A360365 a(n) = sum of the products of the digits of the first n positive multiples of 3.

3, 9, 18, 20, 25, 33, 35, 43, 57, 57, 66, 84, 111, 119, 139, 171, 176, 196, 231, 231, 249, 285, 339, 353, 388, 444, 452, 484, 540, 540, 567, 621, 702, 702, 702, 702, 703, 707, 714, 714, 720, 732, 750, 756, 771, 795, 799, 815, 843, 843, 858, 888, 933, 945, 975, 1023, 1030, 1058, 1107

Offset

1,1

Links

Table of n, a(n) for n=1..59.

Formula

a((10^n-1)/3) = (1/836)*(15*(19*45^n-63) + 44*3^((3*n)/2)*(sqrt(3)*sin((Pi*n)/6) + 15*cos((Pi*n)/6))).

G.f. of the subsequence a((10^n-1)/3): 9*(2 - 32*x + 135*x^2)/((1 - x)*(1 - 45*x)*(1 - 9*x + 27*x^2)).

a((10^n-1)/3) = 55*a((10^(n-1)-1)/3) - 486*a((10^(n-2)-1)/3) + 1647*a((10^(n-3)-1)/3) - 1215*a((10^(n-4)-1)/3) for n > 4.

Example

a(4) = 20 since the first 4 positive multiples of 3 are 3,6,9 and 12 and the sum of the product of their digits is 3 + 6 + 9 + 1*2 = 20.

Mathematica

Accumulate[Times @@@ IntegerDigits[Range[3, 999 , 3]]]

Prog

(PARI) a(n)={sum(k=1, n, vecprod(digits(3*k)))} \\ Andrew Howroyd, Feb 09 2023
(Python)
from math import prod
def A360365(n): return sum(prod(int(d) for d in str(m)) for m in range(3, 3*n+1, 3)) # Chai Wah Wu, Feb 28 2023

Crossrefs

Cf. A061076, A061077, A061078.

Sequence in context: A261952 A066006 A323199 * A310332 A210991 A112559

Adjacent sequences: A360362 A360363 A360364 * A360366 A360367 A360368

Keyword

nonn,base

Author

Luca Onnis, Feb 09 2023

Status

approved