A017089 -id:A017089 - cp's OEIS frontend

A017095 a(n) = (8*n + 2)^7.

128, 10000000, 612220032, 8031810176, 52523350144, 230539333248, 781250000000, 2207984167552, 5455160701056, 12151280273024, 24928547056768, 47829690000000, 86812553324672, 150363025899136, 250226879128704, 402271083010688, 627485170000000, 953133216331392, 1414067010444416

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 (8,-28,56,-70,56,-28,8,-1).

Cf. A013665, A016819, A017089 (8n+2), A001015 (n^7).

[(8*n+2)^7: n in [0..30]]; // Vincenzo Librandi, Jul 12 2011

Table[(8*n + 2)^7, {n, 0, 20}] (* Amiram Eldar, Apr 24 2023 *)

From Amiram Eldar, Apr 24 2023: (Start)
a(n) = A017089(n)^7.
a(n) = 2^7*A016819(n).
Sum_{n>=0} 1/a(n) = 61*Pi^7/47185920 + 127*zeta(7)/32768. (End)

A017097 a(n) = (8*n + 2)^9.

Original entry on oeis.org

512, 1000000000, 198359290368, 5429503678976, 60716992766464, 406671383849472, 1953125000000000, 7427658739644928, 23762680013799936, 66540410775079424, 167619550409708032, 387420489000000000, 833747762130149888, 1689478959002692096, 3251948521156637184, 5987402799531080192

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, A016821, A017089.

[(8*n+2)^9: n in [0..20]]; // Vincenzo Librandi, Jul 12 2011

(8*Range[0,30]+2)^9 (* Harvey P. Dale, Dec 15 2011 *)

a(0)=512, a(1)=1000000000, a(2)=198359290368, a(3)=5429503678976, a(4)=60716992766464, a(5)=406671383849472, a(6)=1953125000000000, a(7)=7427658739644928, a(8)=23762680013799936, a(9)=66540410775079424, 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, Dec 15 2011
From Amiram Eldar, Apr 24 2023: (Start)
a(n) = A017089(n)^9.
a(n) = 2^9*A016821(n).
Sum_{n>=0} 1/a(n) = 277*Pi^9/8455716864 + 511*zeta(9)/524288. (End)

A017099 a(n) = (8*n + 2)^11.

Original entry on oeis.org

2048, 100000000000, 64268410079232, 3670344486987776, 70188843638032384, 717368321110468608, 4882812500000000000, 24986644000165537792, 103510234140112521216, 364375289404334925824, 1127073856954876807168, 3138105960900000000000, 8007313507497959524352

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 (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Cf. A013669, A016823, A017089.

[(8*n+2)^11: n in [0..20]]; // Vincenzo Librandi, Jul 12 2011

(8*Range[0,20]+2)^11 (* Harvey P. Dale, Dec 12 2012 *)

From Amiram Eldar, Apr 24 2023: (Start)
a(n) = A017089(n)^11.
a(n) = 2^11*A016823(n).
Sum_{n>=0} 1/a(n) = 50521*Pi^11/60881161420800 + 2047*zeta(11)/8388608. (End)
a(n) = 12*a(n-1) - 66*a(n-2) + 220*a(n-3) - 495*a(n-4) + 792*a(n-5) - 924*a(n-6) + 792*a(n-7) - 495*a(n-8) + 220*a(n-9) - 66*a(n-10) + 12*a(n-11) - a(n-12). - Wesley Ivan Hurt, Jan 20 2024

A173773 a(3n) = 8n+2, a(3n+1) = 2n+1, a(3n+2) = 8n+6.

Original entry on oeis.org

2, 1, 6, 10, 3, 14, 18, 5, 22, 26, 7, 30, 34, 9, 38, 42, 11, 46, 50, 13, 54, 58, 15, 62, 66, 17, 70, 74, 19, 78, 82, 21, 86, 90, 23, 94, 98, 25, 102, 106, 27, 110, 114, 29, 118, 122, 31, 126, 130, 33, 134, 138, 35, 142, 146, 37, 150, 154, 39, 158, 162, 41, 166

Offset: 0

Paul Curtz, Nov 26 2010

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A005408, A017089, A017137.

Table[Sequence @@ {8*n+2, 2*n+1,  8*n+6}, {n, 0, 20}] (* Amiram Eldar, Oct 08 2023 *)

G.f.: (1+x)*(2-x+7*x^2-x^3+2*x^4)/(1-x^3)^2.

Sum_{n>=0} (-1)^(n+1)/a(n) = (2-sqrt(2))*Pi/8. - Amiram Eldar, Oct 08 2023

A258415 Array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = (2 + 2^(n-1)(6k - 3 + 2*(-1)^n))/3, n,k >= 1.

Original entry on oeis.org

1, 4, 3, 2, 8, 5, 14, 10, 12, 7, 6, 30, 18, 16, 9, 54, 38, 46, 26, 20, 11, 22, 118, 70, 62, 34, 24, 13, 214, 150, 182, 102, 78, 42, 28, 15, 86, 470, 278, 246, 134, 94, 50, 32, 17, 854, 598, 726, 406, 310, 166, 110, 58, 36, 19

Offset: 1

L. Edson Jeffery, May 29 2015

The sequence is a permutation of the natural numbers.

