cp's OEIS Frontend

Octal games 4.4, 4.41, 4.42, 4.43, 4.6, 4.61, 4.62 and 4.63 have values a(n-1).

"The periodicity was first proved by R. K. Guy in 1949, the sequence necessarily being calculated by hand." [Beasley].

The Twopins/t sequence (see Guy).

Number of closed walks of length n at a vertex of the graph with adjacency matrix [1,1,0,0;0,0,0,1;1,0,0,0;0,0,1,1]. - Paul Barry, Mar 15 2004

a(n+3) = number of n-bit sequences that avoid both 010 and 0110. Example: for n=3, there are 8 3-bit sequences and only 010 fails to qualify, so a(6)=7. - David Callan, Mar 25 2004

a(n) is the number of length n binary words that have an even number of 0's and every 0 is immediately followed by a 1. a(6) = 7 because we have: 010111, 011011, 011101, 101011, 101101, 110101, 111111. - Geoffrey Critzer, Jan 08 2014

a(n) is the number of vertices of the Fibonacci cube Gamma(n-1) having an even number of ones. The Fibonacci cube Gamma(n) can be defined as the graph whose vertices are the binary strings of length n without two consecutive 1's and in which two vertices are adjacent when their Hamming distance is exactly 1. Example: a(4) = 2; indeed, the Fibonacci cube Gamma(3) has the five vertices 000, 010, 001, 100, 101, two of which have an even number of ones. See the E. Munarini et al. reference, p. 323. - Emeric Deutsch, Jun 28 2015

a(n) is the number of even permutations p of 1,2,...,n such that |p(i)-i| <= 1 for i=1,2,...,n. - Dmitry Efimov, Jan 08 2016

This sequence (prefixed with 0) is an autosequence of the first kind, whose second kind companion is (2 followed by abs(A111734)). - Jean-François Alcover, Oct 30 2017

a(n+1) is the number of n-bit sequences such that 1's appear in groups of three or more. Example: for n = 5, a(6) = 7 because we have 00000, 00111, 01110, 11100, 11110, 01111, 11111. Source: exercise 1.11 in I. Stewart. - João Camarneiro, Dec 23 2024

From Gus Wiseman, Feb 12 2019: (Start)

Also the number of ways to split an (n + 1)-cycle into nonempty connected subgraphs with no singletons. For example, the a(1) = 1 through a(5) = 12 partitions are:

{{12}} {{123}} {{1234}} {{12345}} {{123456}}

{{12}{34}} {{12}{345}} {{12}{3456}}

{{14}{23}} {{123}{45}} {{123}{456}}

{{125}{34}} {{1234}{56}}

{{145}{23}} {{1236}{45}}

{{15}{234}} {{1256}{34}}

{{126}{345}}

{{1456}{23}}

{{156}{234}}

{{16}{2345}}

{{12}{34}{56}}

{{16}{23}{45}}

Also the number of non-singleton subsets of {1, ..., (n + 1)} with no cyclically successive elements (cyclically successive means 1 succeeds n + 1). For example, the a(1) = 1 through a(5) = 12 subsets are:

{} {} {} {} {}

{1,3} {1,3} {1,3}

{2,4} {1,4} {1,4}

{2,4} {1,5}

{2,5} {2,4}

{3,5} {2,5}

{2,6}

{3,5}

{3,6}

{4,6}

{1,3,5}

{2,4,6}

(End)

Diagonal sums of A038137. - Paul Barry, Oct 24 2005

From Gary W. Adamson, Oct 28 2010: (Start)

INVERT transform of the aerated Fibonacci sequence (1, 0, 1, 0, 2, 0, 3, 0, 5, ...).

a(n) = term (4,4) in the n-th power of the matrix [0,1,0,0; 0,0,1,0; 0,0,0,1; 1,0,1,1]. (End)

Number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=1, r=3, I={2}. - Vladimir Baltic, Mar 07 2012

Number of compositions of n if the summand 2 is frozen in place or equivalently, if the ordering of the summand 2 does not count. - Gregory L. Simay, Jul 18 2016

a(n) - a(n-2) = number of compositions of n with no 2's = A005251(n+1). - Gregory L. Simay, Jul 18 2016

