A107979 -id:A107979 - cp's OEIS frontend

A105980 Near-repdigit primes with at least two 4's as the repeated digit.

443, 449, 4441, 4447, 44449, 444443, 444449, 444444443, 4444444447, 44444444441, 444444444443, 44444444444444444447, 44444444444444444444444447, 4444444444444444444444444441, 444444444444444444444444444443

Offset: 1

Shyam Sunder Gupta, Apr 29 2005

			a(2)=449 is a term because all digits are equal to 4 except one.

Harvey P. Dale, Table of n, a(n) for n = 1..26

Cf. A105976, A105978, A107979, A105980, A105982, A365592, A365596, A365597, A365598.

Select[Flatten[Table[FromDigits[PadLeft[{n},x,4]],{x,3,30},{n,{1,3,7,9}}]],PrimeQ] (* Harvey P. Dale, Feb 18 2018 *)

More terms from Harvey P. Dale, Feb 18 2018

Name clarified by Robert Price, Sep 10 2023

A180147 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 4x - 3x^2 + 6x^3).

Original entry on oeis.org

1, 7, 31, 139, 607, 2659, 11623, 50827, 222223, 971635, 4248247, 18574555, 81213151, 355086787, 1552539271, 6788138539, 29679651247, 129767784979, 567381262423, 2480750497147, 10846539065983, 47424120180835

Offset: 0

Johannes W. Meijer, Aug 13 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 rook on the eight side and corner squares but on the central square the rook goes berserk and turns into a berserker, see A180140.
On a 3 X 3 chessboard there are 2^9 = 512 ways to go berserk on the central square (we assume here that a berserker might behave like a rook). The berserker is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program. For the central squares the 512 berserkers lead to 42 berserker sequences, see the cross-references for some examples.
The sequence above corresponds to six A[5] vectors with decimal values between 191 and 506. These vectors lead for the corner squares to A180145 and for the side squares to A180146.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4, 3, -6).

Cf. A180141 (corner squares), A180140 (side squares), A180147 (central square).

Cf. Berserker sequences central square [numerical values A[5]]: A000007 [0], A000012 [16], 2*A001835 [17, n>=1 and a(0)=1], A155116 [3], A077829 [7], A000302 [15], 6*A179606 [111, with leading 1 added], 2*A033887 [95, n>=1 and a(0)=1], A180147 [191, this sequence], 2*A180141 [495, n>=1 and a(0)=1], 4*A107979 [383, with leading 1 added].

with(LinearAlgebra): nmax:=22; m:=5; A[5]:=[0,1,0,1,1,1,1,1,1]: A:= Matrix([[0,1,1,1,0,0,1,0,0], [1,0,1,0,1,0,0,1,0], [1,1,0,0,0,1,0,0,1], [1,0,0,0,1,1,1,0,0], A[5], [0,0,1,1,1,0,0,0,1], [1,0,0,1,0,0,0,1,1], [0,1,0,0,1,0,1,0,1], [0,0,1,0,0,1,1,1,0]]): 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);

CoefficientList[Series[(1+3x)/(1-4x-3x^2+6x^3),{x,0,40}],x] (* or *) LinearRecurrence[{4,3,-6},{1,7,31},40] (* Harvey P. Dale, Oct 10 2011 *)

G.f.: (1+3*x)/(1 - 4*x - 3*x^2 + 6*x^3).
a(n) = 4*a(n-1) + 3*a(n-2) - 6*a(n-3) with a(0)=1, a(1)=7 and a(2)=31.
a(n) = -1/2 + (7+6*A)*A^(-n-1)/22 + (7+6*B)*B^(-n-1)/22 with A=(-3+sqrt(33))/12 and B=(-3-sqrt(33))/12.
a(n) = A180146(n) + 3*A180146(n-1) with A180146(-1) = 0.

A365592 Near-repdigit primes with at least two 1's as the repeated digit.

Original entry on oeis.org

113, 1117, 11113, 11117, 11119, 111119, 11111117, 11111119, 111111113, 11111111113, 11111111111111119, 1111111111111111111, 11111111111111111111111, 11111111111111111111117, 111111111111111111111113, 11111111111111111111111111117

Offset: 1

Robert Price, Sep 10 2023

			1117 is a term because all digits are equal to 1 except the last one.

Robert Price, Table of n, a(n) for n = 1..24

Cf. A105976, A105978, A107979, A105980, A105982, A365592, A365596, A365597, A365598.

A268409 a(n) = 4a(n - 1) + 2a(n - 2) for n>1, a(0)=3, a(1)=5.

Original entry on oeis.org

3, 5, 26, 114, 508, 2260, 10056, 44744, 199088, 885840, 3941536, 17537824, 78034368, 347213120, 1544921216, 6874111104, 30586286848, 136093369600, 605546052096, 2694370947584, 11988575894528, 53343045473280, 237349333682176, 1056083425675264

Offset: 0

Ilya Gutkovskiy, Feb 04 2016

In general, the ordinary generating function for the recurrence relation b(n) = r*b(n - 1) + s*b(n - 2), with n>1 and b(0)=k, b(1)=m, is (k - (k*r - m)*x)/(1 - r*x - s*x^2). This recurrence gives the closed form b(n) = (2^(-n - 1)*((k*r - 2*m)*(r - sqrt(r^2 + 4*s))^n + (2*m - k*r)*(sqrt(r^2 + 4*s) + r)^n + k*sqrt(r^2 + 4*s)*(r - sqrt(r^2 + 4*s))^n + k*sqrt(r^2 + 4*s)*(sqrt(r^2 + 4*s) + r)^n))/sqrt(r^2 + 4*s).

Index entries for linear recurrences with constant coefficients, signature (4,2).

Cf. A021001, A084059, A090017, A107979, A164549, A176213, A189741.

[n le 2 select 2*n+1 else 4*Self(n-1)+2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 04 2016

RecurrenceTable[{a[0] == 3, a[1] == 5, a[n] == 4 a[n - 1] + 2 a[n - 2]}, a, {n, 23}]
LinearRecurrence[{4, 2}, {3, 5}, 24]
Table[((18 + Sqrt[6]) (2 - Sqrt[6])^n - (Sqrt[6] - 18) (2 + Sqrt[6])^n)/12, {n, 0, 23}]

Vec((3 - 7*x)/(1 - 4*x - 2*x^2) + O(x^30)) \\ Michel Marcus, Feb 04 2016

G.f.: (3 - 7*x)/(1 - 4*x - 2*x^2).
a(n) = ((18 + sqrt(6))*(2 - sqrt(6))^n - (sqrt(6) - 18)*(2 + sqrt(6))^n)/12.
Lim_{n -> infinity} a(n + 1)/a(n) = 2 + sqrt(6) = A176213.
a(n) = 3*A090017(n+1) -7*A090017(n). - R. J. Mathar, Mar 12 2017

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A105980 Near-repdigit primes with at least two 4's as the repeated digit.

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A180147 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + 3*x)/(1 - 4*x - 3*x^2 + 6*x^3).

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A365592 Near-repdigit primes with at least two 1's as the repeated digit.

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A268409 a(n) = 4*a(n - 1) + 2*a(n - 2) for n>1, a(0)=3, a(1)=5.

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A180147 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 4x - 3x^2 + 6x^3).

A268409 a(n) = 4a(n - 1) + 2a(n - 2) for n>1, a(0)=3, a(1)=5.