A055237 Sums of two powers of 5.

2, 6, 10, 26, 30, 50, 126, 130, 150, 250, 626, 630, 650, 750, 1250, 3126, 3130, 3150, 3250, 3750, 6250, 15626, 15630, 15650, 15750, 16250, 18750, 31250, 78126, 78130, 78150, 78250, 78750, 81250, 93750, 156250, 390626, 390630, 390650, 390750, 391250, 393750, 406250, 468750, 781250

Offset: 0

Author

Henry Bottomley, Jun 22 2000

Keywords

• nonn
• easy
• tabl

Links

• T. D. Noe, Rows n = 0..100 of triangle, flattened
• Michael Penn, Fives and squares., YouTube video, 2021.

Crossrefs

Cf. A052216.

Programs

Mathematica

t = 5^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* T. D. Noe, Oct 09 2011 *)
Total/@Tuples[5^Range[0,9],2]//Union (* Harvey P. Dale, Jan 29 2017 *)

Python

from math import isqrt
def A055237(n): return 5**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+5**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 08 2025

Formula

a(n) = 5^(n-trinv(n))+5^trinv(n), where trinv(n) = floor((1+sqrt(1+8*n))/2) = A002262(n) and n-trinv(n) = A003056(n).

Regarded as a triangle T(n, k) = 5^n + 5^k, so as a sequence a(n) = 5^A002262(n) + 5^A003056(n).