I. J. Kennedy - cp's OEIS frontend

User: I. J. Kennedy

I. J. Kennedy has authored 3 sequences.

A125268 Numbers that end with decimal digit 1, 3, 7, or 9 and that produce only composite numbers when any of the digits 0,1,...,9 is inserted anywhere in them (including at the beginning or end).

25011, 52647, 72753, 122313, 168699, 283251, 324021, 598041, 783441, 804131, 837207, 924807, 1247241, 1905759, 2514819, 3461101, 3514077, 3617389, 3905817, 4112913, 4142139, 4203151, 4229871, 4283679, 4531907, 4628827, 4828443, 5380413, 5478091, 5632671, 5714889, 5818569, 5989269, 5990961

Offset: 1

I. J. Kennedy, Jan 15 2007

Since digit 0 can be inserted at the beginning of a term, each term must be composite.

Max Alekseyev, Table of n, a(n) for n = 1..972 (all terms below 10^8)

Cf. A003459, A124665

filter:= proc(n) local x,y,d,t;
  x:= n; y:= 0;
  for d from 0 to ilog10(n)+1 do
     for t from 0 to 9 do
       if isprime(10^(d+1)*x+10^d*t + y) then return false fi;
     od;
     t:= x mod 10;
     y:= y + 10^d*t;
     x:= (x-t)/10;
  od;
  true
end proc:
select(filter, [seq(seq(10*i+j,j=[1,3,7,9]),i=0..10^6)]); # Robert Israel, Sep 12 2016

{ printA125268(U=8) = my(v,t); v=vector(10^U); forprime(p=11,10^(U+1), if(p<=U,v[p]=p); for(i=1,#Str(p), t=(p\10^i) * 10^(i-1) + (p%10^(i-1)); if(#Str(t)==#Str(p)-1,v[t]=p););); forstep(n=1,10^U,2, if(n%10==5||v[n],next); print1(n,", ");); } \\ prints terms below 10^U, by Max Alekseyev, Sep 12 2016

Corrected and extended by Robert G. Wilson v, Jan 26 2007

Removed incorrect terms and extended by Max Alekseyev, Sep 12 2016

A015818 Number of solutions of +- 1 +- 2 +- ... +- (n-1) +- n = 0 in which the partial sums +- 1 +- ... +- k (1<=k<=n) are all distinct.

Original entry on oeis.org

1, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 10, 14, 0, 0, 36, 40, 0, 0, 134, 258, 0, 0, 702, 1040, 0, 0, 4170, 5996, 0, 0, 23642, 36616, 0, 0, 140500, 217002, 0, 0, 852132, 1374692, 0, 0, 5411800, 8852230, 0, 0, 35764246, 56370054, 0, 0, 232969442, 376479130, 0, 0, 1555855594, 2534308444

Offset: 0

Leroy Quet, Christian G. Bower and I. J. Kennedy, Nov 26 2002

nonn

If n==1 or 2 (mod 4) then a(n)=0.

			For n=4 there are 2 solutions: +1-2-3+4=0 and -1+2+3-4=0.

Leroy Quet, Integers Arranged: Q & Puzzle

a(n) <= A063865(n).

issol(i, n) = {b = binary(i); while(length(b) < n, b = concat(0, b)); if (! sum(k=1, n, if (b[k], k, -k)), vsp = []; lastnb = 0; for (j=1, n, vsp = Set(concat(vsp, sum(k=1, j, if (b[k], k, -k)))); if (#vsp == lastnb, return (0)); lastnb = #vsp;); return (1););}
a(n) = if ((!n) || ((n % 4) != 1) && ((n % 4) != 2), sum(i=0, 2^n-1, issol(i, n)));  \\ Michel Marcus, May 22 2014

a(36)-a(46) from Ray Chandler, Nov 29 2008

a(47)-a(58) from Sean A. Irvine, Dec 13 2018

A008466 a(n) = 2^n - Fibonacci(n+2).

Original entry on oeis.org

0, 0, 1, 3, 8, 19, 43, 94, 201, 423, 880, 1815, 3719, 7582, 15397, 31171, 62952, 126891, 255379, 513342, 1030865, 2068495, 4147936, 8313583, 16655823, 33358014, 66791053, 133703499, 267603416, 535524643, 1071563515, 2143959070, 4289264409, 8580707127

Offset: 0

N. J. A. Sloane, I. J. Kennedy, Eric W. Weisstein

Toss a fair coin n times; a(n) is number of possible outcomes having a run of 2 or more heads.
Also the number of binary words of length n with at least two neighboring 1 digits. For example, a(4)=8 because 8 binary words of length 4 have two or more neighboring 1 digits: 0011, 0110, 0111, 1011, 1100, 1101, 1110, 1111 (cf. A143291). - Alois P. Heinz, Jul 18 2008
Equivalently, number of solutions (x_1, ..., x_n) to the equation x_1*x_2 + x_2*x_3 + x_3*x_4 + ... + x_{n-1}*x_n = 1 in base-2 lunar arithmetic. - N. J. A. Sloane, Apr 23 2011
Row sums of triangle A153281 = (1, 3, 8, 19, 43, ...). - Gary W. Adamson, Dec 23 2008
a(n-1) is the number of compositions of n with at least one part >= 3. - Joerg Arndt, Aug 06 2012
One less than the cardinality of the set of possible numbers of (leaf-) nodes of AVL trees with height n (cf. A143897, A217298). a(3) = 4-1, the set of possible numbers of (leaf-) nodes of AVL trees with height 3 is {5,6,7,8}. - Alois P. Heinz, Mar 20 2013
a(n) is the number of binary words of length n such that some prefix contains three more 1's than 0's or two more 0's than 1's. a(4) = 8 because we have: {0,0,0,0}, {0,0,0,1}, {0,0,1,0}, {0,0,1,1}, {0,1,0,0}, {1,0,0,0}, {1,1,1,0}, {1,1,1,1}. - Geoffrey Critzer, Dec 30 2013
With offset 0: antidiagonal sums of P(j,n) array of j-th partial sums of Fibonacci numbers. - Luciano Ancora, Apr 26 2015

