0,2

Erdos conjectured that n^2 - 1 = k! has a solution iff n is 5, 11 or 71 (when k is 4, 5 or 7).

Second order linear recurrences y(m)=2y(m-1)+A005563(n)y(m-2),y(0)=y(1)=1, have closed form solutions involving only powers of integers. - Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 08 2001

Number of edges in the join of two cycle graphs, both of order n, C_n * C_n - Roberto E. Martinez II (remartin(AT)fas.harvard.edu), Jan 07 2002

Let k be a positive integer, M_n be the n X n matrix m_(i,j)=k^abs(i-j) then det(M_n)=(-1)^(n-1)*a(k-1)^(n-1) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 28 2002

Also numbers n such that 4n + 4 is a square. - Cino Hilliard (hillcino368(AT)gmail.com), Dec 18 2003

The function sqrt(x^2 + 1), starting with 1, produces an integer after n(n+2) iterations. - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 19 2004

a(n) mod 3 = 0 iff n mod 3 > 0: a(A008585(n)) = 2; a(A001651(n)) = 0; a(n) mod 3 = 2*(1-A079978(n)). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 16 2006

a(n)=A067725/3 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 06 2007

A129296(n) = number of divisors of a(n+1) that are not greater than n. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 09 2007

Sequence allows us to find X values of the equation: X^3 + X^2 = Y^2. To find Y values: b(n)=n(n+1)(n+2). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 06 2007

Sequence allows us to find X values of the equation: X + (X + 1)^2 + (X + 2)^3 = Y^2. To prove that X = n^2 + 2n: Y^2 = X + (X + 1)^2 + (X + 2)^3 = X^3 + 7*X^2 + 15X + 9 = (X + 1)(X^2 + 6X + 9) = (X + 1)*(X + 3)^2 it means: (X + 1) must be a perfect square, so X = k^2 - 1 with k>=1. we can put: k = n + 1, which gives: X = n^2 + 2n and Y = (n + 1)(n^2 + 2n + 3). - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Nov 12 2007

Comment from R. Guy (rkg(AT)cpsc.ucalgary.ca), Feb 01 2008 (Start):

This is also the number of moves that it takes n frogs to swap places with n toads on a strip of 2n+1 squares (or positions, or lilypads) where a move is a single slide or jump, illustrated for n = 2, a(n) = 8 by

T T - F F

T - T F F

T F T - F

T F T F -

T F - F T

- F T F T

F - T F T

F F T - T

F F - T T

I was alerted to this by the Holton article, but on consulting Singmaster's sources, I find that the puzzle goes back at least to 1867.

Probably the first to publish the number of moves for n of each animal was Edouard Lucas in 1883. (End)

For n>0: A143053(a(n)) = A000290(n+1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 20 2008

a(n+1)=terms of rank 0,1,3,6,10=A000217 of A120072 (3,8,5,15,). [From Paul Curtz (bpcrtz(AT)free.fr), Oct 28 2008]

For all n=A (0,3,8,15,24,..), X=[A000027] (1,2,3,4,5,...,), Y=[A000012] (1,1,1,1,...,) we have the Pell's equation X^2-A*Y^2=1. Example: 1-0*1=1; 2^2-3*1=1; 3^2-8*1=1; 4^2-15*1=1; and so on. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Mar 08 2009]

A053186(a(n)) = 2*n. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 20 2009]

Final digit belongs to a periodic sequence: 0, 3, 8, 5, 4, 5, 8, 3, 0, 9. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009] [Comment edited by N. J. A. Sloane, Sep 24 2009]

Comment from A. K. Devaraj (dkandadai(AT)yahoo.com), Sep 18 2009: (Start)

Let f(x) be a polynomial in x. Then f(x + n*f(x)) is congruent to 0 (mod(f(x)); here n belongs to N.

There is nothing interesting in the quotients f(x + n*f(x))/f(x) when x belongs to Z.

However, when x is irrational these quotients consist of two parts, a) rational integers and b) integer multiples of x.

The present sequence represents the non-integer part when the polynomial is x^2 + x + 1 and x = sqrt(2),

f(x+n*f(x))/f(x) = A056108(n) + a(n)*sqrt(2).

(End)

For n>=1, a(n) is the number for which 1/a(n) = 0.0101... (A000035) in base (n+1). [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), Sep 27 2009]

R. K. Guy, Unsolved Problems in Theory of Numbers, Section D25.

Derek Holton, Math in School, 37 #1 (Jan 2008) 20-22,

Edouard Lucas, Recreations Mathematiques, Gauthier-Villars, Vol. 2 (1883) 141-143.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Eric Weisstein's World of Mathematics, Near-Square Prime

Index entries for sequences related to linear recurrences with constant coefficients

G.f.: ( 3 - x ) / ( 1 - x )^3.

C(n+1, 1)*C(n+3, 1). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 06 2005

A002378(a(n))=A002378(n)*A002378(n+1); e.g. A002378(15)=240=12*20 - Charlie Marion (charliemath(AT)verizon.net), Dec 29 2003

a(n)=sum(sum(j-k, j=2..n),k=0..n), n>=1. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007

a(n)==A134582(n+1)/4 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 01 2008

a(n)=Real((n+1+i)^2) [From Gerald Hillier (adr.rabbicat(AT)gmail.com), Oct 12 2008]

G.f.: x*(1+x)/exp(x) . [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 03 2009]

a(n) = (n!+(n+1)!)/(n-1)!..n>0 [From Gary Detlefs (gdetlefs(AT)aol.com), Aug 10 2009]

[seq(2*binomial(n, 2)-n, n=2..43)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 25 2006

a:=n->sum (n, j=3..n): seq(a(n), n=2..43); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 21 2007

a:=n->sum(sum(j-k, j=2..n), k=0..n): seq(a(n), n=1..42); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 11 2007

a:=n->sum(sum(3, j=3..n)/3, k=1..n): seq(a(n), n=2..43); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 30 2007

A005563:=(-3+z)/(z-1)**3; [S. Plouffe in his 1992 dissertation.]

with(combinat, fibonacci):seq(fibonacci(3, i)-2, i=1..42); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 20 2008

a:=n->sum(1+sum(1, k=4..n), k=1..n):seq(a(n), n=2...43); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 11 2008

restart: G(x):=x*(1+x)/exp(x): f[0]:=G(x): for n from 1 to 43 do f[n]:=diff(f[n-1], x) od: x:=0: seq(abs(f[n]), n=2..43); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 03 2009]

Table[(m^2 - 1), {m, 42}] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 21 2007

(Other) sage: [lucas_number1(3, n, 1) for n in xrange(1, 43)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2009]

A column of triangle A102537.

a(n+1), n>=2, first column (used for the Lyman series of the hydrogen atom) of triangle A120070.

Cf. A013468, A007531, A062196, A002378, A000290(n) = a(n-1) + 1.

Cf. A046092, A067725, A123865, A123866, A123867, A123868.

Cf. A028560.

Adjacent sequences: A005560 A005561 A005562 this_sequence A005564 A005565 A005566

Sequence in context: A086959 A083656 A013648 this_sequence A132411 A147998 A164003

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).