A094865 -id:A094865 - cp's OEIS frontend

A217778 Expansion of (1-x)^2(1-3x)/((1-3x+x^2)(1-5x+5x^2)).

1, 3, 10, 34, 117, 407, 1429, 5055, 17986, 64278, 230473, 828391, 2982825, 10754459, 38811802, 140165322, 506449789, 1830590295, 6618524221, 23933966743, 86562282258, 313102489406, 1132598701585, 4097213146599, 14822370816337, 53623952036787

Offset: 0

Philippe Deléham, Mar 24 2013

A diagonal of the square array A217770.

Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Cf. A217770.

m:=26; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)^2*(1-3*x)/((1-3*x+x^2)*(1-5*x+5*x^2)))); // Bruno Berselli, Mar 28 2013

LinearRecurrence[{8, -21, 20, -5}, {1, 3, 10, 34}, 26] (* Bruno Berselli, Mar 28 2013 *)
CoefficientList[Series[(1-x)^2(1-3x)/((1-3x+x^2)(1-5x+5x^2)),{x,0,30}],x] (* Harvey P. Dale, Sep 26 2023 *)

makelist(expand(((3+sqrt(5))*(5+sqrt(5))^n-(3-sqrt(5))*(5-sqrt(5))^n+(1+sqrt(5))*(3+sqrt(5))^n-(1-sqrt(5))*(3-sqrt(5))^n)/(4*2^n*sqrt(5))), n, 0, 25); /* Bruno Berselli, Mar 28 2013 */

G.f.: (1-5*x+7*x^2-3*x^3)/(1-8*x+21*x^2-20*x^3+5*x^4).
a(n) = A081567(n) - A094865(n).
a(n) = A217770(n+1,n).
a(n) = 8*a(n-1) -21*a(n-2) +20*a(n-3) -5*a(n-4) for n>3, a(0)=1, a(1)=3, a(2)=10, a(3)=34.
a(n) = ((3+r)*(5+r)^n-(3-r)*(5-r)^n+(1+r)*(3+r)^n-(1-r)*(3-r)^n)/(4*r*2^n), where r=sqrt(5). [Bruno Berselli, Mar 28 2013]

A005024 Number of walks of length 2n+8 in the path graph P_9 from one end to the other.

Original entry on oeis.org

8, 43, 196, 820, 3264, 12597, 47652, 177859, 657800, 2417416, 8844448, 32256553, 117378336, 426440955, 1547491404, 5610955132, 20332248992, 73645557469, 266668876540, 965384509651, 3494279574288, 12646311635088, 45764967830976

Offset: 1

N. J. A. Sloane

W. Feller, An Introduction to Probability Theory and its Applications, 3rd ed, Wiley, New York, 1968, p. 96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. J. Everett, P. R. Stein, The combinatorics of random walk with absorbing barriers, Discrete Math. 17 (1977), no. 1, 27-45.
C. J. Everett, P. R. Stein, The combinatorics of random walk with absorbing barriers, Discrete Math. 17 (1977), no. 1, 27-45. [Annotated scanned copy]
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Cf. A005023. Truncated version of A094865.

I:=[8, 43, 196, 820]; [n le 4 select I[n] else 8*Self(n-1)-21*Self(n-2)+20*Self(n-3)-5*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 08 2013

a:=k->sum(binomial(8+2*k,10*j+k-2),j=ceil((2-k)/10)..floor((10+k)/10))-sum(binomial(8+2*k,10*j+k-1),j=ceil((1-k)/10)..floor((9+k)/10)): seq(a(k),k=1..28);
A005024:=-(-8+21*z-20*z**2+5*z**3)/(5*z**2-5*z+1)/(z**2-3*z+1); # conjectured by Simon Plouffe in his 1992 dissertation

CoefficientList[Series[(-5 z^3 + 20 z^2 - 21 z + 8)/((z^2 - 3 z + 1) (5 z^2 - 5 z + 1)), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 27 2011 *)
CoefficientList[Series[(1 / x) (1 / (1 - 8 x + 21 x^2 - 20 x^3 + 5 x^4) - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 08 2013 *)

x='x+O('x^66); Vec(-1+1/((1-3*x+x^2)*(1-5*x+5*x^2))) \\ Joerg Arndt, May 01 2013

From Emeric Deutsch, Apr 02 2004: (Start)
G.f. (assuming a(0)=1): 1/(1 - 8x + 21x^2 - 20x^3 + 5x^4) - 1.
a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4). (End)
a(k) = sum(binomial(8+2k, 10j+k-2)-binomial(8+2k, 10j+k-1), j=-infinity..infinity) (a finite sum).

