A195020 -id:A195020 - cp's OEIS frontend

A001106 9-gonal (or enneagonal or nonagonal) numbers: a(n) = n(7n-5)/2.

0, 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, 559, 651, 750, 856, 969, 1089, 1216, 1350, 1491, 1639, 1794, 1956, 2125, 2301, 2484, 2674, 2871, 3075, 3286, 3504, 3729, 3961, 4200, 4446, 4699, 4959, 5226, 5500, 5781, 6069, 6364

Offset: 0

N. J. A. Sloane

Sequence found by reading the line from 0, in the direction 0, 9, ... and the parallel line from 1, in the direction 1, 24, ..., in the square spiral whose vertices are the generalized 9-gonal (enneagonal) numbers A118277. Also sequence found by reading the same lines in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - Omar E. Pol, Sep 10 2011
Number of ordered pairs of integers (x,y) with abs(x) < n, abs(y) < n and x+y <= n. - Reinhard Zumkeller, Jan 23 2012
Partial sums give A007584. - Omar E. Pol, Jan 15 2013

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and William A. Tedeschi, Table of n, a(n) for n = 0..10000 (1000 terms were computed by T. D. Noe)
S. Barbero, U. Cerruti and N. Murru, Transforming Recurrent Sequences by Using the Binomial and Invert Operators, J. Int. Seq. 13 (2010) # 10.7.7, section 4.4.
C. K. Cook and M. R. Bacon, Some polygonal number summation formulas, Fib. Q., 52 (2014), 336-343.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 343
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Nonagonal Number.
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A093564 ((7, 1) Pascal, column m=2). Partial sums of A016993.

Cf. A131875, A057655, A069099, A244646.

a001106 n = length [(x,y) | x <- [-n+1..n-1], y <- [-n+1..n-1], x + y <= n]
-- Reinhard Zumkeller, Jan 23 2012

a001106 n = n*(7*n-5) `div` 2 -- James Spahlinger, Oct 18 2012

Table[n(7n - 5)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 9}, 50] (* Harvey P. Dale, Nov 06 2011 *)
(* For Mathematica 10.4+ *) Table[PolygonalNumber[RegularPolygon[9], n], {n, 0, 43}] (* Arkadiusz Wesolowski, Aug 27 2016 *)
PolygonalNumber[9,Range[0,50]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 19 2019 *)

a(n)=n*(7*n-5)/2 \\ Charles R Greathouse IV, Jun 10 2011

def aList(): # Intended to compute the initial segment of the sequence, not isolated terms.
     x, y = 1, 1
     yield 0
     while True:
         yield x
         x, y = x + y + 7, y + 7
A001106 = aList()
print([next(A001106) for i in range(49)]) # Peter Luschny, Aug 04 2019

a(n) = (7*n - 5)*n/2.
G.f.: x*(1+6*x)/(1-x)^3. - Simon Plouffe in his 1992 dissertation.
a(n) = n + 7*A000217(n-1). - Floor van Lamoen, Oct 14 2005
Starting (1, 9, 24, 46, 75, ...) gives the binomial transform of (1, 8, 7, 0, 0, 0, ...). - Gary W. Adamson, Jul 22 2007
Row sums of triangle A131875 starting (1, 9, 24, 46, 75, 111, ...). A001106 = binomial transform of (1, 8, 7, 0, 0, 0, ...). - Gary W. Adamson, Jul 22 2007
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0) = 0, a(1) = 1, a(2) = 9. - Jaume Oliver Lafont, Dec 02 2008
a(n) = 2*a(n-1) - a(n-2) + 7. - Mohamed Bouhamida, May 05 2010
a(n) = a(n-1) + 7*n - 6 (with a(0) = 0). - Vincenzo Librandi, Nov 12 2010
a(n) = A174738(7n). - Philippe Deléham, Mar 26 2013
a(7*a(n) + 22*n + 1) =  a(7*a(n) + 22*n) + a(7*n+1). - Vladimir Shevelev, Jan 24 2014
E.g.f.: x*(2 + 7*x)*exp(x)/2. - Ilya Gutkovskiy, Jul 28 2016
a(n+2) + A000217(n) = (2*n+3)^2. - Ezhilarasu Velayutham, Mar 18 2020
Product_{n>=2} (1 - 1/a(n)) = 7/9. - Amiram Eldar, Jan 21 2021
Sum_{n>=1} 1/a(n) = A244646. - Amiram Eldar, Nov 12 2021
a(n) = A000217(3*n-2) - (n-1)^2. - Charlie Marion, Feb 27 2022
a(n) = 3*A000217(n) + 2*A005563(n-2). In general, if P(k,n) = the n-th k-gonal number, then P(m*k,n) = m*P(k,n) + (m-1)*A005563(n-2). - Charlie Marion, Feb 21 2023

