A108851 -id:A108851 - cp's OEIS frontend

A126473 Number of strings over a 5 symbol alphabet with adjacent symbols differing by three or less.

1, 5, 23, 107, 497, 2309, 10727, 49835, 231521, 1075589, 4996919, 23214443, 107848529, 501037445, 2327695367, 10813893803, 50238661313, 233396326661, 1084301290583, 5037394142315, 23402480441009, 108722104190981, 505095858086951, 2346549744920747

Offset: 0

R. H. Hardin, Dec 27 2006

[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

Shanzhen Gao and Keh-Hsun Chen, Tackling Sequences From Prudent Self-Avoiding Walks, FCS'14, The 2014 International Conference on Foundations of Computer Science.
S. Gao and H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks, 2010.
Index entries for linear recurrences with constant coefficients, signature (4,3).

Cf. 5 symbol differing by two or less A126392, one or less A057960.
Cf. Red king sequences side squares [numerical value A[5]]: A086347 [495], A179598 [239], A126473 [367], A123347 [335], A179602 [95], A154964 [31], A015448 [327], A152187 [27], A003947 [325], A108981 [11], A007483 [2]. - Johannes W. Meijer, Aug 01 2010
Cf. A055099.

with(LinearAlgebra): nmax:=19; m:=2; A[5]:= [1,0,1,1,0,1,1,1,1]: A:=Matrix([[0,1,0,1,1,0,0,0,0],[1,0,1,1,1,1,0,0,0],[0,1,0,0,1,1,0,0,0],[1,1,0,0,1,0,1,1,0],A[5],[0,1,1,0,1,0,0,1,1],[0,0,0,1,1,0,0,1,0],[0,0,0,1,1,1,1,0,1],[0,0,0,0,1,1,0,1,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Aug 01 2010
# second Maple program:
a:= n-> (M-> M[1,2]+M[2,2])(<<0|1>, <3|4>>^n):
seq(a(n), n=0..24);  # Alois P. Heinz, Jun 28 2021

LinearRecurrence[{4, 3}, {1, 5}, 24] (* Jean-François Alcover, Dec 10 2024 *)

a(n)=([0,1; 3,4]^n*[1;5])[1,1] \\ Charles R Greathouse IV, May 10 2016

From Johannes W. Meijer, Aug 01 2010: (Start)
G.f.: (1+x)/(1-4*x-3*x^2).
a(n) = 4*a(n-1) + 3*a(n-2) with a(0) = 1 and a(1) = 5.
a(n) = ((1+3/sqrt(7))/2)*(A)^(-n) + ((1-3/sqrt(7))/2)*(B)^(-n) with A = (-2 + sqrt(7))/3 and B = (-2-sqrt(7))/3.
Lim_{k->oo} a(n+k)/a(k) = (-1)^(n+1)*A000244(n)/(A015530(n)*sqrt(7)-A108851(n))
(End)
a(n) = A015330(n)+A015330(n+1). - R. J. Mathar, May 09 2023

Edited by Johannes W. Meijer, Aug 10 2010

A015530 Expansion of x/(1 - 4x - 3x^2).

Original entry on oeis.org

0, 1, 4, 19, 88, 409, 1900, 8827, 41008, 190513, 885076, 4111843, 19102600, 88745929, 412291516, 1915403851, 8898489952, 41340171361, 192056155300, 892245135283, 4145149007032, 19257331433977, 89464772757004

Offset: 0

Olivier Gérard

Let b(1)=1, b(k) = floor(b(k-1)) + 3/b(k-1); then for n>1, b(n) = a(n)/a(n-1). - Benoit Cloitre, Sep 09 2002
In general, x/(1 - a*x - b*x^2) has a(n) = Sum_{k=0..floor((n-1)/2)} C(n-k-1,k)*b^k*a^(n-2k-1). - Paul Barry, Apr 23 2005
Pisano period lengths: 1, 2, 1, 4, 24, 2, 21, 4, 3, 24, 40, 4, 84, 42, 24, 8, 288, 6, 18, 24, ... . - R. J. Mathar, Aug 10 2012
This is the Lucas sequence U(4,-3). - Bruno Berselli, Jan 09 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Lucyna Trojnar-Spelina and Iwona Włoch, On Generalized Pell and Pell-Lucas Numbers, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.
Wikipedia, Lucas sequence.
Index entries for linear recurrences with constant coefficients, signature (4,3).

