A261973 -id:A261973 - cp's OEIS frontend

A261972 The first of three consecutive positive integers the sum of the squares of which is equal to the sum of the squares of four consecutive positive integers.

25, 361, 5041, 70225, 978121, 13623481, 189750625, 2642885281, 36810643321, 512706121225, 7141075053841, 99462344632561, 1385331749802025, 19295182152595801, 268747218386539201, 3743165875258953025, 52135575035238803161, 726154884618084291241

Offset: 1

Colin Barker, Sep 07 2015

For the first of the corresponding four consecutive positive integers, see A157088.

			25 is in the sequence because 25^2 + 26^2 + 27^2 = 2030 = 21^2 + 22^2 + 23^2 + 24^2.

Colin Barker, Table of n, a(n) for n = 1..873
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Cf. A157088, A261973, A261974.

LinearRecurrence[{15,-15,1},{25,361,5041},20] (* Harvey P. Dale, Jul 16 2025 *)

Vec(-x*(x^2-14*x+25)/((x-1)*(x^2-14*x+1)) + O(x^40))

a(n) = 15*a(n-1)-15*a(n-2)+a(n-3) for n>3.
G.f.: -x*(x^2-14*x+25) / ((x-1)*(x^2-14*x+1)).
a(n) = (-2-(7-4*sqrt(3))^n*(-2+sqrt(3))+(2+sqrt(3))*(7+4*sqrt(3))^n)/2. - Colin Barker, Mar 05 2016

A261974 The first of eleven consecutive positive integers the sum of the squares of which is equal to the sum of the squares of three consecutive positive integers.

Original entry on oeis.org

67, 3307, 152275, 7001563, 321919843, 14801311435, 680538406387, 31289965382587, 1438657869192835, 66146972017488043, 3041322054935257363, 139834667555004350875, 6429353385475264883107, 295610421064307180272267, 13591650015572655027641395

Offset: 1

Colin Barker, Sep 07 2015

For the first of the corresponding three consecutive positive integers, see A261973.

			67 is in the sequence because 67^2 + ... + 77^2 = 57134 = 137^2 + 138^2 + 139^2.

Colin Barker, Table of n, a(n) for n = 1..601
Index entries for linear recurrences with constant coefficients, signature (47,-47,1).

Cf. A157088, A261972, A261973.

LinearRecurrence[{47,-47,1},{67,3307,152275},20] (* Harvey P. Dale, Jul 02 2016 *)

Vec(x*(5*x^2-158*x-67)/((x-1)*(x^2-46*x+1)) + O(x^40))

a(n) = 47*a(n-1)-47*a(n-2)+a(n-3) for n>3.
G.f.: x*(5*x^2-158*x-67) / ((x-1)*(x^2-46*x+1)).
a(n) = -5+3*sqrt(3/11)*(23+4*sqrt(33))^(-n)*(-1+(23+4*sqrt(33))^(2*n)). - Colin Barker, Mar 03 2016

cp's OEIS Frontend

A261972 The first of three consecutive positive integers the sum of the squares of which is equal to the sum of the squares of four consecutive positive integers.

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