A195019 Multiples of 3 and of 4 interleaved: a(2n-1) = 3n, a(2n) = 4n.

Original entry on oeis.org

3, 4, 6, 8, 9, 12, 12, 16, 15, 20, 18, 24, 21, 28, 24, 32, 27, 36, 30, 40, 33, 44, 36, 48, 39, 52, 42, 56, 45, 60, 48, 64, 51, 68, 54, 72, 57, 76, 60, 80, 63, 84, 66, 88, 69, 92, 72, 96, 75, 100, 78, 104, 81, 108, 84, 112, 87, 116, 90, 120, 93, 124, 96, 128

Offset: 1

Omar E. Pol, Sep 07 2011, Sep 12 2011

First differences of A195020.

a(n) is also the length of the n-th edge of a square spiral in which the first two edges are the legs of the primitive Pythagorean triple [3, 4, 5]. The spiral contains infinitely many Pythagorean triples in which the hypotenuses are the positives A008587. Zero together with partial sums give A195020; the vertices of the spiral.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean triangles and triples
Eric Weisstein's World of Mathematics, Pythagorean Triple
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008585, A008586, A026741, A195020-A195029, A195031, A195033, A195035.

[((n-3)*(-1)^n+7*n+3)/4: n in [1..60]]; // Vincenzo Librandi, Sep 12 2011

Table[((n-3)*(-1)^n + 7*n + 3)/4, {n,1,50}] (* G. C. Greubel, Aug 19 2017 *)

a(n)=(n+1)\2*(4-n%2)  \\ M. F. Hasler, Sep 08 2011

pair(3*n, 4*n).
a(2*n-1) = 3*n, a(2*n) = 4*n. - M. F. Hasler, Sep 08 2011
G.f.: x*(3+4*x) / ( (x-1)^2*(1+x)^2 ). - R. J. Mathar, Sep 09 2011
From Bruno Berselli, Sep 12 2011: (Start)
a(n) = ((n-3)*(-1)^n + 7*n + 3)/4.
a(n) + a(n+1) = A047355(n+2). (End)
E.g.f.: (1/4)*((3 + 7*x)*exp(x) - (3 + x)*exp(-x)). - G. C. Greubel, Aug 19 2017

A022264 a(n) = n(7n - 1)/2.

Original entry on oeis.org

0, 3, 13, 30, 54, 85, 123, 168, 220, 279, 345, 418, 498, 585, 679, 780, 888, 1003, 1125, 1254, 1390, 1533, 1683, 1840, 2004, 2175, 2353, 2538, 2730, 2929, 3135, 3348, 3568, 3795, 4029, 4270, 4518, 4773, 5035, 5304, 5580, 5863, 6153, 6450, 6754, 7065, 7383

Offset: 0

N. J. A. Sloane

Sequence found by reading the line from 0, in the direction 0, 13, ..., and the parallel line from 3, in the direction 3, 30, ..., in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - Omar E. Pol, Sep 09 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Leo Tavares, Illustration: Crysta-gons
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A022265.
Cf. sequences listed in A254963.
Cf. similar sequences listed in A022288.
Cf. A016742, A049450, A174738, A195019, A195020.
Cf. A005449, A000384.

