A346006 Successive numbers arising from the Moessner construction of the sequence of fourth powers on page 64 of Conway-Guy's "Book of Numbers".

0, 1, 4, 6, 4, 16, 32, 24, 8, 81, 108, 54, 12, 256, 256, 96, 16, 625, 500, 150, 20, 1296, 864, 216, 24, 2401, 1372, 294, 28, 4096, 2048, 384, 32, 6561, 2916, 486, 36, 10000, 4000, 600, 40, 14641, 5324, 726, 44, 20736, 6912, 864, 48, 28561, 8788, 1014, 52, 38416, 10976, 1176, 56, 50625, 13500, 1350, 60

Offset

0,3

Comments

a(4*k+1) = (k+1)^2 for k >= 0.

References

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996. Sequence can be obtained by reading the successive circled numbers in the second display on page 64.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (0,0,0,5,0,0,0,-10,0,0,0,10,0,0,0,-5,0,0,0,1).

Formula

Let b=4. If n == -i (mod b) for 0 <= i < b, then a(n) = binomial(b,i+1)*((n+i)/b)^(i+1).

G.f.: x*(1 + 4*x + 6*x^2 + 4*x^3 + 11*x^4 + 12*x^5 - 6*x^6 - 12*x^7 + 11*x^8 - 12*x^9 - 6*x^10 + 12*x^11 + x^12 - 4*x^13 + 6*x^14 - 4*x^15)/((1 - x)^5*(1 + x)^5*(1 + x^2)^5). - Andrew Howroyd, Nov 12 2025

Maple

f:=proc(n, b) local i;
for i from 0 to b-1 do
if ((n+i) mod b) = 0 then return(binomial(b, i+1)*((n+i)/b)^(i+1)); fi;
od;
end;
[seq(f(n, 3), n=0..60)];

Prog

(Python)
from sympy import binomial
def A346006(n):
    i = (4-n)%4
    return binomial(4, i+1)*((n+i)//4)**(i+1) # Chai Wah Wu, Jul 25 2021
(PARI) a(n) = { my(r=(-n)%4); binomial(4, r+1)*((n+r)/4)^(r+1) } \\ Andrew Howroyd, Nov 12 2025

Crossrefs

Cf. A125714, A346004, A346005, A346595.

Sequence in context: A143174 A032241 A065748 * A019077 A019245 A019191

Adjacent sequences: A346003 A346004 A346005 * A346007 A346008 A346009

Keyword

nonn

Author

N. J. A. Sloane, Jul 25 2021

Status

approved