

A106521


Numbers m such that Sum_{k=0..10} (m+k)^2 is a square.


10



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 a 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 Pelltype 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(np)  x(n2p) + 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^2x^32*x^4)/(1x)/(120*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(n1)+20*a(n2) 20*a(n3)a(n4)+a(n5).  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(n2)  a(n4) + 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=[ 42*(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