[n*(7*n-1)/2 : n in [0..50]]; // Wesley Ivan Hurt, Dec 04 2016

[seq(binomial(7*n,2)/7, n=0..37)]; # Zerinvary Lajos, Jan 02 2007

Table[n (7*n - 1)/2, {n, 0, 40}] (* Zerinvary Lajos, Jul 10 2009 *)

a(n)=n*(7*n-1)/2 \\ Charles R Greathouse IV, Mar 08 2013

a(n) = C(7*n,2)/7, n >= 0. - Zerinvary Lajos, Jan 02 2007
a(n) = A049450(n) + A000217(n). - Reinhard Zumkeller, Oct 09 2008
a(n) = 7*n + a(n-1) - 4 for n > 0, a(0)=0. - Vincenzo Librandi, Aug 04 2010
a(n) = (2*n)^2 - n*(n+1)/2 = A016742(n) - A000217(n). - Philippe Deléham, Mar 08 2013
a(n) = A174738(7*n+2). - Philippe Deléham, Mar 26 2013
G.f.: x*(3 + 4*x)/(1 - x)^3. - R. J. Mathar, Aug 04 2016
a(n) = A000217(4*n-1) - A000217(3*n-1). - Bruno Berselli, Oct 17 2016
a(n) = (1/5) * Sum_{i=n..(6*n-1)} i. - Wesley Ivan Hurt, Dec 04 2016
E.g.f.: (1/2)*x*(7*x + 6)*exp(x). - G. C. Greubel, Aug 19 2017
a(n) = A005449(n) + A000384(n). See Crysta-gons illustration. - Leo Tavares, Nov 21 2021

A144555 a(n) = 14*n^2.

Original entry on oeis.org

0, 14, 56, 126, 224, 350, 504, 686, 896, 1134, 1400, 1694, 2016, 2366, 2744, 3150, 3584, 4046, 4536, 5054, 5600, 6174, 6776, 7406, 8064, 8750, 9464, 10206, 10976, 11774, 12600, 13454, 14336, 15246, 16184, 17150, 18144, 19166, 20216, 21294, 22400, 23534, 24696

Offset: 0

N. J. A. Sloane, Jan 01 2009

Sequence found by reading the line from 0, in the direction 0, 14, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Also sequence found by reading the same line and direction in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - Omar E. Pol, Sep 10 2011

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

See also A033428, A033429, A033581, A033582, A033583, A033584, ... and A249327 for the whole table.
Cf. A000290, A001105, A064761, A118277, A152742, A195019, A195020.
Cf. A008596, A195145.

Table[14*n^2, {n, 0, 45}] (* Amiram Eldar, Feb 03 2021 *)

A144555(n)=14*n^2 \\ M. F. Hasler, Oct 31 2014

a(n) = 14*A000290(n) = 7*A001105(n) = 2*A033582(n). - Omar E. Pol, Jan 01 2009
a(n) = a(n-1) + 14*(2*n-1), with a(0) = 0. - Vincenzo Librandi, Nov 25 2010
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/84.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/168.
Product_{n>=1} (1 + 1/a(n)) = sqrt(14)*sinh(Pi/sqrt(14))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(14)*sin(Pi/sqrt(14))/Pi. (End)
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 14*x*(1 + x)/(1-x)^3.
E.g.f.: 14*x*(1 + x)*exp(x).
a(n) = n*A008596(n) = A195145(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A024966 7 times triangular numbers: 7n(n+1)/2.

Original entry on oeis.org

0, 7, 21, 42, 70, 105, 147, 196, 252, 315, 385, 462, 546, 637, 735, 840, 952, 1071, 1197, 1330, 1470, 1617, 1771, 1932, 2100, 2275, 2457, 2646, 2842, 3045, 3255, 3472, 3696, 3927, 4165, 4410, 4662, 4921, 5187, 5460, 5740, 6027, 6321, 6622

Offset: 0

Joe Keane (jgk(AT)jgk.org), Dec 11 1999

Sequence found by reading the line from 0, in the direction 0, 7, ... and the same line from 0, in the direction 1, 21, ..., in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the main diagonal in the spiral. - Omar E. Pol, Sep 09 2011
Also sequence found by reading the same line mentioned above in the square spiral whose vertices are the generalized enneagonal numbers A118277. Axis perpendicular to A195145 in the same spiral. - Omar E. Pol, Sep 18 2011
Sequence provides all integers m such that 56*m + 49 is a square. - Bruno Berselli, Oct 07 2015
Sum of the numbers from 3*n to 4*n. - Wesley Ivan Hurt, Dec 22 2015

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

Cf. A000217, A001106, A002378, A022264, A022265, A028895, A028896, A033996.
Cf. A045943, A046092, A069099, A118277, A174738, A179986, A186029, A193053.
Cf. A195019, A195020, A195145, A218471.

