0,2

Albert H. Beiler, Recreations in the Theory of Numbers, Dover Publ., 2nd Ed., NY, 1966, Chapter XV, 'On The Square', p. 139.

W. Penney, On the final digits of squares, Amer. Math. Monthly, 67 (1960), 1000-1002.

Index entries for sequences related to final digits of numbers

a(n) = (83 + 27*(-1)^n + 9*2^(1 + n) + (-1)^n*2^(2 + n) + 9*5^(2 + n) + (-1)^n*5^(2 + n) + 2^(1 + n)*5^(2 + n))/ 72.

a(n+8) = 130 a(n+6) - 3129 a(n+4) + 13000 a(n+2) - 10000 a(n) for n >= 1

G.f.: (1 - 4*x - 68*x^2 + 59*x^3 + 723*x^4 - 5*x^5 - 1700*x^6 - 500*x^7)/(1 - 10*x - 30*x^2 + 300*x^3 + 129*x^4 - 1290*x^5 - 100*x^6 + 1000*x^7)

Any square ends with one of 0, 1, 4, 5, 6, 9, so a(1) = 6.

A square may end with 22 different two-digit combinations: 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, 96. E.g. number ending with 14 can not be square etc. See also A075821, A075823.

-(-6+38*z+241*z^2-594*z^3-1285*z^4+1600*z^5+1500*z^6)/((-1+z)*(5*z-1)*(2*z+1)*(2*z-1)*(5*z+1)*(10*z-1)*(z+1)); #from Bruno Salvy

a[n_] := (83 - 27*(-1)^n + 9*2^(n) - (-1)^n*2^(1 + n) + 9*5^(1 + n) - (-1)^n*5^(1 + n) + 2^(n)*5^(1 + n))/72; Table[ Floor[ a[n]], {n, 0, 20}]

(* Or *) a[0] = 1; a[1] = 6; a[2] = 22; a[3] = 159; a[4] = 1044; a[5] = 9121; a[6] = 78132; a[7] = 748719; a[8] = 7161484; a[n_] := 130 a[n - 2] - 3129 a[n - 4] + 13000 a[n - 6] - 10000 a[n - 8]; Table[ a[n], {n, 0, 20}]

(* Or *) CoefficientList[ Series[(1 - 4*x - 68*x^2 + 59*x^3 + 723*x^4 - 5*x^5 - 1700*x^6 - 500*x^7)/(1 - 10*x - 30*x^2 + 300*x^3 + 129*x^4 - 1290*x^5 - 100*x^6 + 1000*x^7), {x, 0, 20}], x] (from Robert G. Wilson v Nov 27 2004)

Cf. A036688, A023105, A039300-A039306, A075821, A075823.

Adjacent sequences: A000990 A000991 A000992 this_sequence A000994 A000995 A000996

Sequence in context: A002137 A009361 A075759 this_sequence A028406 A090372 A009366

nonn,easy,nice

njas