A185055 Number of representations of 5^(2n) as a sum a^2 + b^2 + c^2 with 0 < a <= b <= c.

0, 0, 2, 14, 76, 388, 1950, 9762, 48824, 244136, 1220698, 6103510, 30517572, 152587884, 762939446, 3814697258, 19073486320, 95367431632, 476837158194, 2384185791006, 11920928955068, 59604644775380, 298023223876942, 1490116119384754, 7450580596923816, 37252902984619128

Offset: 0

Author

Zak Seidov, Mar 02 2012

Keywords

• nonn
• easy

Comments

Corresponding formulas for several first primes:

p=3, a(n)=(3*3^n+2*n+1)/4 (A047926)

p=5, a(n)=(5^n-4*n-1)/8 (A185055)

p=7, a(n)=(7^n-1)/6

p=11, a(n)=(3*11^n+10*n-3)/20

p=13, a(n)=(13^n-4*n-1)/8

p=17, a(n)=(17^n-1)/8

p=19, a(n)=(5*19^n+18*n-5)/36

p=23, a(n)=3*(23^n-1)/22

p=29, a(n)=(29^n-4*n-1)/8

p=31, a(n)=2*(31^n-1)/15

p=37, a(n)=(37^n-4*n-1)/8

p=41, a(n)=(41^n-1)/8

p=43, a(n)=(11*43^n+42*n-11)/84

p=47, a(n)=3*(47^n-1)/23.

General formulas for a(n) depend on p mod 8 as follows:

p = 1 mod 8, a(n)=(p^n-1)/8

p = 3 mod 8, a(n)=((p + 1)*p^n + 4*(p - 1)*n - (p + 1))/(8*(p - 1))

p = 5 mod 8, a(n)=(p^n-4*n-1)/8

p = 7 mod 8, a(n)=((p + 1)*(p^n - 1))/(8*(p - 1)).

Examples

			a(2)=2 because 25^2 = 9^2+12^2+20^2 = 12^2+15^2+16^2.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (7,-11,5).

Crossrefs

Cf. A014827, A047926.

Formula

a(n) = (5^n-4n-1)/8.

From Chai Wah Wu, Jun 07 2024: (Start)

a(n) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3) for n > 2.

G.f.: -2*x^2/((x - 1)^2*(5*x - 1)). (End)

a(n) = 2 * A014827(n-1) for n >= 2. - Alois P. Heinz, Jun 07 2024