A094196 Indices of the start of a string of 24 consecutive squares whose sum is a square.

1, 9, 20, 25, 44, 76, 121, 197, 304, 353, 540, 856, 1301, 2053, 3112, 3597, 5448, 8576, 12981, 20425, 30908, 35709, 54032, 84996, 128601, 202289, 306060, 353585, 534964, 841476, 1273121, 2002557, 3029784, 3500233, 5295700, 8329856, 12602701

Offset: 1

Author

Lekraj Beedassy, May 25 2004

Keywords

• nonn
• easy

Comments

The sequence could also include -11, -8 and -4; and if N is in the sequence, then so is -23-N.

Equivalently, 24*a(n)^2 + 552*a(n) + 4324 is a square.

All sequences of this type (i.e., sequences with fixed offset k, and a discernible pattern: k=0...23 for this sequence, k=0...22 for A269447, k=0..1 for A001652) can be extended 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

• Daniel Mondot, Table of n, a(n) for n = 1..106
• K. S. Brown, Sum of Consecutive Nth Powers Equals an Nth Power
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,10,-10,0,0,0,0,-1,1).

Crossrefs

Cf. A001032, A001652, A106521, A269447.

Programs

Mathematica

LinearRecurrence[{1,0,0,0,0,10,-10,0,0,0,0,-1,1},{1,9,20,25,44,76,121,197,304,353,540,856,1301},60] (* Harvey P. Dale, Oct 10 2011 *)

PARI

for(n=1,15000000,if(issquare(sum(j=n,n+23,j^2)),print1(n,","))) \\ Klaus Brockhaus, Jun 01 2004

Formula

Recurrence: a(n+12) = 10a(n+6) - a(n) + 92.

O.g.f.: x*(-1-8*x-11*x^2-5*x^3-19*x^4-32*x^5-35*x^6+4*x^7+3*x^8+x^9+3*x^10+4*x^11+4*x^12) / ((-1+x) * (1-10*x^6+x^12)). - R. J. Mathar, Dec 02 2007

a(0)=1, a(1)=9, a(2)=20, a(3)=25, a(4)=44, a(5)=76, a(6)=121, a(7)=197, a(8)=304, a(9)=353, a(10)=540, a(11)=856, a(12)=1301; thereafter a(n) = a(n-1)+10*a(n-6)-10*a(n-7)-a(n-12)+a(n-13). - Harvey P. Dale, Oct 10 2011

a(1)=1, a(2)=9, a(3)=20, a(4)=25, a(5)=44, a(6)=76, a(7)=121, a(8)=197, a(9)=304, a(10)=353, a(11)=540, a(12)=856; a(n)=10*a(n-6)-a(n-12) + 92 for n>12. - Daniel Mondot, Aug 05 2016

Extensions

More terms from Don Reble (djr(AT)hotmail.com) and Klaus Brockhaus, Jun 01 2004