Appears in A179596, A126473 and A179597. - Johannes W. Meijer, Aug 01 2010

Cf. A080042: Lucas sequence V(4,-3).

I:=[0, 1]; [n le 2 select I[n] else 4*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 19 2012

LinearRecurrence[{4,3},{0,1},30] (* Vincenzo Librandi, Jun 19 2012 *)

x='x+O('x^30); concat([0], Vec(x/(1-4*x-3*x^2))) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,4,-3) for n in range(0, 23)]# Zerinvary Lajos, Apr 23 2009

a(n) = 4*a(n-1) + 3*a(n-2).
a(n) = (A086901(n+2) - A086901(n+1))/6. - Ralf Stephan, Feb 01 2004
a(n) = Sum_{k=0..floor((n-1)/2)} C(n-k-1, k)*3^k*4^(n-2k-1). - Paul Barry, Apr 23 2005
a(n) = ((2+sqrt(7))^n - (2-sqrt(7))^n)/sqrt(28). Offset 1. a(3)=19. - Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009
From Johannes W. Meijer, Aug 01 2010: (Start)
Limit(a(n+k)/a(k), k=infinity) = A108851(n)+a(n)*sqrt(7).
Limit(A108851(n)/a(n), n=infinity) = sqrt(7). (End)
G.f.: x*G(0) where G(k)= 1 + (4*x+3*x^2)/(1 - (4*x+3*x^2)/(4*x + 3*x^2 + 1/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jul 28 2012
G.f.: G(0)*x/(2-4*x), where G(k)= 1 + 1/(1 - x*(7*k-4)/(x*(7*k+3) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 16 2013

A179596 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2x - 11x^2 - 6*x^3).

Original entry on oeis.org

1, 3, 17, 73, 351, 1607, 7513, 34809, 161903, 751783, 3493353, 16227737, 75393055, 350251335, 1627192697, 7559508409, 35119644495, 163157037671, 757987215241, 3521419711833, 16359641017343, 76002822156295, 353090213774361

Offset: 0

Johannes W. Meijer, Jul 28 2010; edited Jun 21 2013

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 center square the king goes crazy and turns into a red king.
On a 3 X 3 chessboard there are 2^9 = 512 ways to go crazy on the center square (off the center the piece behaves like a normal king). The red king is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the corner squares the 512 red kings lead to 47 different red king sequences, see the overview of the red king sequences.
The sequence above corresponds to four A[5] vectors with decimal [binary] values 367 [101 101 111], 463 [111 001 111], 487 [111 100 111] and 493 [111 101 101]. These vectors lead for the side squares to A126473 and for the central square to A179597.
This sequence belongs to a family of sequences with g.f. (1+x)/(1 - 2*x - (k+8)*x^2 - 2*k*x^3). Red king sequences that are members of this family are A083424 (k=0), A179604 (k=1), A179600 (k=2), A179596 (k=3; this sequence) and A086346 (k=4). Other members of this family are A015528 (k=5) and A179608 (k=-4).

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Charles Krauthammer, Did Chess Make Him Crazy?, Time, April 26, 2005.
Johannes W. Meijer, The red king sequences.
Index entries for linear recurrences with constant coefficients, signature (2,11,6).

Cf. A180140 (berserker sequences).