[ (7*n^2 + 7*n)/2 : n in [0..50] ]; // Wesley Ivan Hurt, Jun 09 2014

[seq(7*binomial(n,2), n=1..44)]; # Zerinvary Lajos, Nov 24 2006

7 Table[n (n + 1)/2, {n, 0, 43}] (* or *)
Table[Sum[i, {i, 3 n, 4 n}], {n, 0, 43}] (* or *)
Table[SeriesCoefficient[7 x/(1 - x)^3, {x, 0, n}], {n, 0, 43}] (* Michael De Vlieger, Dec 22 2015 *)
7*Accumulate[Range[0,50]] (* or *) LinearRecurrence[{3,-3,1},{0,7,21},50] (* Harvey P. Dale, Jul 20 2025 *)

x='x+O('x^100); concat(0, Vec(7*x/(1-x)^3)) \\ Altug Alkan, Dec 23 2015

a(n) = (7/2)*n*(n+1).
G.f.: 7*x/(1-x)^3.
a(n) = (7*n^2 + 7*n)/2 = 7*A000217(n). - Omar E. Pol, Dec 12 2008
a(n) = a(n-1) + 7*n with n > 0, a(0)=0. - Vincenzo Librandi, Nov 19 2010
a(n) = A069099(n+1) - 1. - Omar E. Pol, Oct 03 2011
a(n) = a(-n-1), a(n+2) = A193053(n+2) + 2*A193053(n+1) + A193053(n). - Bruno Berselli, Oct 21 2011
From Philippe Deléham, Mar 26 2013: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0) = 0, a(1) = 7, a(2) = 21.
a(n) = A174738(7*n+6).
a(n) = A179986(n) + n = A186029(n) + 2*n = A022265(n) + 3*n = A022264(n) + 4*n = A218471(n) + 5*n = A001106(n) + 6*n. (End)
a(n) = Sum_{i=3*n..4*n} i. - Wesley Ivan Hurt, Dec 22 2015
E.g.f.: (7/2)*x*(x+2)*exp(x). - G. C. Greubel, Aug 19 2017
From Amiram Eldar, Feb 25 2022: (Start)
Sum_{n>=1} 1/a(n) = 2/7.
Sum_{n>=1} (-1)^(n+1)/a(n) = (2/7)*(2*log(2) - 1). (End)
From Amiram Eldar, Feb 21 2023: (Start)
Product_{n>=1} (1 - 1/a(n)) = -(7/(2*Pi))*cos(sqrt(15/7)*Pi/2).
Product_{n>=1} (1 + 1/a(n)) = (7/(2*Pi))*cosh(Pi/(2*sqrt(7))). (End)

A185019 a(n) = n(14n-3).

Original entry on oeis.org

0, 11, 50, 117, 212, 335, 486, 665, 872, 1107, 1370, 1661, 1980, 2327, 2702, 3105, 3536, 3995, 4482, 4997, 5540, 6111, 6710, 7337, 7992, 8675, 9386, 10125, 10892, 11687, 12510, 13361, 14240, 15147, 16082, 17045, 18036, 19055, 20102, 21177, 22280, 23411, 24570

Offset: 0

Bruno Berselli, Oct 14 2011 - based on remarks and sequences by Omar E. Pol

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms: An origin of A185019.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A013656, A033429, A049452, A049453, A147296, A195025.
Cf. A000384, A195023, A195024, A135703.
Cf. A195020 (vertices of the numerical spiral in link).

Magma
```
[n*(14*n-3): n in [0..42]];
```

I:=[0,11,50]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

CoefficientList[Series[x (11 + 17 x)/(1 - x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)

PARI
```
for(n=0, 42, print1(n*(14*n-3)", "));
```

G.f.: x*(11+17*x)/(1-x)^3.
a(n) = A195025(-n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. - Wesley Ivan Hurt, Dec 18 2020
From Elmo R. Oliveira, Dec 30 2024: (Start)
E.g.f.: exp(x)*x*(11 + 14*x).
a(n) = n + A195023(n). (End)

A193053 a(n) = (14n(n+3) + (2n-5)(-1)^n + 21)/16.

