1,2

For n>=2, least m>=1 such that f(m,n)=0 where f(m,n)=sum(i=0,m,sum(k=0,i,(-1)^k*(floor(i/n^k)-n*floor(i/n^(k+1))))) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 02 2004

For n=>3, the a(n)-th row of Pascal's triangle always contains a triple forming an arithmetic progression. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 03 2004

For n>1: a(n)=A143053(A000290(n)), A143054(a(n))=A000290(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 20 2008

Except for the first term of [A008865] and [A000027], if X=[A013648], Y=[A000027], A= [A008865], we have, for all other terms, Pell's equation: [A013648]^2 - [A008865]*[A000027]^2=1; (X^2-A*Y^2=1); example: 3^2-2*2^2=1; 8^2-7*3^2=1; 120^2-119*11^2=1; and so on. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 11 2009]

a(n+1) = A101986(n)-A101986(n-1) = A160805(n)-A160805(n-1). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 26 2009]

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 29 2009: (Start)

Let C = 1 + sqrt(2) = 2.414213...; and 1/C = .414213...

Then a(n) = (n + 1 + 1/C) * (n + 1 - C). Example: a(6) = 34 =

(7 + .414...) * (7 - 2.414...) (End)

Except for the first term, a(n)=2*n+a(n-1)+1 (with a(1)=2) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 07 2009]

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

G.f.: (x-5*x^2+2*x^3)/(-1+3*x-3*x^2+x^3). [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Oct 17 2008]

a(n)=2*n+a(n-1)-1 (with a(1)=-1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 29 2009]

For n=2, a(2)=2*2-1-1=2; n=3, a(3)=2*3+2-1=7; n=4, a(4)=2*4+7-1=14 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 29 2009]

with(combinat, fibonacci):seq(fibonacci(3, i)-3, i=1..47); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 20 2008

a:=n->sum(k-1, k=0..n):seq(a(n)+sum(k, k=2..n), n=1..54); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 10 2008

s=-1; lst={}; Do[s+=n-3; AppendTo[lst, s], {n, 2, 6!, 2}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 04 2008]

sage: [lucas_number1(3, n, 2) for n in xrange(1, 43)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 03 2008

(PARI) {for(n=1, 47, print1(n^2-2, ", "))} [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Oct 17 2008]

(Other) sage: [lucas_number1(3, n, 2) for n in xrange(1, 48)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 16 2009]

Cf. A145067 (Zero followed by partial sums of A008865). [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Oct 17 2008]

Cf. A000027, A013648 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 11 2009]

Sequence in context: A018363 A161702 A087324 this_sequence A018392 A051640 A119354

Adjacent sequences: A008862 A008863 A008864 this_sequence A008866 A008867 A008868

sign,easy,new

N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy