A017053 -id:A017053 - cp's OEIS frontend

A179986 Second 9-gonal (or nonagonal) numbers: a(n) = n(7n+5)/2.

0, 6, 19, 39, 66, 100, 141, 189, 244, 306, 375, 451, 534, 624, 721, 825, 936, 1054, 1179, 1311, 1450, 1596, 1749, 1909, 2076, 2250, 2431, 2619, 2814, 3016, 3225, 3441, 3664, 3894, 4131, 4375, 4626, 4884, 5149, 5421, 5700, 5986, 6279, 6579, 6886

Offset: 0

Bruno Berselli, Jan 13 2011

This sequence is a bisection of A118277 (even part).
Sequence found by reading the line from 0, in the direction 0, 19... and the line from 6, in the direction 6, 39,..., in the square spiral whose vertices are the generalized 9-gonal numbers A118277. - Omar E. Pol, Jul 24 2012
The early part of this sequence is a strikingly close approximation to the early part of A100752. - Peter Munn, Nov 14 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. second k-gonal numbers: A005449 (k=5), A014105 (k=6), A147875 (k=7), A045944 (k=8), this sequence (k=9), A033954 (k=10), A062728 (k=11), A135705 (k=12).

Cf. A001106, A022264, A022265, A024966, A055998, A100752, A174738, A186029, A218471.

[n*(7*n+5)/2: n in [0..50]]; // Bruno Berselli, Sep 23 2016

I:=[0, 6, 19]; [n le 3 select I[n] else 3*Self(n-1) -3*Self(n-2) +Self(n-3): n in [1..60]]; // Vincenzo Librandi, Oct 15 2012

f[n_] := n (7 n + 5)/2; f[Range[0, 60]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011*)
LinearRecurrence[{3, -3, 1}, {0, 6, 19}, 60] (* or *) Array[(#(7# + 5))/2&, 60, 0] (* Harvey P. Dale, Aug 19 2011 *)
CoefficientList[Series[x (6 + x)/(1 - x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Oct 15 2012 *)

a(n)=n*(7*n+5)/2 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: x*(6 + x)/(1 - x)^3.
a(n) = Sum_{i=0..(n-1)} A017053(i) for n>0.
a(-n) = A001106(n).
Sum_{i=0..n} (a(n)+i)^2 = ( Sum_{i=(n+1)..2*n} (a(n)+i)^2 ) + 21*A000217(n)^2 for n>0.
a(n) = a(n-1)+7*n-1 for n>0, with a(0)=0. - Vincenzo Librandi, Feb 05 2011
a(0)=0, a(1)=6, a(2)=19; for n>2, a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Aug 19 2011
a(n) = A174738(7n+5). - Philippe Deléham, Mar 26 2013
a(n) = A001477(n) + 2*A000290(n) + 3*A000217(n). - J. M. Bergot, Apr 25 2014
a(n) = A055998(4*n) - A055998(3*n). - Bruno Berselli, Sep 23 2016
E.g.f.: (x/2)*(12 + 7*x)*exp(x). - G. C. Greubel, Aug 19 2017

A134504 a(n) = Fibonacci(7n + 6).

Original entry on oeis.org

8, 233, 6765, 196418, 5702887, 165580141, 4807526976, 139583862445, 4052739537881, 117669030460994, 3416454622906707, 99194853094755497, 2880067194370816120, 83621143489848422977, 2427893228399975082453

Offset: 0

Artur Jasinski, Oct 28 2007

Index entries for linear recurrences with constant coefficients, signature (29,1).

Cf. A000045, A001906, A001519, A033887, A015448, A014445, A033888, A033889, A033890, A033891, A102312, A099100, A134490, A134491, A134492, A134493, A134494, A134495, A103134, A134497, A134498, A134499, A134500, A134501, A134502, A134503, A134504.

[Fibonacci(7*n +6): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Table[Fibonacci[7n+6], {n, 0, 30}]
LinearRecurrence[{29,1},{8,233},20] (* Harvey P. Dale, Jul 21 2021 *)

a(n)=fibonacci(7*n+6) \\ Charles R Greathouse IV, Jun 11 2015

G.f.: (-8-x) / (-1 + 29*x + x^2). - R. J. Mathar, Jul 04 2011
a(n) = A000045(A017053(n)). - Michel Marcus, Nov 08 2013
a(n) = 29*a(n-1) + a(n-2). - Wesley Ivan Hurt, Mar 15 2023

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011

A155724 Triangle read by rows: T(n, k) = 2nk + n + k - 4.

Original entry on oeis.org

0, 3, 8, 6, 13, 20, 9, 18, 27, 36, 12, 23, 34, 45, 56, 15, 28, 41, 54, 67, 80, 18, 33, 48, 63, 78, 93, 108, 21, 38, 55, 72, 89, 106, 123, 140, 24, 43, 62, 81, 100, 119, 138, 157, 176, 27, 48, 69, 90, 111, 132, 153, 174, 195, 216, 30, 53, 76, 99, 122, 145, 168, 191, 214, 237, 260

Offset: 1

Vincenzo Librandi, Jan 25 2009

			Triangle begins:
   0;
   3,  8;
   6, 13, 20;
   9, 18, 27, 36;
  12, 23, 34, 45,  56;
  15, 28, 41, 54,  67,  80;
  18, 33, 48, 63,  78,  93, 108;
  21, 38, 55, 72,  89, 106, 123, 140;
  24, 43, 62, 81, 100, 119, 138, 157, 176;
  27, 48, 69, 90, 111, 132, 153, 174, 195, 216;

Vincenzo Librandi, Rows n = 1..100, flattened

All terms are in A155723.
Cf. A008585, A016885, A017053, A020705, A154685, A155722.
Cf. A162261 (row sums).
Columns k: A008585 (k=1), A016885 (k=2), A017053 (k=3), 9*A020705 (k=4).
Diagonals include: A139570, A181510, A271625.

/* Triangle: */ [[2*m*n+m+n-4: m in [1..n]]: n in [1..10]]; // Bruno Berselli, Aug 31 2012

Flatten[Table[2 n m + m + n - 4, {n, 10}, {m, n}]] (* Vincenzo Librandi, Mar 01 2012 *)

def A155724(n,k): return 2*n*k+n+k-4
print(flatten([[A155724(n,k) for k in range(1,n+1)] for n in range(1,16)])) # G. C. Greubel, Jan 21 2025

T(n, k) = A154685(n, k) - 8. - L. Edson Jeffery, Oct 12 2012
2*T(n, k) + 9 = (2*k+1)*(2*n+1). - Vincenzo Librandi, Nov 18 2012
From G. C. Greubel, Jan 21 2025: (Start)
T(2*n-1, n) = 4*n^2 + n - 5 (main diagonal).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (1/4)*( 4*(-1)^(n+1)*n^2 + 2*(2-3*(-1)^n)*n - 7*(1-(-1)^n)).
G.f.: x*y*(3*x + 3*y - 4*x*y)/((1-x)*(1-y))^2. (End)

Edited by N. J. A. Sloane, Jun 23 2010

A155704 Triangle read by rows where T(m,n)=2mn + m + n + 10.

Original entry on oeis.org

14, 17, 22, 20, 27, 34, 23, 32, 41, 50, 26, 37, 48, 59, 70, 29, 42, 55, 68, 81, 94, 32, 47, 62, 77, 92, 107, 122, 35, 52, 69, 86, 103, 120, 137, 154, 38, 57, 76, 95, 114, 133, 152, 171, 190, 41, 62, 83, 104, 125, 146, 167, 188, 209, 230, 44, 67, 90, 113, 136, 159, 182

Offset: 1

Vincenzo Librandi, Jan 25 2009

The numbers 2*T(m,n)-19 = (2*m+1)*(2*n+1) are not prime.

First column: A016789, second column: A016873, third column: A017053, fourth column: A017221. - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
14;
17, 22;
20, 27, 34;
23, 32, 41, 50;
26, 37, 48, 59, 70;
29, 42, 55, 68, 81, 94;
32, 47, 62, 77, 92, 107, 122;
35, 52, 69, 86, 103, 120, 137, 154;
38, 57, 76, 95, 114, 133, 152, 171, 190;
41, 62, 83, 104, 125, 146, 167, 188, 209, 230;

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A153041, A016789, A016873, A017053, A017221.

[2*n*k + n + k + 10: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k +  10; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

A001526 a(n) = (7n+1)(7*n+6).

Original entry on oeis.org

6, 104, 300, 594, 986, 1476, 2064, 2750, 3534, 4416, 5396, 6474, 7650, 8924, 10296, 11766, 13334, 15000, 16764, 18626, 20586, 22644, 24800, 27054, 29406, 31856, 34404, 37050, 39794, 42636, 45576, 48614, 51750, 54984, 58316, 61746, 65274, 68900, 72624, 76446

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016993, A017053.

a[n_] := (7*n + 1)*(7*n + 6); Array[a, 40, 0] (* Amiram Eldar, Feb 19 2023 *)

a(n)=(7*n+1)*(7*n+6) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 98*n + a(n-1) with a(0)=6. - Vincenzo Librandi, Nov 12 2010
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A016993(n)*A017053(n).
Sum_{n>=0} 1/a(n) = cot(Pi/7)*Pi/35 = 0.186388....
Product_{n>=0} (1 - 1/a(n)) = cosec(Pi/7)*cos(sqrt(29)*Pi/14).
Product_{n>=0} (1 + 1/a(n)) = cosec(Pi/7)*cos(sqrt(3/7)*Pi/2). (End)
G.f.: -2*(3+43*x+3*x^2)/(x-1)^3. - R. J. Mathar, Apr 23 2024
From Elmo R. Oliveira, Oct 25 2024: (Start)
E.g.f.: exp(x)*(6 + 49*x*(2 + x)).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A017054 a(n) = (7*n + 6)^2.

Original entry on oeis.org

36, 169, 400, 729, 1156, 1681, 2304, 3025, 3844, 4761, 5776, 6889, 8100, 9409, 10816, 12321, 13924, 15625, 17424, 19321, 21316, 23409, 25600, 27889, 30276, 32761, 35344, 38025, 40804, 43681, 46656

Offset: 0

N. J. A. Sloane

If Y is a fixed 2-subset of a (7n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007

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

Cf. A017053 (7*n+6).

[(7*n+6)^2: n in [0..40]]; // Vincenzo Librandi, Jul 10 2011

(7*Range[0,30]+6)^2 (* or *) LinearRecurrence[{3,-3,1},{36,169,400},40] (* Harvey P. Dale, Apr 28 2016 *)

a(n)=(7*n+6)^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=36, a(1)=169, a(2)=400. - Harvey P. Dale, Apr 28 2016
Sum_{n>=0} 1/a(n) = psi'(6/7)/49 = 0.04223032499681527770... - R. J. Mathar, May 07 2024
G.f.: -(36+61*x+x^2)/(x-1)^3 . - R. J. Mathar, May 07 2024

A017055 a(n) = (7*n + 6)^3.

Original entry on oeis.org

216, 2197, 8000, 19683, 39304, 68921, 110592, 166375, 238328, 328509, 438976, 571787, 729000, 912673, 1124864, 1367631, 1643032, 1953125, 2299968, 2685619, 3112136, 3581577, 4096000, 4657463, 5268024

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 (4,-6,4,-1).

Cf. A017053 (7*n+6).

[(7*n+6)^3: n in [0..40]]; // Vincenzo Librandi, Jul 10 2011

(7*Range[0,30]+6)^3 (* or *) LinearRecurrence[{4,-6,4,-1},{216,2197,8000,19683},30] (* Harvey P. Dale, Mar 06 2019 *)

G.f.: ( 216 + 1333*x + 508*x^2 + x^3 ) / (x-1)^4. - R. J. Mathar, Aug 01 2014

A017056 a(n) = (7*n + 6)^4.

Original entry on oeis.org

1296, 28561, 160000, 531441, 1336336, 2825761, 5308416, 9150625, 14776336, 22667121, 33362176, 47458321, 65610000, 88529281, 116985856, 151807041, 193877776, 244140625, 303595776, 373301041

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 (5,-10,10,-5,1).

Cf. A017053 (7*n+6).

[(7*n+6)^4: n in [0..40]]; // Vincenzo Librandi, Jul 10 2011

G.f.: -(1296 + 22081*x + 30155*x^2 + 4091*x^3 + x^4)/(x-1)^5. - R. J. Mathar, Jul 14 2016

A017057 a(n) = (7*n + 6)^5.

Original entry on oeis.org

7776, 371293, 3200000, 14348907, 45435424, 115856201, 254803968, 503284375, 916132832, 1564031349, 2535525376, 3939040643, 5904900000, 8587340257, 12166529024, 16850581551, 22877577568, 30517578125

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 (6,-15,20,-15,6,-1).

Cf. A017053 (7*n+6).

[(7*n+6)^5: n in [0..40]]; // Vincenzo Librandi, Jul 10 2011

(7*Range[0,20]+6)^5 (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{7776,371293,3200000,14348907,45435424,115856201},20] (* Harvey P. Dale, Dec 20 2016 *)

G.f.: (x^5 + 32762*x^4 + 562782*x^3 + 1088882*x^2 + 324637*x + 7776)/(x-1)^6. [Colin Barker, Sep 17 2012]

A017058 a(n) = (7*n + 6)^6.

Original entry on oeis.org

46656, 4826809, 64000000, 387420489, 1544804416, 4750104241, 12230590464, 27680640625, 56800235584, 107918163081, 192699928576, 326940373369, 531441000000, 832972004929, 1265319018496, 1870414552161

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 (7, -21, 35, -35, 21, -7, 1).

Cf. A017053 (7*n+6).

[(7*n+6)^6: n in [0..40]]; // Vincenzo Librandi, Jul 10 2011

(7*Range[0,20]+6)^6 (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{46656,4826809,64000000,387420489,1544804416,4750104241,12230590464},20] (* Harvey P. Dale, Sep 04 2015 *)

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(0)=46656, a(1)=4826809, a(2)=64000000, a(3)=387420489, a(4)=1544804416, a(5)=4750104241, a(6)=12230590464. - Harvey P. Dale, Sep 04 2015

cp's OEIS Frontend

A179986 Second 9-gonal (or nonagonal) numbers: a(n) = n*(7*n+5)/2.

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A134504 a(n) = Fibonacci(7n + 6).

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A155724 Triangle read by rows: T(n, k) = 2*n*k + n + k - 4.

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A155704 Triangle read by rows where T(m,n)=2*m*n + m + n + 10.

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A001526 a(n) = (7*n+1)*(7*n+6).

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A017054 a(n) = (7*n + 6)^2.

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A017055 a(n) = (7*n + 6)^3.

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A179986 Second 9-gonal (or nonagonal) numbers: a(n) = n(7n+5)/2.

A155724 Triangle read by rows: T(n, k) = 2nk + n + k - 4.

A155704 Triangle read by rows where T(m,n)=2mn + m + n + 10.

A001526 a(n) = (7n+1)(7*n+6).