A308215 a(n) is the multiplicative inverse of A001844(n+1) modulo A001844(n); where A001844 is the sequence of centered square numbers.

0, 2, 12, 11, 39, 28, 82, 53, 141, 86, 216, 127, 307, 176, 414, 233, 537, 298, 676, 371, 831, 452, 1002, 541, 1189, 638, 1392, 743, 1611, 856, 1846, 977, 2097, 1106, 2364, 1243, 2647, 1388, 2946, 1541, 3261, 1702, 3592, 1871, 3939, 2048

Offset

0,2

Comments

The sequence explores the relationship between the terms of A001844, the sums of consecutive squares. The sequence is an interleaving of A054552 (a number spiral arm) and (A001844-n). The gap between the lower values of A308215 and the upper values of A308217 increase by 3n; each successive gap increasing by 6.

Links

Table of n, a(n) for n=0..45.

Daniel Hoyt, Graph of A308215 and A308217 in relation to A001844

Formula

a(n) satisfies a(n)*(2*n*(n+1)+1) == 1 (mod 2*n*(n-1)+1).

Conjectures from Colin Barker, May 16 2019: (Start)

G.f.: x*(2 + 12*x + 5*x^2 + 3*x^3 + x^4 + x^5) / ((1 - x)^3*(1 + x)^3).

a(n) = (3 + (-1)^n + 2*(2+(-1)^n)*n + 2*(3+(-1)^n)*n^2) / 4 for n>0.

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.

(End)

Prog

(Python)
import gmpy2
sos = [] # sum of squares
a=0
b=1
for i in range(50):
    c = a**2 + b**2
    sos.append(c)
    a +=1
    b +=1
ls = []
for i in range(len(sos)-1):
    c = gmpy2.invert(sos[i+1], sos[i])
    ls.append(int(c))
print(ls)
(PARI) f(n) = 2*n*(n+1)+1; \\ A001844
a(n) = lift(1/Mod(f(n+1), f(n))); \\ Michel Marcus, May 16 2019

Crossrefs

Cf. A001844, A033951, A054552, A308217.

Sequence in context: A343645 A245281 A358505 * A216478 A181060 A171446

Adjacent sequences: A308212 A308213 A308214 * A308216 A308217 A308218

Keyword

nonn

Author

Daniel Hoyt, May 15 2019

Status

approved