Original entry on oeis.org

1, 5, 10, 17, 26, 36, 49, 62, 79, 95, 116, 135, 160, 182, 211, 236, 269, 297, 334, 365, 406, 440, 485, 522, 571, 611, 664, 707, 764, 810, 871, 920, 985, 1037, 1106, 1161, 1234, 1292, 1369, 1430, 1511, 1575, 1660, 1727, 1816, 1886, 1979, 2052, 2149, 2225, 2326

Offset: 0

Bruno Berselli, Oct 20 2011 - based on remarks and sequences by Omar E. Pol

For an origin of this sequence, see the numerical spiral illustrated in the Links section.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A195020 (vertices of the numerical spiral in link).

Cf. A001106, A022264, A033572, A144555, A152760, A158482, A158485, A185019, A195021, A195023-A195030, A195320, A198017 [incomplete list].

[(14*n*(n+3)+(2*n-5)*(-1)^n+21)/16: n in [0..50]];

Table[(14*n*(n + 3) + (2*n - 5)*(-1)^n + 21)/16, {n, 0, 50}] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{1,5,10,17,26},60] (* Harvey P. Dale, Jun 19 2020 *)

for(n=0, 50, print1((14*n*(n+3)+(2*n-5)*(-1)^n+21)/16", "));

O.g.f.: (1 + 4*x + 3*x^2 - x^3)/((1 + x)^2*(1 - x)^3).
E.g.f.: (1/16)*((21 + 56*x + 14*x^2)*exp(x) - (5 + 2*x)*exp(-x)). - G. C. Greubel, Aug 19 2017
a(n) = A195020(n) + n + 1.
a(n)   - a(-n-1)  = A047336(n+1).
a(n+1) - a(-n)    = A113804(n+1).
a(n+2) - a(n)     = A047385(n+3).
a(n+4) - a(n)     = A113803(n+4).
a(2*n) + a(2*n-1) = A069127(n+1).
a(2*n) - a(2*n-1) = A016813(n).
a(2*n+1) - a(2*n) = A016777(n+1).
a(n+2) + 2*a(n+1) + a(n) = A024966(n+2).

A195320 7 times hexagonal numbers: a(n) = 7n(2*n-1).

Original entry on oeis.org

0, 7, 42, 105, 196, 315, 462, 637, 840, 1071, 1330, 1617, 1932, 2275, 2646, 3045, 3472, 3927, 4410, 4921, 5460, 6027, 6622, 7245, 7896, 8575, 9282, 10017, 10780, 11571, 12390, 13237, 14112, 15015, 15946, 16905, 17892, 18907, 19950, 21021, 22120, 23247, 24402, 25585

Offset: 0

Omar E. Pol, Sep 18 2011

Sequence found by reading the line from 0, in the direction 0, 7, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277.

Also sequence found by reading the same line (mentioned above) in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the one of the semi-diagonals of the square spiral, which is related to the primitive Pythagorean triple [3, 4, 5]. - Omar E. Pol, Oct 13 2011

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

Bisection of A024966.

Cf. A000384, A118277, A152746, A152750, A185019, A193053, A195019, A195020, A198017, A316466.

[7*n*(2*n-1): n in [0..50]]; // Vincenzo Librandi, Sep 28 2011

Table[7*n*(2*n - 1), {n, 0, 50}] (* Paolo Xausa, Jan 29 2025 *)
7*PolygonalNumber[6, Range[0, 50]] (* Paolo Xausa, Jan 29 2025 *)

a(n)=7*n*(2*n-1) \\ Charles R Greathouse IV, Sep 28 2015

a(n) = 14*n^2 - 7*n = 7*A000384(n).
G.f.: -7*x*(1+3*x)/(x-1)^3. - R. J. Mathar, Sep 27 2011
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: 7*exp(x)*x*(2*x + 1).
a(n) = A316466(n) - n = A024966(2*n+1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A195014 Vertex number of a square spiral whose edges have length A195013.

