A016761 -id:A016761 - cp's OEIS frontend

A016953 a(n) = (6*n + 3)^9.

19683, 387420489, 38443359375, 794280046581, 7625597484987, 46411484401953, 208728361158759, 756680642578125, 2334165173090451, 6351461955384057, 15633814156853823, 35452087835576229, 75084686279296875, 150094635296999121, 285544154243029527, 520411082988487293

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A016761, A016945, A016946, A016947, A016948, A016949, A016950, A016951, A016952.

Subsequence of A001017.

[(6*n+3)^9: n in [0..20]]; // Vincenzo Librandi, May 06 2011

(6*Range[0,20]+3)^9  (* Harvey P. Dale, Jan 19 2012 *)

a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10). - Harvey P. Dale, Jan 19 2012
From Amiram Eldar, Mar 30 2022: (Start)
a(n) = A016945(n)^9 = A016947(n)^3.
a(n) = 3^9*A016761(n).
Sum_{n>=0} 1/a(n) = 511*zeta(9)/10077696.
Sum_{n>=0} (-1)^n/a(n) = 277*Pi^9/162533081088. (End)

A016749 a(n) = (2*n)^9.

Original entry on oeis.org

0, 512, 262144, 10077696, 134217728, 1000000000, 5159780352, 20661046784, 68719476736, 198359290368, 512000000000, 1207269217792, 2641807540224, 5429503678976, 10578455953408, 19683000000000, 35184372088832, 60716992766464, 101559956668416, 165216101262848

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A016761.

[(2*n)^9: n in [0..20]]; // Vincenzo Librandi, Sep 05 2011

A016749:=n->(2*n)^9: seq(A016749(n), n=0..30); # Wesley Ivan Hurt, Sep 15 2018

Table[(2n)^9, {n, 0, 40}] (* Stefan Steinerberger, Apr 08 2006 *)
LinearRecurrence[{10,-45,120,-210,252,-210,120,-45, 10, -1}, {0,512, 262144, 10077696, 134217728, 1000000000, 5159780352, 20661046784, 68719476736, 198359290368}, 20] (* Harvey P. Dale, Jan 13 2013 *)

vector(30, n, n--; (2*n)^9) \\ G. C. Greubel, Sep 15 2018

a(n) = 10*a(n-1)-45*a(n-2)+ 120*a(n-3)- 210*a(n-4)+252*a(n-5)-210*a(n-6)+120*a(n-7)-45*a(n-8)+10*a(n-9)-a(n-10). - Harvey P. Dale, Jan 13 2013
From Amiram Eldar, Oct 11 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(9)/512.
Sum_{n>=1} (-1)^(n+1)/a(n) = 255*zeta(9)/131072. (End)

More terms from Stefan Steinerberger, Apr 08 2006

A016833 a(n) = (4n+2)^9.

Original entry on oeis.org

512, 10077696, 1000000000, 20661046784, 198359290368, 1207269217792, 5429503678976, 19683000000000, 60716992766464, 165216101262848, 406671383849472, 922190162669056, 1953125000000000, 3904305912313344, 7427658739644928, 13537086546263552, 23762680013799936

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A013667, A016761, A016825.

(4*Range[0,20]+2)^9 (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{512,10077696,1000000000,20661046784,198359290368,1207269217792,5429503678976,19683000000000,60716992766464,165216101262848},20] (* Harvey P. Dale, May 26 2018 *)

From Amiram Eldar, Apr 21 2023: (Start)
a(n) = A016825(n)^9.
a(n) = 2^9*A016761(n).
Sum_{n>=0} 1/a(n) = 511*zeta(9)/262144.
Sum_{n>=0} (-1)^n/a(n) = 277*Pi^9/4227858432. (End)

A017121 a(n) = (8*n + 4)^9.

Original entry on oeis.org

262144, 5159780352, 512000000000, 10578455953408, 101559956668416, 618121839509504, 2779905883635712, 10077696000000000, 31087100296429568, 84590643846578176, 208215748530929664, 472161363286556672, 1000000000000000000, 1999004627104432128, 3802961274698203136

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A013667, A016761, A016833, A017113.

