A167963 -id:A167963 - cp's OEIS frontend

A168029 a(n) = n*(n^6 + 1)/2.

0, 1, 65, 1095, 8194, 39065, 139971, 411775, 1048580, 2391489, 5000005, 9743591, 17915910, 31374265, 52706759, 85429695, 134217736, 205169345, 306110025, 446935879, 640000010, 900544281, 1247178955, 1702412735, 2293235724, 3051757825, 4015905101

Offset: 0

N. J. A. Sloane, Dec 11 2009

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Sequences of the form n*(n^m + 1)/2: A001477 (m=0), A000217 (m=1), A006003 (m=2), A027441 (m=3), A021003 (m=4), A167963 (m=5), this sequence (m=6), A168067 (m=7), A168116 (m=8), A168118 (m=9), A168119 (m=10).

[n*(n^6+1)/2: n in [0..40]]; // Vincenzo Librandi, Dec 10 2014

CoefficientList[Series[x(1 +57x +603x^2 +1198x^3 +603x^4 +57x^5 +x^6)/ (1-x)^8, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 10 2014 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1}, {0,1,65,1095,8194,39065, 139971,411775}, 41] (* Harvey P. Dale, Jan 24 2019 *)

[n*(n^6+1)/2 for n in range(41)] # G. C. Greubel, Jan 12 2023

G.f.: x*(1+57*x+603*x^2+1198*x^3+603*x^4+57*x^5+x^6)/(1-x)^8. - Vincenzo Librandi, Dec 10 2014

E.g.f.: (x/2)*(2 +63*x +301*x^2 +350*x^3 +140*x^4 +21*x^5 +x^6)*exp(x). - G. C. Greubel, Jan 12 2023

More terms from Vincenzo Librandi, Dec 10 2014

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

Original entry on oeis.org

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

Thomas Scheuerle, Mar 06 2023

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

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).

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

Cf. A167963 (bisection).

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

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)))

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

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).

cp's OEIS Frontend

A168029 a(n) = n*(n^6 + 1)/2.

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

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