Cf. Red king sequences corner squares [decimal value A[5]]: A086346 [495], A015525 [239], A179596 [367], A179600 [335], A015524 [95], A083858 [31], A179604 [327], A015523 [27], A179610 [85], A083424 [325], A015521 [11], A007482 [2], A014335 [16].

nmax:=22; m:=1; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:=[1,1,0,0,1,0,1,1,0]: A[5]:= [1,0,1,1,0,1,1,1,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: A:=Matrix([A[1],A[2],A[3],A[4],A[5], A[6],A[7],A[8],A[9]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

LinearRecurrence[{2,11,6},{1,3,17},30] (* Harvey P. Dale, May 18 2011 *)

Vec((1+x)/(1-2*x-11*x^2-6*x^3)+O(x^99)) \\ Charles R Greathouse IV, Jul 16 2011

G.f.: (1+x)/(1 - 2*x - 11*x^2 - 6*x^3).
a(n) = 2*a(n-1) + 11*a(n-2) + 6*a(n-3) with a(0)=1, a(1)=3 and a(2)=17.
a(n) = (-1)^(-n)*2^(n+1)/9 + ((49+17*sqrt(7))*A^(-n) + (49-17*sqrt(7))*B^(-n))/126 with A = (-2+sqrt(7))/3 and B = (-2-sqrt(7))/3.
Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n+1)*A000244(n)/(A015530(n)*sqrt(7) - A108851(n)).

A179597 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 5x + 2x^2)/(1 - 2x - 11x^2 - 6*x^3).

Original entry on oeis.org

1, 7, 27, 137, 613, 2895, 13355, 62233, 288741, 1342175, 6233899, 28964169, 134554277, 625117807, 2904117675, 13491856889, 62679715045, 291194561919, 1352817130667, 6284852732713, 29197861274277, 135646005392399

Offset: 0

Johannes W. Meijer, Jul 28 2010, Aug 10 2010

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.
For the central square 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 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 side squares to A126473.
This sequence belongs to a family of sequences with g.f. (1 + (k+2)*x + (2*k-4)*x^2)/(1 - 2*x - (k+8)*x^2 - (2*k)*x^3). Red king sequences that are members of this family are A179607 (k=0), A179605 (k=1), A179601 (k=2), A179597 (k=3; this sequence) and A086348 (k=4). Another member of this family is A179609 (k = -4).

Index entries for linear recurrences with constant coefficients, signature (2,11,6).

Red king sequences central square [numerical value A[5]]: A086348 [495], A179599 [239], A179597 [367], A179601 [335], A179603 [95], A154964 [31], A179605 [327], A179606 [27], A179611 [15], A179607 [325], A015521 [11], A007483 [2], A000012 [16], A000007 [0].

with(LinearAlgebra): nmax:=21; m:=5; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:=[1,1,0,0,1,0,1,1,0]: A[5]:= [1,0,1,1,0,1,1,1,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: A:=Matrix([A[1],A[2],A[3],A[4],A[5], A[6],A[7],A[8],A[9]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

LinearRecurrence[{2,11,6},{1,7,27},30] (* Harvey P. Dale, Mar 01 2015 *)

G.f.: (1 + 5*x + 2*x^2)/(1 - 2*x - 11*x^2 - 6*x^3).
a(n) = 2*a(n-1) + 11*a(n-2) + 6*a(n-3) with a(0) = 1, a(1) = 7 and a(2) = 27.
a(n) = 8*(-1/2)^(-n+1)/9 + ((7+11*sqrt(7))*A^(-n-1) + (7-11*sqrt(7))*B^(-n-1))/126 with A = (-2+sqrt(7))/3 and B = (-2-sqrt(7))/3.
Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n+1)*(A000244(n)/(A015530(n)*sqrt(7) - A108851(n))).

A201730 Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 5, 1, 0, 14, 6, 0, 0, 41, 26, 1, 0, 0, 122, 100, 10, 0, 0, 0, 365, 363, 63, 1, 0, 0, 0, 1094, 1274, 322, 14, 0, 0, 0, 0, 3281, 4372, 1462, 116, 1, 0, 0, 0, 0, 9842, 14760, 6156, 744, 18, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 04 2011

Riordan array ((1-2x)/(1-4x+3x^2),x^2/(1-4x+3x^2)).

A007318*A201701 as lower triangular matrices.

			Triangle begins:
1
2, 0
5, 1, 0
14, 6, 0, 0
41, 26, 1, 0, 0
122, 100, 10, 0, 0, 0
365, 363, 63, 1, 0, 0, 0

Cf. A007051 (1st column), A261064 (2nd column).