In general, the number of compositions of n with summand k frozen in place is equal to the number of compositions of n with only summands 1,...,k,2k. - Gregory L. Simay, May 10 2017

In the same way that the sum of any two alternating terms of A006498 produces a term from A000045 (the Fibonacci sequence), so it could be thought of as a "meta-Fibonacci," and the sum of any two alternating terms of A013979 produces a term from A000930 (Narayana’s cows), so it could analogously be called "meta-Narayana’s cows," this sequence embeds (can generate) A000931 (the Padovan sequence), as the odd terms of A000931 are generated by the sum of successive elements (e.g. 1+2=3, 2+3=5, 3+6=9, 6+10=16) and its even terms are generated by the difference of "supersuccessive" (second-order successive or "alternating," separated by a single other term) terms (e.g. 10-3=7, 18-6=12, 31-10=21, 55-18=37) — or, equivalently, adding "supersupersuccessive" terms (separated by 2 other terms, e.g. 1+6=7, 2+10=12, 3+18=21, 6+31=37) — so it could be dubbed the "metaPadovan." - Michael Cohen and Yasuyuki Kachi, Jun 13 2024

Table starts

.1......2.........4...........7.............12...............21

.1......7........20..........57............156..............438

.1.....21........93.........453...........2050.............9516

.1.....65.......436........3617..........26635...........201960

.1....200......2043.......28888.........346501..........4299802

.1....616......9573......230726........4507281.........91513161

.1...1897.....44857.....1842766.......58634265.......1948262831

.1...5842....210190....14717828......762745363......41475009100

.1..17991....984904...117548611.....9922198586.....882920599454

.1..55405...4615043...938839259...129073371922...18795714385533

.1.170625..21625074..7498337380..1679056577224..400125250527550

.1.525456.101330329.59887849061.21842080049186.8517908622924528

Table starts

.1......2........4..........7..........12...........21............37

.1......6.......18.........39..........96..........225...........543

.1.....18.......78........281........1072.........4033.........15255

.1.....51......331.......1774.......10505........59127........341648

.1....142.....1400......11385......101600.......872437.......7640541

.1....405.....5949......73136......989947.....12867295.....170792866

.1...1157....25277.....472638.....9702349....191707377....3862325048

.1...3289...107353....3050285....94937311...2851740545...87205791997

.1...9344...455938...19667721...927766394..42351530487.1964627314453

.1..26580..1936500..126807809..9067631046.628917557052

.1..75621..8224877..817795362.88653591037

.1.215076.34933256.5274346426

Table starts

....2.......4.........7..........12............21..............37

....4......16........49.........144...........441............1369

....8......64.......316........1404..........6768...........33893

...16.....256......2032.......13452.........99721..........795741

...32....1024.....13045......128628.......1492864........19468046

...64....4096.....83737.....1228512......22289912.......477128662

..128...16384....537496....11733712.....333124565.....11711612310

..256...65536...3450100...112065936....4978704008....287687887135

..512..262144..22145617..1070316016...74410715409...7067105036501

.1024.1048576.142149013.10222334864.1112149145053.173620295413143

Table starts

....2.......4........7.........12..........21.............37...............65

....4......16.......49........144.........441...........1369.............4225

....8......64......292.......1164........5238..........25046...........116100

...16.....256.....1723.......8496.......50024.........357323..........2482591

...32....1024....10327......65160......532565........6204967.........68121839

...64....4096....61996.....515560.....6110500......118571483.......2076231513

..128...16384...371641....4075336....69943253.....2239578131......61652076124

..256...65536..2227333...32031600...783072552....41236726541....1785011303305

..512..262144.13350748..251533888..8759983583...764615054191...52081392909734

.1024.1048576.80027347.1976926440.98440457351.14279876468131.1531637258052071

T(n,k) is the number of lattice points in k*P where P is a polytope of dimension n+2 in R^(n+2) whose vertices are lattice points, and therefore for each n it is an Ehrhart polynomial of degree n+2. This confirms the empirical formulas for the rows. - Robert Israel, Mar 21 2021