Original entry on oeis.org

0, 2, 5, 9, 15, 21, 30, 38, 50, 60, 75, 87, 105, 119, 140, 156, 180, 198, 225, 245, 275, 297, 330, 354, 390, 416, 455, 483, 525, 555, 600, 632, 680, 714, 765, 801, 855, 893, 950, 990, 1050, 1092, 1155, 1199, 1265, 1311, 1380, 1428, 1500, 1550, 1625, 1677

Offset: 0

Omar E. Pol, Sep 09 2011

Zero together with the partial partial sums of A195013.
Second bisection is 2, 9, 21, 38, 60, 87, 119, ...: A005476. - Omar E. Pol, Sep 25 2011
Number of pairs (x,y) with even x in {0,...,n}, odd y in {0,...,3n}, and xClark Kimberling, Jul 02 2012

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

Cf. A005476, A028895, A195013, A195020.

[(10*n^2 + 18*n + 3 + (2*n - 3)*(-1)^n)/16 : n in [0..50]]; // Vincenzo Librandi, Oct 26 2014

LinearRecurrence[{1,2,-2,-1,1},{0,2,5,9,15},60] (* Harvey P. Dale, May 20 2019 *)

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: f(x)/g(x), where f(x) = 2*x + 3*x^2 and g(x) = (1+x)^2 * (1-x)^3. - Clark Kimberling, Jul 02 2012
a(n) = (10*n^2 + 18*n + 3 + (2*n - 3)*(-1)^n)/16. - Luce ETIENNE, Aug 11 2014

A198017 a(n) = n(7n + 11)/2 + 1.

Original entry on oeis.org

1, 10, 26, 49, 79, 116, 160, 211, 269, 334, 406, 485, 571, 664, 764, 871, 985, 1106, 1234, 1369, 1511, 1660, 1816, 1979, 2149, 2326, 2510, 2701, 2899, 3104, 3316, 3535, 3761, 3994, 4234, 4481, 4735, 4996, 5264, 5539, 5821, 6110, 6406, 6709, 7019, 7336, 7660, 7991

Offset: 0

Bruno Berselli, Oct 21 2011 - based on remarks and sequences by Omar E. Pol

First bisection of A193053 (see also the numerical spiral illustrated in the Links section).

The inverse binomial transform yields 1, 9, 7, 0, 0 (0 continued).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A195020 (vertices of the numerical spiral in link).
Cf. A017005 (first differences).
Cf. A069099, A001106.
Cf. A001106, A022264, A033572, A144555, A152760, A158482, A158485, A185019, A193053, A195021, A195023-A195030, A195320 [incomplete list].

Magma
```
[n*(7*n+11)/2+1: n in [0..47]];
```

Table[(n(7n+11))/2+1,{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{1,10,26},60] (* Harvey P. Dale, Mar 03 2013 *)

for(n=0, 47, print1(n*(7*n+11)/2+1", "));

G.f.: (1 + 7*x - x^2)/(1-x)^3.
a(n) = A195020(2*n) + 2*n + 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = 2*a(n-1) - a(n-2) + 7.
From Elmo R. Oliveira, Dec 24 2024: (Start)
E.g.f.: exp(x)*(2 + 18*x + 7*x^2)/2.
a(n) = n + A001106(n+1). (End)

cp's OEIS Frontend

A001106 9-gonal (or enneagonal or nonagonal) numbers: a(n) = n*(7*n-5)/2.

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A195019 Multiples of 3 and of 4 interleaved: a(2*n-1) = 3*n, a(2*n) = 4*n.

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A022264 a(n) = n*(7*n - 1)/2.

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A144555 a(n) = 14*n^2.

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A024966 7 times triangular numbers: 7*n*(n+1)/2.

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A185019 a(n) = n*(14*n-3).

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A193053 a(n) = (14*n*(n+3) + (2*n-5)*(-1)^n + 21)/16.

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A001106 9-gonal (or enneagonal or nonagonal) numbers: a(n) = n(7n-5)/2.

A195019 Multiples of 3 and of 4 interleaved: a(2n-1) = 3n, a(2n) = 4n.

A022264 a(n) = n(7n - 1)/2.

A024966 7 times triangular numbers: 7n(n+1)/2.

A185019 a(n) = n(14n-3).

A193053 a(n) = (14n(n+3) + (2n-5)(-1)^n + 21)/16.

A195320 7 times hexagonal numbers: a(n) = 7n(2*n-1).

A198017 a(n) = n(7n + 11)/2 + 1.