Search: id:A008865
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%I A008865
%S A008865 1,2,7,14,23,34,47,62,79,98,119,142,167,194,223,254,287,322,359,
%T A008865 398,439,482,527,574,623,674,727,782,839,898,959,1022,1087,1154,
%U A008865 1223,1294,1367,1442,1519,1598,1679,1762,1847,1934,2023,2114,2207
%V A008865 -1,2,7,14,23,34,47,62,79,98,119,142,167,194,223,254,287,322,359,
%W A008865 398,439,482,527,574,623,674,727,782,839,898,959,1022,1087,1154,
%X A008865 1223,1294,1367,1442,1519,1598,1679,1762,1847,1934,2023,2114,2207
%N A008865 n^2 - 2.
%C A008865 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
%C A008865 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
%C A008865 For n>1: a(n)=A143053(A000290(n)), A143054(a(n))=A000290(n). - Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 20 2008
%C A008865 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]
%C A008865 a(n+1) = A101986(n)-A101986(n-1) = A160805(n)-A160805(n-1). [From Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), May 26 2009]
%C A008865 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 29 2009: 
               (Start)
%C A008865 Let C = 1 + sqrt(2) = 2.414213...; and 1/C = .414213...
%C A008865 Then a(n) = (n + 1 + 1/C) * (n + 1 - C). Example: a(6) = 34 =
%C A008865 (7 + .414...) * (7 - 2.414...) (End)
%C A008865 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]
%H A008865 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Near-SquarePrime.html">Near-Square Prime</a>
%F A008865 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]
%F A008865 a(n)=2*n+a(n-1)-1 (with a(1)=-1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 29 2009]
%e A008865 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]
%p A008865 with(combinat, fibonacci):seq(fibonacci(3, i)-3,i=1..47); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Mar 20 2008
%p A008865 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
%t A008865 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]
%o A008865 sage: [lucas_number1(3,n,2) for n in xrange(1,43)] - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jul 03 2008
%o A008865 (PARI) {for(n=1, 47, print1(n^2-2, ","))} [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), 
               Oct 17 2008]
%o A008865 (Other) sage: [lucas_number1(3,n,2) for n in xrange(1, 48)] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), May 16 2009]
%Y A008865 Cf. A145067 (Zero followed by partial sums of A008865). [From Klaus Brockhaus 
               (klaus-brockhaus(AT)t-online.de), Oct 17 2008]
%Y A008865 Cf. A000027, A013648 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 11 2009]
%Y A008865 Sequence in context: A018363 A161702 A087324 this_sequence A018392 A051640 
               A119354
%Y A008865 Adjacent sequences: A008862 A008863 A008864 this_sequence A008866 A008867 
               A008868
%K A008865 sign,easy,new
%O A008865 1,2
%A A008865 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy

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