A361263 Numbers of the form k*(k^5 +- 1)/2.

0, 1, 31, 33, 363, 366, 2046, 2050, 7810, 7815, 23325, 23331, 58821, 58828, 131068, 131076, 265716, 265725, 499995, 500005, 885775, 885786, 1492986, 1492998, 2413398, 2413411, 3764761, 3764775, 5695305, 5695320, 8388600, 8388616, 12068776, 12068793, 17006103, 17006121, 23522931, 23522950

Offset: 1

Author

Thomas Scheuerle, Mar 06 2023

Keywords

• nonn
• easy

Comments

Integer solutions of x + y = (x - y)^6. If x = a(n) then y = a(n - (-1)^n).

Links

• Winston de Greef, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1).

Crossrefs

Cf. A006003, A021003, A027441, A057587, A057590, A135503, A168029.

Cf. A167963 (bisection).

Programs

Maple

map(k -> (k*(k^5-1)/2, k*(k^5+1)/2), [$1..100]);

PARI

concat(0, Vec(x^2*(1+30*x-4*x^2+150*x^3+6*x^4+150*x^5-4*x^6+30*x^7+x^8)/((1-x)^7*(1+x)^6) + O(x^100)))

Python

def A361263(n): return (k:=n+1>>1)*(k**5+1-((n&1)<<1))>>1 # Chai Wah Wu, Mar 22 2023

Formula

G.f.: x^2*(1+30*x-4*x^2+150*x^3+6*x^4+150*x^5-4*x^6+30*x^7+x^8) / ((1-x)^7*(1+x)^6).

a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - 15*a(n-4) + 15*a(n-5) + 20*a(n-6) - 20*a(n-7) - 15*a(n-8) + 15*a(n-9) + 6*a(n-10) - 6*a(n-11) - a(n-12) + a(n-13).