Search: id:A008844
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%I A008844
%S A008844 1,25,841,28561,970225,32959081,1119638521,38034750625,
%T A008844 1292061882721,43892069261881,1491038293021225,50651409893459761,
%U A008844 1720656898084610641,58451683124983302025,1985636569351347658201
%N A008844 Squares of sequence A001653: y^2 such that x^2 - 2*y^2 = -1 for some 
               x.
%C A008844 Numbers simultaneously square and centered square. E.g. a(1)=25 because 
               25 is the fourth centered square number and the fifth square number. 
               - Steven Schlicker (schlicks(AT)gvsu.edu), Apr 24 2007
%C A008844 Solutions to A007913(x)=A007913(2x-1) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 07 2002
%D A008844 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, 
               p. 256.
%D A008844 Fielder, Daniel C.; Special integer sequences controlled by three parameters. 
               Fibonacci Quart 6 1968 64-70.
%D A008844 Problem 47, Amer. Math. Monthly, 4 (1897), 25-28.
%D A008844 S. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, 
               submitted.
%H A008844 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HexagonalSquareNumber.html">Link to a section of The World of Mathematics.</
               a>
%F A008844 a(n)=A078522(n)+1. a(n)=ceiling(A*B^n) where A=(3+2*sqrt(2))/8 and B=17+12*sqrt(2). 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 19 2003
%F A008844 G.f.: (1-10x+x^2)/((1-x)(1-34x+x^2)).
%F A008844 a(n) = ceiling(A046176(n)/sqrt(2)) - Helge Robitzsch (hrobi(AT)math.uni-goettingen.de), 
               Jul 28 2000
%F A008844 a(n+1)=17*a(n)-4+6*sqrt(8*a(n)^2-4*a(n)). - Richard Choulet (richardchoulet(AT)yahoo.fr), 
               Sep 14 2007
%F A008844 Define x(n) + y(n)*sqrt(8) = (4+sqrt(8))*(3+sqrt(8))^n, s(n) = (y(n)+1)/
               2; then a(n) = (1/2)*(2+4*(s(n)^2-s(n))) - Steven Schlicker (schlicks(AT)gvsu.edu), 
               Apr 24 2007
%p A008844 CP := n -> 1+1/2*4*(n^2-n): N:=10: u:=3: v:=1: x:=4: y:=1: k_pcp:=[1]: 
               for i from 1 to N do tempx:=x; tempy:=y; x:=tempx*u+8*tempy*v: y:=tempx*v+tempy*u: 
               s:=(y+1)/2: k_pcp:=[op(k_pcp),CP(s)]: end do: k_pcp; - Steven Schlicker 
               (schlicks(AT)gvsu.edu), Apr 24 2007
%o A008844 (PARI) a(n)=if(n<0,0,sqr(subst(poltchebi(n+1)+poltchebi(n),x,3)/4))
%Y A008844 Cf. A000290, A001844, A007913.
%Y A008844 Sequence in context: A142998 A122142 A151557 this_sequence A159332 A139163 
               A167257
%Y A008844 Adjacent sequences: A008841 A008842 A008843 this_sequence A008845 A008846 
               A008847
%K A008844 nonn
%O A008844 0,2
%A A008844 N. J. A. Sloane (njas(AT)research.att.com).
%E A008844 Entry edited by N. J. A. Sloane (njas(AT)research.att.com), Sep 14 2007

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