, A158953.

[16*n - 4: n in [1..60]]; // Vincenzo Librandi, Jul 24 2011

Range[12, 1000, 16] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)

a(n)=16*n-4 \\ Charles R Greathouse IV, Jul 10 2016

G.f.: 4*x*(3+x)/(1-x)^2. - Colin Barker, Jan 09 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + log(3 - 2*sqrt(2)))/(16*sqrt(2)). - Amiram Eldar, Sep 01 2024
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: 4*(exp(x)*(4*x - 1) + 1).
a(n) = 2*a(n-1) - a(n-2) for n > 2.
a(n) = 4*A004767(n-1) = 2*A017137(n-1) = A017113(2*n-1). (End)

A103747 Trajectory of 2 under repeated application of the map n -> A102370(n).

Original entry on oeis.org

2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 254, 258, 262, 266, 270, 274, 278, 282, 286, 290, 294, 298, 302, 306, 310, 314, 382, 386, 390, 394, 398, 402, 406, 410, 414, 418, 422

Offset: 1

Benoit Cloitre and David Applegate, Mar 25 2005

Although it initially appears that a(n)-8n is the 16-periodic sequence {-2,-6,-10,-14,-18,-22,-26,-30,-34,-38,-42,-46,-50,-54,6,2}, this pattern eventually breaks down. However, the first divergence occurs beyond the first 400 million terms.

(a(n)) agrees with the 16-periodic sequence up to a(2^67-1) = 2^70 - 70, but then diverges with a(2^67) = 2^71 - 2. - Charlie Neder, Feb 07 2019

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps], preprint, 2005.
David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
Index entries for sequences which agree for a long time but are different

Cf. A102370, A132417.

Trajectories of other numbers: A103192 (1), A103621 (7), A158953 (12), A159887 (29).

a103747 n = a103747_list !! (n-1)
a103747_list = iterate (fromInteger . a102370) 2
-- Reinhard Zumkeller, Jul 21 2012

Edited by Peter Munn, Jan 13 2024

A103621 Trajectory of 7 under repeated application of the map n -> A102370(n).

Original entry on oeis.org

7, 9, 11, 13, 23, 25, 27, 61, 71, 73, 75, 77, 87, 89, 91, 125, 135, 137, 139, 141, 151, 153, 155, 189, 199, 201, 203, 205, 215, 217, 219, 253, 263, 265, 267, 269, 279, 281, 283, 317, 327, 329, 331, 333, 343, 345, 347, 381, 391, 393, 395, 397, 407, 409, 411, 445

Offset: 1

Philippe Deléham, Mar 31 2005

Initially, first differences are 8-periodic: 2,2,2,10,2,2,34,10. [Unsigned comment made accurate by Peter Munn, Jan 13 2024]

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Cf. A102370.

Trajectories of other numbers A103192 (1), A103747 (2), A158953 (12), A159887 (29).

f[n_] := Block[{k = 1, s = 0, l = Max[2, Floor[ Log[2, n + 1] + 2]]}, While[k < l, If[ Mod[n + k, 2^k] == 0, s = s + 2^k]; k++ ]; s + n]; NestList[f, 7, 55] (* Robert G. Wilson v, Mar 30 2005 *)

Conjectures from Chai Wah Wu, Feb 01 2018: (Start)
a(n) = a(n-1) + a(n-8) - a(n-9) for n > 9.
G.f.: x*(3*x^8 + 34*x^7 + 2*x^6 + 2*x^5 + 10*x^4 + 2*x^3 + 2*x^2 + 2*x + 7)/(x^9 - x^8 - x + 1). (End)
The above conjectures are incompatible with A102370(2^37-37) = 2^38-3. - Peter Munn, Jan 13 2024

More terms from Robert G. Wilson v, Mar 30 2005

A159887 Trajectory of 29 under repeated application of the map n -> A102370(n).

Original entry on oeis.org

29, 39, 41, 43, 45, 55, 57, 59, 93, 103, 105, 107, 109, 119, 121, 251, 285, 295, 297, 299, 301, 311, 313, 315, 349, 359, 361, 363, 365, 375, 377, 507, 541, 551, 553, 555, 557, 567, 569, 571, 605, 615, 617, 619, 621, 631, 633, 763, 797, 807, 809, 811, 813, 823, 825

Offset: 1

Philippe Deléham, Apr 25 2009

Not the same as A159888: see the comments in A159888.

The divergence from A159888 follows from Theorem 3.1 in the Applegate, Cloitre, Deléham and Sloane link: in general, the first differences of an A102370 trajectory cannot be a cycle. - Peter Munn, Jan 14 2024

David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.
Index entries for sequences which agree for a long time but are different

Cf. A102370, A159888.

Trajectories of other numbers: A103192 (1), A103747 (2), A103621 (7), A158953 (12).

Missing term 617 inserted by Georg Fischer, Nov 28 2023

cp's OEIS Frontend

A098502 a(n) = 16*n - 4.

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A103747 Trajectory of 2 under repeated application of the map n -> A102370(n).

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A103621 Trajectory of 7 under repeated application of the map n -> A102370(n).

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A159887 Trajectory of 29 under repeated application of the map n -> A102370(n).

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