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A106521
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Numbers a(n) such that Sum[k=0..10, (a(n)+k)^2 ] is square.
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7
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18, 38, 456, 854, 9192, 17132, 183474, 341876, 3660378, 6820478, 73024176, 136067774, 1456823232, 2714535092, 29063440554, 54154634156, 579811987938, 1080378148118, 11567176318296, 21553408328294
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OFFSET
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1,1
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COMMENTS
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Equivalently, 11*a(n)^2 + 110*a(n) + 385 is square.
11*((m+5)^2+10) is a square iff the second factor is divisible by 11 and the quotient is a square, i.e. iff m = 11 k - 4 or m = 11 k - 6 and 11 k^2 +/- 2 k + 1 is a square. Thus a(n)=(7,5,5,7,7,5,5,7,...) (mod 11), repeating with period 4 and the values are obtained by solving these Pell-type equations, e.g. using http://www.alpertron.com.ar/QUAD.HTM. The corresponding recurrence equations (see PARI code) should make it possible to prove the conjectured g.f. - M. F. Hasler, Jan 27 2008
All sequences of this type (i.e. sequences with fixed offset k, and a discernible pattern: k=0...10 for this sequence, k=0..1 for A001652) can be continued using a formula such as x(n) = a*x(n-p) - x(n-2p) + b, where a and b are various constants, and p is the period of the series. Alternatively 'p' can be considered the number of concurrent series. - Daniel Mondot, Aug 05 2016
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (1,20,-20,-1,1).
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FORMULA
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G.f.: 2*x*(9+10*x+29*x^2-x^3-2*x^4)/(1-x)/(1-20*x^2+x^4). - Vladeta Jovovic, May 31 2005
a(1)=18, a(2)=38, a(3)=456, a(4)=854, a(5)=9192; thereafter a(n)=a(n-1)+20*a(n-2)- 20*a(n-3)-a(n-4)+a(n-5). - Harvey P. Dale, May 07 2011
a(n) = A198949(n+1)-5. - Bruno Berselli, Feb 12 2012
a(1)=18, a(2)=38, a(3)=456, a(4)=854; thereafter a(n) = 20*a(n-2) - a(n-4) + 90. - Daniel Mondot, Aug 05 2016
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EXAMPLE
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Since 18^2 + 19^2 + ... + 28^2 = 5929 = 77^2, 18 is in the sequence. - Michael B. Porter, Aug 07 2016
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MATHEMATICA
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LinearRecurrence[{1, 20, -20, -1, 1}, {18, 38, 456, 854, 9192}, 30] (* Harvey P. Dale, May 07 2011 *)
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PROG
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(PARI) A106521(n)={local(xy=[ -4-2*(n%2); 11], PQRS=[10, 3; 33, 10], KL=[45; 165]); until(0>=n-=2, xy=PQRS*xy+KL); xy[1]} \\ M. F. Hasler, Jan 27 2008
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CROSSREFS
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Cf. A001032, A094196.
Sequence in context: A084585 A132761 A079862 * A070686 A043118 A039295
Adjacent sequences: A106518 A106519 A106520 * A106522 A106523 A106524
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KEYWORD
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nonn,easy
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AUTHOR
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Ralf Stephan, May 30 2005
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EXTENSIONS
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Edited and extended by M. F. Hasler, Jan 27 2008
G.f. adapted to the offset by Bruno Berselli, May 16 2011
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STATUS
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approved
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