A201730 := proc(n,k)
    (1-2*x)/(1-4*x+(3-y)*x^2) ;
    coeftayl(%,y=0,k) ;
    coeftayl(%,x=0,n) ;
end proc:
seq(seq(A201730(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Dec 06 2011

m = 13;
(* DELTA is defined in A084938 *)
DELTA[Join[{2, 1/2, 3/2}, Table[0, {m}]], Join[{0, 1/2, -1/2}, Table[0, {m}]], m] // Flatten (* Jean-François Alcover, Feb 19 2020 *)

G.f.: (1-2x)/(1-4x+(3-y)*x^2).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A139011(n), A000079(n), A007051(n), A006012(n), A001075(n), A081294(n), A001077(n), A084059(n), A108851(n), A084128(n), A081340(n), A084132(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
Sum_{k, k>+0} T(n+k,k) = A081704(n) .
T(n,k) = 3*T(n-1,k)+ Sum_{j>0} T(n-1-j,k-1).
T(n,k) = 4*T(n-1,k)+ T(n-2,k-1) - 3*T(n-2,k) with T(0,0)=1, T(1,0)= 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if n

A146963 a(n) = ((3 + sqrt(7))^n + (3 - sqrt(7))^n)/2.

Original entry on oeis.org

1, 3, 16, 90, 508, 2868, 16192, 91416, 516112, 2913840, 16450816, 92877216, 524361664, 2960415552, 16713769984, 94361788800, 532743192832, 3007735579392, 16980927090688, 95870091385344, 541258694130688

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Nov 03 2008

nonn

Binomial transform of A108851.

Inverse binomial transform of A146964.

Vincenzo Librandi, Table of n, a(n) for n = 0..158
Index entries for linear recurrences with constant coefficients, signature (6,-2).

Cf. A098158, A108851, A146964.

a:=[1,3];; for n in [3..25] do a[n]:=6*a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Jan 08 2020

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((3+r7)^n+(3-r7)^n)/2: n in [0..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 05 2008

seq(coeff(series((1-3*x)/(1-6*x+2*x^2), x, n+1), x, n), n = 0..25); # G. C. Greubel, Jan 08 2020

Transpose[NestList[Join[{Last[#],6Last[#]-2First[#]}]&,{1,3},25]] [[1]]  (* or *) CoefficientList[Series[(1-3x)/(1-6x+2x^2),{x,0,25}],x]  (* Harvey P. Dale, Apr 11 2011 *)
LinearRecurrence[{6,-2}, {1,3}, 25] (* G. C. Greubel, Jan 08 2020 *)

my(x='x+O('x^25)); Vec((1-3*x)/(1-6*x+2*x^2)) \\ G. C. Greubel, Jan 08 2020

def A146963_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-3*x)/(1-6*x+2*x^2) ).list()
A146963_list(25) # G. C. Greubel, Jan 08 2020

From Philippe Deléham and Klaus Brockhaus, Nov 05 2008: (Start)
a(n) = 6*a(n-1) - 2*a(n-2) with a(0)=1, a(1)=3.
G.f.: (1-3*x)/(1-6*x+2*x^2). (End)
a(n) = (Sum_{k=0..n} A098158(n,k)*3^(2*k)*7^(n-k))/3^n. - Philippe Deléham, Nov 06 2008
E.g.f.: exp(3*x)*cosh(sqrt(7)*x). - G. C. Greubel, Jan 08 2020
a(n) = A154244(n)-3*A154244(n-1). - R. J. Mathar, Jan 25 2023

Extended beyond a(7) by Klaus Brockhaus, Nov 05 2008

Edited by Klaus Brockhaus, Jul 16 2009

A191347 Array read by antidiagonals: ((floor(sqrt(n)) + sqrt(n))^k + (floor(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 1, 1, 0, 8, 7, 4, 2, 1, 0, 16, 17, 10, 8, 2, 1, 0, 32, 41, 28, 32, 9, 2, 1, 0, 64, 99, 76, 128, 38, 10, 2, 1, 0, 128, 239, 208, 512, 161, 44, 11, 2, 1, 0, 256, 577, 568, 2048, 682, 196, 50, 12, 3, 1