Better definition from Emeric Deutsch, Apr 02 2004

A094825 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 1, s(2n) = 7.

Original entry on oeis.org

1, 7, 35, 153, 624, 2444, 9333, 35055, 130207, 479941, 1759616, 6427032, 23412105, 85121783, 309062619, 1121050449, 4063463728, 14721293860, 53313308477, 193023319071, 698715633111, 2528895064637, 9152032060800, 33118656195888

Offset: 3

Herbert Kociemba, Jun 15 2004

nonn

Michael De Vlieger, Table of n, a(n) for n = 3..1792
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Cf. A094865 (partial sums).

LinearRecurrence[{8,-21,20,-5},{1,7,35,153},30] (* Harvey P. Dale, Jan 16 2015 *)

a(n) = (1/5)*Sum_{r=1..9} sin(r*Pi/10)*sin(7*r*Pi/10)*(2*cos(r*Pi/10))^(2n).
a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4).
G.f.: x^3*(1-x)/( (1-3*x+x^2)*(1-5*x+5*x^2) ).
a(n) = -A001906(n)/2 + A020876(n)/10. - R. J. Mathar, Jun 24 2011

A217593 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=1 or if k-n >= 9, T(0,k) = 1 for k = 0..8, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 5, 0, 0, 0, 1, 5, 9, 5, 0, 0, 0, 1, 6, 14, 14, 0, 0, 0, 0, 1, 7, 20, 28, 14, 0, 0, 0, 0, 0, 8, 27, 48, 42, 0, 0, 0, 0, 0, 0, 8, 35, 75, 90, 42, 0, 0, 0, 0, 0, 0, 0, 43, 110, 165, 132, 0, 0, 0, 0, 0, 0, 0, 0, 43, 153, 275, 297, 132, 0, 0, 0, 0, 0, 0, 0, 0, 0, 196, 428, 572, 429, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 18 2013

			Square array begins :
1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, ...
0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 0, 0, ...
0, 0, 2, 5, 9, 14, 20, 27, 35, 43, 43, 0, 0, ...
0, 0, 0, 5, 14, 28, 75, 110, 153, 196, 196, 0, 0, ....
0, 0, 0, 0, 14, 42, 90, 165, 275, 428, 624, 820, 820, 0, 0, ...
...
Square array, read by rows, with 0 omitted:
1, 1, 1, 1, 1, 1, 1, 1, 1
1, 2, 3, 4, 5, 6, 7, 8, 8
2, 5, 9, 14, 20, 27, 35, 43, 43
5, 14, 28, 48, 75, 110, 153, 196, 196
14, 42, 90, 165, 275, 428, 624, 820, 820
42, 132, 297, 572, 1000, 1624, 2444, 3264, 3264
132, 429, 1001, 2001, 3625, 6069, 9333, 12597, 12597
429, 1430, 3431, 7056, 13125, 22458, 35055, 47652, 47652
...

A hexagon arithmetic of E. Lucas.

T(n,n)   = A033191(n).
T(n,n+1) = A033191(n+1).
T(n,n+2) = A033190(n+1).
T(n,n+3) = A094667(n+1).
T(n,n+4) = A093131(n+1) = A030191(n).
T(n,n+5) = A094788(n+2).
T(n,n+6) = A094825(n+3).
T(n,n+7) = T(n,n+8) = A094865(n+3).
Sum_{k, 0<=k<=n} T(n-k,k) = A178381(n).

cp's OEIS Frontend

A217778 Expansion of (1-x)^2*(1-3*x)/((1-3*x+x^2)*(1-5*x+5*x^2)).

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A005024 Number of walks of length 2n+8 in the path graph P_9 from one end to the other.

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A094825 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 1, s(2n) = 7.

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A217593 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=1 or if k-n >= 9, T(0,k) = 1 for k = 0..8, T(n,k) = T(n-1,k) + T(n,k-1).

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A217778 Expansion of (1-x)^2(1-3x)/((1-3x+x^2)(1-5x+5x^2)).