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[Empirical] a(base,n) = a(base-1,n) + 7^(n-1) for base >= 3n-2; a(base,n) = a(base-1,n) + 7^(n-1)-2 when base = 3n-3.

From Johannes W. Meijer, Aug 01 2010: (Start)

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 or 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.

For the side squares the 512 red kings lead to 47 different red king sequences, see the cross-references for some examples.

The sequence above corresponds to four A[5] vectors with the decimal [binary] values 367 [1,0,1,1,0,1,1,1,1], 463 [1,1,1,0,0,1,1,1,1], 487 [1,1,1,1,0,0,1,1,1] and 493 [1,1,1,1,0,1,1,0,1]. These vectors lead for the corner squares to A179596 and for the central square to A179597.

This sequence belongs to a family of sequences with g.f. (1+x)/(1-4*x-k*x^2). Red king sequences that are members of this family are A003947 (k=0), A015448 (k=1), A123347 (k=2), A126473 (k=3; this sequence) and A086347 (k=4). Other members of this family are A000351 (k=5), A001834 (k=-1), A111567 (k=-2), A048473 (k=-3) and A053220 (k=-4)

Inverse binomial transform of A154244. (End)

Equals the INVERT transform of A055099: (1, 4, 14, 50, 178, ...). - Gary W. Adamson, Aug 14 2010

Number of one-sided n-step walks taking steps from {E, W, N, NE, NW}. - Shanzhen Gao, May 10 2011

For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,2,3,4} containing no subwords 00 and 11. - Milan Janjic, Jan 31 2015

Let f = floor and c = ceiling. For x > 1, define four sequences as functions of x, as follows:

p1(0) = f(x), p1(n) = f(x*p1(n-1));

p2(0) = f(x), p2(n) = c(x*p2(n-1)) if n is odd and p2(n) = f(x*p1(n-1)) if n is even;

p3(0) = c(x), p3(n) = f(x*p3(n-1)) if n is odd and p3(n) = c(x*p3(n-1)) if n is even;

p4(0) = c(x), p4(n) = c(x*p4(n-1)).

The present sequence is given by a(n) = p3(n).

Following the terminology at A214986, call the four sequences power floor, power floor-ceiling, power ceiling-floor, and power ceiling sequences. In the table below, a sequence is identified with an A-numbered sequence if they appear to agree except possibly for initial terms. Notation: S(t)=sqrt(t), r = (1+S(5))/2 = golden ratio, and Limit = limit of p3(n)/p2(n).

x ......p1..... p2..... p3..... p4.......Limit

r.......A000045 A000045 A000045 A000045..r

r^2.....A001519 A001654 A061646 A001906..-1+S(5)

r^3.....A024551 A001076 A015448 A049652..-1+S(5)

r^4.....A049685 A157335 A214992 A004187..-19+9*S(5)

r^5.....A214993 A049666 A015457 A214994...(-9+5*S(5))/2

r^6.....A007805 A156085 A214995 A049660..-151+68*S(5)

1+S(2)..A024537 A000129 A001333 A048739..S(2)

2+S(2)..A007052 A214996 A214997 A007070..(1+S(2))/2

1+S(3)..A057960 A002605 A028859 A077846..(1+S(3))/2

2+S(3)..A001835 A109437 A214998 A001353..-4+3*S(3)

S(5)....A214999 A215091 A218982 A218983..1.26879683...

2+S(5)..A024551 A001076 A015448 A049652..-1+S(5)

2+S(6)..A218984 A090017 A123347 A218985..S(3/2)

2+S(7)..A218986 A015530 A126473 A218987..(1+S(7))/3

2+S(8)..A218988 A057087 A086347 A218989..(1+S(2))/2

3+S(8)..A001653 A084158 A218990 A001109..-13+10*S(2)

3+S(10).A218991 A005668 A015451 A218992..-2+S(10)

...

Properties of p1, p2, p3, p4:

(1) If x > 2, the terms of p2 and p3 interlace: p2(0) < p3(0) < p2(1) < p3(1) < p2(2) < p3(2)... Also, p1(n) <= p2(n) <= p3(n) <= p4(n) <= p1(n+1) for all x>0 and n>=0.