Offset: 0

Charles L. Hohn, May 31 2011

			1, 0,  0,   0,    0,    0,     0,      0,       0,        0,        0, ...
1, 1,  2,   4,    8,   16,    32,     64,     128,      256,      512, ...
1, 1,  3,   7,   17,   41,    99,    239,     577,     1393,     3363, ...
1, 1,  4,  10,   28,   76,   208,    568,    1552,     4240,    11584, ...
1, 2,  8,  32,  128,  512,  2048,   8192,   32768,   131072,   524288, ...
1, 2,  9,  38,  161,  682,  2889,  12238,   51841,   219602,   930249, ...
1, 2, 10,  44,  196,  872,  3880,  17264,   76816,   341792,  1520800, ...
1, 2, 11,  50,  233, 1082,  5027,  23354,  108497,   504050,  2341691, ...
1, 2, 12,  56,  272, 1312,  6336,  30592,  147712,   713216,  3443712, ...
1, 3, 18, 108,  648, 3888, 23328, 139968,  839808,  5038848, 30233088, ...
1, 3, 19, 117,  721, 4443, 27379, 168717, 1039681,  6406803, 39480499, ...
1, 3, 20, 126,  796, 5028, 31760, 200616, 1267216,  8004528, 50561600, ...
1, 3, 21, 135,  873, 5643, 36477, 235791, 1524177,  9852435, 63687141, ...
1, 3, 22, 144,  952, 6288, 41536, 274368, 1812352, 11971584, 79078912, ...
1, 3, 23, 153, 1033, 6963, 46943, 316473, 2133553, 14383683, 96969863, ...
...

Row 1 is A000007, row 2 is A011782, row 3 is A001333, row 4 is A026150, row 5 is A081294, row 6 is A001077, row 7 is A084059, row 8 is A108851, row 9 is A084128, row 10 is A081341, row 11 is A005667, row 13 is A141041.
Row 3*2 is A002203, row 4*2 is A080040, row 5*2 is A155543, row 6*2 is A014448, row 8*2 is A080042, row 9*2 is A170931, row 11*2 is A085447.
Cf. A191348 which uses ceiling() in place of floor().

T(n, k) = if (n==0, k==0, my(x=sqrtint(n)); sum(i=0, (k+1)\2, binomial(k, 2*i)*x^(k-2*i)*n^i));
matrix(9,9, n, k, T(n-1,k-1)) \\ Michel Marcus, Aug 22 2019

T(n, k) = if (k==0, 1, if (k==1, sqrtint(n), T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2));
matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 22 2019

For each row n>=0 let T(n,0)=1 and T(n,1)=floor(sqrt(n)), then for each column k>=2: T(n,k)=T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 22 2019

T(n, k) = Sum_{i=0..floor((k+1)/2)} binomial(k, 2*i)*floor(sqrt(n))^(k-2*i)*n^i for n > 0, with T(0, 0) = 1 and T(0, k) = 0 for k > 0. - Michel Marcus, Aug 23 2019

Showing 1-7 of 7 results.

cp's OEIS Frontend

A126473 Number of strings over a 5 symbol alphabet with adjacent symbols differing by three or less.

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A015530 Expansion of x/(1 - 4*x - 3*x^2).

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A179596 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2*x - 11*x^2 - 6*x^3).

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A179597 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 5*x + 2*x^2)/(1 - 2*x - 11*x^2 - 6*x^3).

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A201730 Triangle T(n,k), read by rows, given by (2,1/2,3/2,0,0,0,0,0,0,0,...) DELTA (0,1/2,-1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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A146963 a(n) = ((3 + sqrt(7))^n + (3 - sqrt(7))^n)/2.

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A191347 Array read by antidiagonals: ((floor(sqrt(n)) + sqrt(n))^k + (floor(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

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A015530 Expansion of x/(1 - 4x - 3x^2).

A179596 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2x - 11x^2 - 6*x^3).

A179597 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 5x + 2x^2)/(1 - 2x - 11x^2 - 6*x^3).