A156676 -id:A156676 - cp's OEIS frontend

A017137 a(n) = 8*n + 6.

6, 14, 22, 30, 38, 46, 54, 62, 70, 78, 86, 94, 102, 110, 118, 126, 134, 142, 150, 158, 166, 174, 182, 190, 198, 206, 214, 222, 230, 238, 246, 254, 262, 270, 278, 286, 294, 302, 310, 318, 326, 334, 342, 350, 358, 366, 374, 382, 390, 398, 406, 414, 422, 430

Offset: 0

N. J. A. Sloane, Dec 11 1996

First differences of A002943. - Aaron David Fairbanks, May 13 2014

			G.f. = 6 + 14*x + 22*x^2 + 30*x^3 + 38*x^4 + 46*x^5 + 54*x^6 + 62*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000290, A002943, A004767, A004770, A008590, A017101, A017113, A017245, A156676.

Cf. A016825, A017629, A047595 (complement), A089911.

a017137 = (+ 6) . (* 8)  -- Reinhard Zumkeller, Jul 05 2013

[8*n+6: n in [0..60]]; // Vincenzo Librandi, Jun 07 2011

A017137:=n->8*n+6; seq(A017137(n), n=0..50); # Wesley Ivan Hurt, May 13 2014

Range[6, 1000, 8] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)
8Range[0,60]+6 (* or *) LinearRecurrence[{2,-1},{6,14},60] (* Harvey P. Dale, Nov 14 2021 *)

a(n) = 8*n+6; \\ Michel Marcus, Sep 17 2015

Vec((6+2*x)/(1-x)^2 + O(x^100)) \\ Altug Alkan, Oct 23 2015

a(n) = 2*A004767(n) = A000290(A017245(n)) - A156676(n+1). - Reinhard Zumkeller, Jul 13 2010
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Jun 07 2011
A089911(3*a(n)) = 4. - Reinhard Zumkeller, Jul 05 2013
From Michael Somos, May 15 2014: (Start)
G.f.: (6 + 2*x)/(1 - x)^2.
E.g.f.: (6 + 8*x)*exp(x). (End)
Sum_{n>=0} (-1)^n/a(n) = (Pi + log(3-2*sqrt(2)))/(8*sqrt(2)). - Amiram Eldar, Dec 11 2021
a(n) = A016825(2*n+1). - Elmo R. Oliveira, Apr 12 2025

A017245 a(n) = 9*n + 7.

Original entry on oeis.org

7, 16, 25, 34, 43, 52, 61, 70, 79, 88, 97, 106, 115, 124, 133, 142, 151, 160, 169, 178, 187, 196, 205, 214, 223, 232, 241, 250, 259, 268, 277, 286, 295, 304, 313, 322, 331, 340, 349, 358, 367, 376, 385, 394, 403, 412, 421, 430, 439, 448, 457, 466, 475, 484

Offset: 0

N. J. A. Sloane

Numbers whose digital root is 7. - Halfdan Skjerning, Mar 15 2018

Tanya Khovanova, Recursive Sequences
J. Laroche and N. J. A. Sloane, Correspondence, 1977
Leo Tavares, Illustration: Triple Triangular Frames
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008591, A017233.

[9*n+7: n in [0..60]]; // Vincenzo Librandi, Apr 30 2015

Range[7, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
Table[9 n + 7, {n, 0, 70}] (* or *) CoefficientList[Series[(7 + 2 x)/(1 - x)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Apr 30 2015 *)
LinearRecurrence[{2,-1},{7,16},60] (* Harvey P. Dale, Jul 30 2024 *)

vector(100,n,9*n-2) \\ Derek Orr, Apr 30 2015

a(n)^2 = A156676(n+1) + A017137(n). - Reinhard Zumkeller, Jul 13 2010
From Vincenzo Librandi, Apr 30 2015: (Start)
G.f.: (7+2*x)/(1-x)^2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. (End)
E.g.f.: exp(x)*(7 + 9*x). - Stefano Spezia, Dec 08 2024

A156772 a(n) = 729*n - 198.

Original entry on oeis.org

531, 1260, 1989, 2718, 3447, 4176, 4905, 5634, 6363, 7092, 7821, 8550, 9279, 10008, 10737, 11466, 12195, 12924, 13653, 14382, 15111, 15840, 16569, 17298, 18027, 18756, 19485, 20214, 20943, 21672, 22401, 23130, 23859, 24588, 25317, 26046

Offset: 1

Vincenzo Librandi, Feb 15 2009

The identity (6561*n^2 - 3564*n + 485)^2 - (81*n^2 - 44*n + 6)*(729*n - 198)^2 = 1 can be written as A156774(n)^2 - A156676(n)*a(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A156676, A156774.

I:=[531, 1260]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]];

LinearRecurrence[{2,-1}, {531,1260}, 40]

a(n)=729*n-198 \\ Charles R Greathouse IV, Dec 23 2011

[9*(81*n -22) for n in [1..50]] # G. C. Greubel, Jun 19 2021

a(n) = 2*a(n-1) - a(n-2).
G.f.: x*(531 + 198*x)/(1-x)^2.
E.g.f.: 9*(22 - (22 - 81*x)*exp(x)). - G. C. Greubel, Jun 19 2021

A156774 a(n) = 6561n^2 - 3564n + 485.

Original entry on oeis.org

485, 3482, 19601, 48842, 91205, 146690, 215297, 297026, 391877, 499850, 620945, 755162, 902501, 1062962, 1236545, 1423250, 1623077, 1836026, 2062097, 2301290, 2553605, 2819042, 3097601, 3389282, 3694085, 4012010, 4343057, 4687226

Offset: 0

Vincenzo Librandi, Feb 15 2009

The identity (6561*n^2 - 3564*n + 485)^2 - (81*n^2 - 44*n + 6)*(729*n - 198)^2 = 1 can be written as a(n)^2 - A156676(n)*A156772(n)^2 = 1 for n>0.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A156676, A156772.

I:=[485, 3482, 19601]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];

LinearRecurrence[{3,-3,1},{485,3482,19601},40]
Table[6561n^2-3564n+485,{n,0,30}] (* Harvey P. Dale, Dec 09 2020 *)

a(n)= 6561*n^2-3564*n+485 \\ Charles R Greathouse IV, Dec 23 2011

[485 -3564*n +6561*n^2 for n in (0..40)] # G. C. Greubel, Jun 21 2021

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (485 + 2027*x + 10610*x^2)/(1-x)^3.
E.g.f.: (485 + 2997*x + 6561*x^2)*exp(x). - G. C. Greubel, Jun 21 2021

Edited by Charles R Greathouse IV, Jul 25 2010

cp's OEIS Frontend

A017137 a(n) = 8*n + 6.

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A017245 a(n) = 9*n + 7.

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A156772 a(n) = 729*n - 198.

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A156774 a(n) = 6561*n^2 - 3564*n + 485.

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A156774 a(n) = 6561n^2 - 3564n + 485.