The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A008844 Squares of sequence A001653: y^2 such that x^2 - 2*y^2 = -1 for some x. 17
 1, 25, 841, 28561, 970225, 32959081, 1119638521, 38034750625, 1292061882721, 43892069261881, 1491038293021225, 50651409893459761, 1720656898084610641, 58451683124983302025, 1985636569351347658201, 67453191674820837076801, 2291422880374557112953025 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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, Apr 24 2007 Solutions to A007913(x)=A007913(2x-1). - Benoit Cloitre, Apr 07 2002 From Ant King, Nov 09 2011: (Start) Indices of positive hexagonal numbers that are also perfect squares. As n increases, this sequence is approximately geometric with common ratio r = lim_{n -> infinity} a(n)/a(n-1) = (1 + sqrt(2))^4 = 17 + 12 * sqrt(2). (End) Also indices of hexagonal numbers (A000384) which are also centered octagonal numbers (A016754). - Colin Barker, Jan 25 2015 Also positive integers x in the solutions to 4*x^2 - 8*y^2 - 2*x + 8*y - 2 = 0, the corresponding values of y being A253826. - Colin Barker, Jan 25 2015 Squares that are sum of two consecutive squares: y^2 = (k + 1)^2 + k^2 is equivalent to x^2 - 2*y^2 = -1 with x = 2*k + 1. - Jean-Christophe Hervé, Nov 11 2015 REFERENCES A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 256. LINKS Muniru A Asiru, Table of n, a(n) for n = 0..100 Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70. M. A. Gruber, Artemas Martin, A. H. Bell, J. H. Drummond, A. H. Holmes and H. C. Wilkes, Problem 47, Amer. Math. Monthly, 4 (1897), 25-28. Giovanni Lucca, Integer Sequences and Circle Chains Inside a Circular Segment, Forum Geometricorum, Vol. 18 (2018), 47-55. S. C. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Mathematics Magazine,  Vol. 84, No. 5, December 2011 Eric Weisstein's World of Mathematics, Hexagonal Square Number. Index entries for linear recurrences with constant coefficients, signature (35,-35,1). FORMULA From Benoit Cloitre, Jan 19 2003: (Start) a(n) = A078522(n) + 1. a(n) = ceiling(A*B^n) where A = (3 + 2*sqrt(2))/8 and B = 17 + 12*sqrt(2). (End) G.f.: (1-10x+x^2)/((1-x)(1-34x+x^2)). a(n) = ceiling(A046176(n)/sqrt(2)). - Helge Robitzsch (hrobi(AT)math.uni-goettingen.de), Jul 28 2000 a(n+1) = 17*a(n) - 4 + 12*sqrt(2*a(n)^2 - a(n)). - Richard Choulet, Sep 14 2007 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, Apr 24 2007 From Ant King, Nov 09 2011: (Start) a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3). a(n) = 34*a(n-1) - a(n-2) - 8. a(n) = 1/8 * ((1 + sqrt(2))^(4*n-2) + (1 - sqrt(2))^(4*n-2) + 2). a(n) = ceiling((1/8) * (1 + sqrt(2))^(4*n-2)). (End) From Ravi Kumar Davala, May 26 2013: (Start) a(n+2) = 577*a(n) - 144 + 408*sqrt(2*a(n)^2 - a(n)). a(n+m) = A001333(4*m)*a(n) - (A000129(2*m))^2 + A000129(4*m)*sqrt(2*a(n)^2 - a(n)). a(n+m) = (1/2)*A002203(4*m)*a(n) - (A000129(2*m))^2 + A000129(4*m)*sqrt(2*a(n)^2 - a(n)). a(n+1)*a(n-1) = (a(n)+4)^2. (End) a(n) = A001652(n)^2 + A046090(n)^2. - César Aguilera, Jan 15 2018 Lim_{n -> infinity} a(n)/a(n-1) = A156164. - César Aguilera, Jan 28 2018 sqrt(2*a(n))-1 = A002315(n). - Ezhilarasu Velayutham, Apr 05 2019 EXAMPLE From Ravi Kumar Davala, May 26 2013: (Start) A001333(0)=1, A001333(4)=17, A001333(8)=577, A000129(0)=0, A000129(2)=2, A000129(4)=12, A000129(8)=408 so clearly a(n+m)=A001333(4*m)*a(n)-(A000129(2*m))^2+A000129(4*m)*sqrt(2*a(n)^2-a(n)), with m=1,2 is true. A002203(0)=2, A002203(4)=34, A002203(8)=1154 so clearly a(n+m)=(1/2)*A002203(4*m)*a(n)-(A000129(2*m))^2+A000129(4*m)*sqrt(2*a(n)^2-a(n)) is true for m=1,2 a(n+1)*a(n-1) = (a(n)+4)^2 , with n=1 is 841*1=(25+4)^2, for n=2 , 28561*25=(841+4)^2. (End) 1 = 1 + 0, 25 = 16 + 9, 841 = 29^2 = 21^2 + 20^2 = 441 + 400. MAPLE 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, Apr 24 2007 MATHEMATICA LinearRecurrence[{35, -35, 1}, {1, 25, 841}, 15] (* Ant King, Nov 09 2011 *) CoefficientList[Series[(1 - 10 x + x^2) / ((1 - x) (1 - 34 x + x^2)), {x, 0, 33}], x] (* Vincenzo Librandi, Jan 20 2018 *) PROG (PARI) a(n)=if(n<0, 0, sqr(subst(poltchebi(n+1)+poltchebi(n), x, 3)/4)) (PARI) vector(40, n, n--; (([5, 2; 2, 1]^n)[1, 1])^2) \\ Altug Alkan, Nov 11 2015 (GAP) a := [1, 25, 841];; for i in [4..10^2] do a[i] := 35*a[i-1] - 35*a[i-2] + a[i-3]; od; a;  # Muniru A Asiru, Jan 17 2018 (MAGMA) I:=[1, 25, 841]; [n le 3 select I[n] else 35*Self(n-1)-35*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jan 20 2018 CROSSREFS Cf. A000290, A001844, A007913, A000384, A046177, A016754, A253826. Sequence in context: A246761 A122142 A151557 * A251925 A181892 A274469 Adjacent sequences:  A008841 A008842 A008843 * A008845 A008846 A008847 KEYWORD nonn,easy AUTHOR EXTENSIONS Entry edited by N. J. A. Sloane, Sep 14 2007 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 12 06:54 EDT 2021. Contains 343820 sequences. (Running on oeis4.)