[(8*n+4)^9: n in [0..15] ]; // Vincenzo Librandi, Jul 21 2011

Table[(8*n + 4)^9, {n, 0, 20}] (* Amiram Eldar, Apr 25 2023 *)

G.f.: 262144*(1+x)*(x^8 + 19672*x^7 + 1736668*x^6 + 19971304*x^5 + 49441990*x^4 + 19971304*x^3 + 1736668*x^2 + 19672*x + 1) / (x-1)^10 . - R. J. Mathar, May 08 2015
From Amiram Eldar, Apr 25 2023: (Start)
a(n) = A017113(n)^9.
a(n) = 2^9*A016833(n) = 2^18*A016761(n).
Sum_{n>=0} 1/a(n) = 511*zeta(9)/134217728.
Sum_{n>=0} (-1)^n/a(n) = 277*Pi^9/2164663517184. (End)

A017337 a(n) = (10*n + 5)^9.

Original entry on oeis.org

1953125, 38443359375, 3814697265625, 78815638671875, 756680642578125, 4605366583984375, 20711912837890625, 75084686279296875, 231616946283203125, 630249409724609375, 1551328215978515625, 3517876291919921875, 7450580596923828125, 14893745087865234375, 28334269484119140625

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A013667, A016761, A017329.

[(10*n+5)^9: n in [0..15]]; // Vincenzo Librandi, Aug 02 2011

(10*Range[0,20]+5)^9 (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1953125,38443359375,3814697265625,78815638671875,756680642578125,4605366583984375,20711912837890625,75084686279296875,231616946283203125,630249409724609375},20] (* Harvey P. Dale, Jul 23 2016 *)

G.f.: 1953125*(x+1)*(x^8 + 19672*x^7 + 1736668*x^6 + 19971304*x^5 + 49441990*x^4 + 19971304*x^3 + 1736668*x^2 + 19672*x + 1)/(x-1)^10. -Colin Barker, Nov 13 2012
From Amiram Eldar, Apr 18 2023: (Start)
a(n) = A017329(n)^9.
a(n) = 5^9 * A016761(n).
Sum_{n>=0} 1/a(n) = 511*zeta(9)/1000000000.
Sum_{n>=0} (-1)^n/a(n) = 277*Pi^9/16128000000000. (End)

A384949 Decimal expansion of Sum_{k>=0} 1/(2*k+1)^9.

Original entry on oeis.org

1, 0, 0, 0, 0, 5, 1, 3, 4, 5, 1, 8, 3, 8, 4, 3, 7, 7, 2, 5, 9, 2, 8, 1, 7, 9, 0, 0, 5, 4, 2, 5, 0, 5, 0, 0, 5, 6, 7, 9, 9, 6, 9, 9, 0, 2, 4, 6, 6, 3, 8, 3, 1, 1, 4, 4, 4, 5, 9, 6, 2, 6, 3, 8, 1, 4, 4, 3, 3, 6, 2, 5, 4, 4, 5, 7, 8, 5, 5, 5, 5, 0, 5, 1, 4, 9, 4, 3, 0, 8, 2, 6, 6, 4, 0, 3, 0, 5, 8, 3

Offset: 1

Jason Bard, Jul 24 2025

			1.00005134518384377259281790054250500567996990246638...

Jason Bard, Table of n, a(n) for n = 1..10000
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 21.

Cf. A013667, A016761.

evalf(sum(1/(2*k+1)^9, k=0..infinity), 120);  # Alois P. Heinz, Jul 26 2025

RealDigits[511*Zeta[9]/512, 10, 100][[1]]

511*zeta(9)/512 \\ Amiram Eldar, Jul 28 2025

Equals 511*Zeta(9)/512 = 511/512 * A013667.

Equals Sum_{n>=0} 1/A016761(n). - Amiram Eldar, Jul 28 2025

cp's OEIS Frontend

A016953 a(n) = (6*n + 3)^9.

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A016749 a(n) = (2*n)^9.

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A016833 a(n) = (4n+2)^9.

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A017121 a(n) = (8*n + 4)^9.

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A017337 a(n) = (10*n + 5)^9.

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A384949 Decimal expansion of Sum_{k>=0} 1/(2*k+1)^9.

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