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A106521 Numbers a(n) such that Sum[k=0..10, (a(n)+k)^2 ] is square. 7
18, 38, 456, 854, 9192, 17132, 183474, 341876, 3660378, 6820478, 73024176, 136067774, 1456823232, 2714535092, 29063440554, 54154634156, 579811987938, 1080378148118, 11567176318296, 21553408328294 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

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).

FORMULA

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

EXAMPLE

Since 18^2 + 19^2 + ... + 28^2 = 5929 = 77^2, 18 is in the sequence. - Michael B. Porter, Aug 07 2016

MATHEMATICA

LinearRecurrence[{1, 20, -20, -1, 1}, {18, 38, 456, 854, 9192}, 30] (* Harvey P. Dale, May 07 2011 *)

PROG

(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

CROSSREFS

Cf. A001032, A094196.

Sequence in context: A084585 A132761 A079862 * A070686 A043118 A039295

Adjacent sequences:  A106518 A106519 A106520 * A106522 A106523 A106524

KEYWORD

nonn,easy

AUTHOR

Ralf Stephan, May 30 2005

EXTENSIONS

Edited and extended by M. F. Hasler, Jan 27 2008

G.f. adapted to the offset by Bruno Berselli, May 16 2011

STATUS

approved

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Last modified January 18 16:40 EST 2019. Contains 319271 sequences. (Running on oeis4.)