(2) If x > 2, the limits L(x) = limit(p/x^n) exist for the four functions p(x), and L1(x) <= L2(x) <= L3(x) <= L4 (x). See the Mathematica programs for plots of the four functions; one of them also occurs in the Odlyzko and Wilf article, along with a discussion of the special case x = 3/2.

(3) Suppose that x = u + sqrt(v) where v is a nonsquare positive integer. If u = f(x) or u = c(x), then p1, p2, p3, p4 are linear recurrence sequences. Is this true for sequences p1, p2, p3, p4 obtained from x = (u + sqrt(v))^q for every positive integer q?

(4) Suppose that x is a Pisot-Vijayaraghavan number. Must p1, p2, p3, p4 then be linearly recurrent? If x is also a quadratic irrational b + c*sqrt(d), must the four limits L(x) be in the field Q(sqrt(d))?

(5) The Odlyzko and Wilf article (page 239) raises three interesting questions about the power ceiling function; it appears that they remain open.

a(n) gives the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6, 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a rook on the four side and four corner (m = 1, 3, 7, 9) squares but on the center square (m = 5) it goes berserk and turns into a berserker. For this sequence, the berserker can move to three of the side squares and three of the corners from the center.

The berserker is one of the Lewis chessmen which were discovered in 1831 on the Isle of Lewis. They are carved from walrus ivory in Scandinavian style of the 12th century. The pawns look like decorated tombstones. The pieces have all human representations with facial expressions varying from gloom to anger. Some of the rooks show men biting their shield in the manner of berserkers. According to Hooper and Whyld none looks happy.

Let A be the adjacency matrix of the graph G, where V(G) = {v1, v2, v3, v4, v5, v6, v7, v8, v9}. Then the (m, k) entry of A^n is the number of different vm-vk walks of length n in G, see the Chartrand reference. In the adjacency matrix A, see the Maple program, the A[1], A[3], A[7] and A[9] vectors represent the rook moves on the corner squares, the A[2], A[4], A[6] and A[8] vectors represent the rook moves on the side squares and the A[5] vector represents the moves of the berserker. On a 3 X 3 chessboard there are 2^9 = 512 ways a berserker could move from the center square (off the center the berserker behaves like a rook) so there are 512 different berserkers.

For the side squares the 512 berserker vectors lead to 42 different sequences, see the overview of berserker sequences. There are 16 berserker vectors that lead to the sequence given above. Their decimal [binary] values are: 111 [001 101 111] , 207 [011 001 111], 231 [011 100 111], 237 [011 101 101], 303 [100 101 111], 363 [101 101 011], 366 [101 101 110], 399 [110 001 111], 423 [110 100 111], 429 [110 101 101], 459 [111 001 011], 462 [111 001 110], 483 [111 100 011], 486 [111 100 110], 489 [111 101 001] and 492 [111 101 100]. These berserker vectors lead for the corner squares to sequence 4*A179606 (with leading term 1 added) and for the central square to sequence 6*A179606 (with leading term 1 added).

This sequence belongs to a family of sequences with GF(x)=(1+x-k*x^2)/(1-3*x+(k-4)*x^2), see A180142.

Row sums are A003688. Diagonal sums are A116413. Product of A007318 and A116413 is A116414. Product of A007318 and A105475.

Subtriangle of triangle given by (0, 3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 18 2012

See A214992 for a discussion of power floor sequence and the power floor function, p1(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(6), and the limit p1(r) = 3.77794213613376987528458445727451673384055973517...

See A214992 for a discussion of power ceiling sequence and the power ceiling function, p4(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 2+sqrt(6), and the limit p4(r) = 5.2127890589687233047437696796862841514303439...

See A218984 for the power floor function, p1(x). For comparison of p4 and p1, limit(p4(r)/p1(r)) = 2*(1+sqrt(6))/5 = 1.379795897113271239278913629882356556786378...

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 or 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.

The sequence above corresponds to 6 red king vectors, i.e., A[5] vectors, with decimal values 335, 359, 365, 455, 461 and 485. These vectors lead for the side squares to A123347 and for the central square to A179601.

The a(n) represent the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.

The sequence above corresponds to 6 red king vectors, i.e., A[5] vectors, with decimal values 335, 359, 365, 455, 461 and 485. These vectors lead for the corner squares to A179600 and for the side squares to A123347.