Theorem: Let v(y) denote the 2-adic valuation of y. For x an odd natural number, let F(x) = (3*x+1)/2^v(3*x+1) (see A075677). Row n of A is the set of all natural numbers m such that v(1+F(4*(2*m-1)-3)) = n.

			Array begins:
.      1     3     5     7     9    11    13    15    17     19
.      4     8    12    16    20    24    28    32    36     40
.      2    10    18    26    34    42    50    58    66     74
.     14    30    46    62    78    94   110   126   142    158
.      6    38    70   102   134   166   198   230   262    294
.     54   118   182   246   310   374   438   502   566    630
.     22   150   278   406   534   662   790   918  1046   1174
.    214   470   726   982  1238  1494  1750  2006  2262   2518
.     86   598  1110  1622  2134  2646  3158  3670  4182   4694
.    854  1878  2902  3926  4950  5974  6998  8022  9046  10070

Cf. A005408, A008586, A017089 (rows 1-3).

Cf. A075677, A257480, A257499.

(* Array: *)
Grid[Table[(2 + 2^(n - 1)*(6*k - 3 + 2*(-1)^n))/3, {n, 10}, {k, 10}]]
(* Array antidiagonals flattened: *)
Flatten[Table[(2 + 2^(n - k)*(6*k - 3 + 2*(-1)^(n - k + 1)))/3, {n, 10}, {k, n}]]

A(n,k) = (1 + A257499(n,k))/2.

A281813 a(0) = 3, a(n) = 8*n + 4 for n > 0.

Original entry on oeis.org

3, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 140, 148, 156, 164, 172, 180, 188, 196, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 284, 292, 300, 308, 316, 324, 332, 340, 348, 356, 364, 372, 380, 388, 396, 404

Offset: 0

Rayan Ivaturi, Jan 30 2017

Consider a 1 X S rectangle on an infinite grid and surround it successively with the minimum number of 1 X 1 tiles: the initial S on step 0, 2S + 6 on step 1, 2S + 14 on step 2, and so on. This sequence is case S = 3. See Ivaturi link for a connection to sieving for primes.

Rayan Ivaturi, Ripple Numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017113.

Other 'ripple sequences': A022144 (s=1), A017089 (s=2).

a(n)=if(n>0, 8*n+4, 3) \\ Charles R Greathouse IV, Feb 07 2017

G.f.: (3 + 6*x - x^2)/(1 - x)^2.
a(n) = A017113(n) for n>0, a(0) = 3.
a(n) = A086570(n+1) for n>=1. - R. J. Mathar, Jun 21 2025

Entry revised by Editors of OEIS, Feb 09 2017

A047473 Numbers that are congruent to {2, 3} mod 8.

Original entry on oeis.org

2, 3, 10, 11, 18, 19, 26, 27, 34, 35, 42, 43, 50, 51, 58, 59, 66, 67, 74, 75, 82, 83, 90, 91, 98, 99, 106, 107, 114, 115, 122, 123, 130, 131, 138, 139, 146, 147, 154, 155, 162, 163, 170, 171, 178, 179, 186, 187, 194, 195, 202, 203, 210, 211, 218, 219, 226, 227, 234

Offset: 1

N. J. A. Sloane

Numbers k such that k and k+2 have the same digital binary sum. - Benoit Cloitre, Dec 01 2002

Also, numbers k such that k*(3*k + 1)/8 + 1/4 is a nonnegative integer. - Bruno Berselli, Feb 14 2017

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

Union of A017089 and A017101.

Cf. A047467, A047468, A047470, A047471.

Flatten[# + {2,3} &/@ (8 Range[0, 30])] (* or *) LinearRecurrence[{1, 1, -1}, {2, 3, 10}, 60] (* Harvey P. Dale, Sep 28 2012 *)

a(n) = 8*n - a(n-1) - 11 for n>1, a(1)=2. - Vincenzo Librandi, Aug 06 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 4*n - 7/2 - 3*(-1)^n/2.
G.f.: x*(2 + x + 5*x^2)/((1 + x)*(1 - x)^2). (End)
a(1)=2, a(2)=3, a(3)=10; for n>3, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Sep 28 2012
Sum_{n>=1} (-1)^(n+1)/a(n) = (2-sqrt(2))*Pi/16 + sqrt(2)*log(sqrt(2)+1)/8 - log(2)/8. - Amiram Eldar, Dec 18 2021

More terms from Vincenzo Librandi, Aug 06 2010

cp's OEIS Frontend

A017095 a(n) = (8*n + 2)^7.

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A017097 a(n) = (8*n + 2)^9.

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A017099 a(n) = (8*n + 2)^11.

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A173773 a(3*n) = 8*n+2, a(3*n+1) = 2*n+1, a(3*n+2) = 8*n+6.

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A258415 Array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = (2 + 2^(n-1)*(6*k - 3 + 2*(-1)^n))/3, n,k >= 1.

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A281813 a(0) = 3, a(n) = 8*n + 4 for n > 0.

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PARI

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A047473 Numbers that are congruent to {2, 3} mod 8.

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A173773 a(3n) = 8n+2, a(3n+1) = 2n+1, a(3n+2) = 8n+6.

A258415 Array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = (2 + 2^(n-1)(6k - 3 + 2*(-1)^n))/3, n,k >= 1.