			From _Gus Wiseman_, Jun 25 2020: (Start)
The a(2) = 1 through a(5) = 19 compositions of n + 1 with at least one part >= 3 are:
  (3)  (4)    (5)      (6)
       (1,3)  (1,4)    (1,5)
       (3,1)  (2,3)    (2,4)
              (3,2)    (3,3)
              (4,1)    (4,2)
              (1,1,3)  (5,1)
              (1,3,1)  (1,1,4)
              (3,1,1)  (1,2,3)
                       (1,3,2)
                       (1,4,1)
                       (2,1,3)
                       (2,3,1)
                       (3,1,2)
                       (3,2,1)
                       (4,1,1)
                       (1,1,1,3)
                       (1,1,3,1)
                       (1,3,1,1)
                       (3,1,1,1)
(End)

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, p. 300, 1968.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 14, Exercise 1.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 301 terms from T. D. Noe)
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2001. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Simon Cowell, A Formula for the Reliability of a d-dimensional Consecutive-k-out-of-n:F System, arXiv preprint arXiv:1506.03580 [math.CO], 2015.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1020
T. Langley, J. Liese, and J. Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011) # 11.4.2.
B. E. Merkel, Probabilities of Consecutive Events in Coin Flipping, Master's Thesis, Univ. Cincinatti, May 11 2011.
D. J. Persico and H. C. Friedman, Another Coin Tossing Problem, Problem 62-6, SIAM Review, 6 (1964), 313-314.
Eric Weisstein's World of Mathematics, Run.
Index entries for sequences related to dismal (or lunar) arithmetic
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Cf. A050227, A050231, A050232, A050233, A056986.
Cf. A153281, A186244 (ternary words), A335457, A335458, A335516.
The non-contiguous version is A335455.
Row 2 of A340156. Column 3 of A109435.

[2^n-Fibonacci(n+2): n in [0..40]]; // Vincenzo Librandi, Apr 27 2015

a:= n-> (<<3|1|0>, <-1|0|1>, <-2|0|0>>^n)[1, 3]:
seq(a(n), n=0..50); # Alois P. Heinz, Jul 18 2008
# second Maple program:
with(combinat): F:=fibonacci; f:=n->add(2^(n-1-i)*F(i),i=0..n-1); [seq(f(n),n=0..50)]; # N. J. A. Sloane, Mar 31 2014

Table[2^n-Fibonacci[n+2],{n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2008 *)
MMM = 30;
For[ M=2, M <= MMM, M++,
vlist = Array[x, M];
cl[i_] := And[ x[i], x[i+1] ];
cl2 = False; For [ i=1, i <= M-1, i++, cl2 = Or[cl2, cl[i]] ];
R[M] = SatisfiabilityCount[ cl2, vlist ] ]
Table[ R[M], {M,2,MMM}]
(* Find Boolean values of variables that satisfy the formula x1 x2 + x2 x3 + ... + xn-1 xn = 1; N. J. A. Sloane, Apr 23 2011 *)
LinearRecurrence[{3,-1,-2},{0,0,1},40] (* Harvey P. Dale, Aug 09 2013 *)
nn=33; a=1/(1-2x); b=1/(1-2x^2-x^4-x^6/(1-x^2));
CoefficientList[Series[b(a x^3/(1-x^2)+x^2a),{x,0,nn}],x] (* Geoffrey Critzer, Dec 30 2013 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+1],Max@@#>2&]],{n,0,10}] (* Gus Wiseman, Jun 25 2020 *)

a(n) = 2^n-fibonacci(n+2) \\ Charles R Greathouse IV, Feb 03 2014

def A008466(n): return 2^n - fibonacci(n+2) # G. C. Greubel, Apr 23 2025

a(1)=0, a(2)=1, a(3)=3, a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3). - Miklos Kristof, Nov 24 2003
G.f.: x^2/((1-2*x)*(1-x-x^2)). - Paul Barry, Feb 16 2004
From Paul Barry, May 19 2004: (Start)
Convolution of Fibonacci(n) and (2^n - 0^n)/2.
a(n) = Sum_{k=0..n} (2^k-0^k)*Fibonacci(n-k)/2.
a(n+1) = Sum_{k=0..n} Fibonacci(k)*2^(n-k).
a(n) = 2^n*Sum_{k=0..n} Fibonacci(k)/2^k. (End)
a(n) = a(n-1) + a(n-2) + 2^(n-2). - Jon Stadler (jstadler(AT)capital.edu), Aug 21 2006
a(n) = 2*a(n-1) + Fibonacci(n-1). - Thomas M. Green, Aug 21 2007
a(n) = term (1,3) in the 3 X 3 matrix [3,1,0; -1,0,1; -2,0,0]^n. - Alois P. Heinz, Jul 18 2008
a(n) = 2*a(n-1) - a(n-3) + 2^(n-3). - Carmine Suriano, Mar 08 2011

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User: I. J. Kennedy

A125268 Numbers that end with decimal digit 1, 3, 7, or 9 and that produce only composite numbers when any of the digits 0,1,...,9 is inserted anywhere in them (including at the beginning or end).

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A015818 Number of solutions of +- 1 +- 2 +- ... +- (n-1) +- n = 0 in which the partial sums +- 1 +- ... +- k (1<=k<=n) are all distinct.

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A008466 a(n) = 2^n - Fibonacci(n+2).

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