A016993 -id:A016993 - 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

A008589 Multiples of 7.

Original entry on oeis.org

0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252, 259, 266, 273, 280, 287, 294, 301, 308, 315, 322, 329, 336, 343, 350, 357, 364, 371, 378

Offset: 0

N. J. A. Sloane, Mar 15 1996

Also the Engel expansion of exp(1/7); cf. A006784 for the Engel expansion definition. - Benoit Cloitre, Mar 03 2002
Complement of A047304; A082784(a(n))=1; A109720(a(n))=0. - Reinhard Zumkeller, Nov 30 2009
The most likely sum of digits to occur when randomly tossing n pairs of (fair) six-sided dice. - Dennis P. Walsh, Jan 26 2012

			For n=2, a(2)=14 because 14 is the most likely sum (of the possible sums 4, 5, ..., 24) to occur when tossing 2 pairs of six-sided dice. - _Dennis P. Walsh_, Jan 26 2012

Ivan Panchenko, Table of n, a(n) for n = 0..200
Edward Brooks, Divisibility by Seven, The Analyst, Vol. 2, No. 5 (Sep., 1875), pp. 129-131.
Tanya Khovanova, Recursive Sequences
Michael Penn, The Luckiest Divisibility Test, YouTube video, 2022.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 319
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008587, A008588, A016993.

a008589 = (* 7)
a008589_list = [0, 7 ..]  -- Reinhard Zumkeller, Jan 25 2013

[7*n : n in [0..50]]; // Wesley Ivan Hurt, Jun 06 2014

A008589:=n->7*n; seq(A008589(n), n=0..50); # Wesley Ivan Hurt, Jun 06 2014

Range[0, 1000, 7] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)
7*Range[0,60] (* Harvey P. Dale, Feb 28 2023 *)

makelist(7*n,n,0,20); /* Martin Ettl, Dec 17 2012 */

a(n)=7*n \\ Charles R Greathouse IV, Jul 10 2016

(floor(a(n)/10) - 2*(a(n) mod 10)) == 0 modulo 7, see A076309. - Reinhard Zumkeller, Oct 06 2002
a(n) = 7*n = 2*a(n-1)-a(n-2); G.f.: 7*x/(x-1)^2. - Vincenzo Librandi, Dec 24 2010
E.g.f.: 7*x*exp(x). - Ilya Gutkovskiy, May 11 2016

A017089 a(n) = 8*n + 2.

Original entry on oeis.org

2, 10, 18, 26, 34, 42, 50, 58, 66, 74, 82, 90, 98, 106, 114, 122, 130, 138, 146, 154, 162, 170, 178, 186, 194, 202, 210, 218, 226, 234, 242, 250, 258, 266, 274, 282, 290, 298, 306, 314, 322, 330, 338, 346, 354, 362, 370, 378, 386, 394, 402, 410, 418, 426

Offset: 0

N. J. A. Sloane, Dec 11 1996

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 33 ).
Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 81 ).
First differences of A002939. - Aaron David Fairbanks, May 13 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A002939, A008589, A008590, A016813, A016993, A017005, A017017, A017077.

a017089 = (+ 2) . (* 8)
a017089_list = [2, 10 ..]  -- Reinhard Zumkeller, Jun 07 2015

[8*n+2: n in [0..60]]; // Vincenzo Librandi, May 28 2011

A017089:=n->8*n+2; seq(A017089(n), n=0..50); # Wesley Ivan Hurt, May 13 2014

Range[2, 1000, 8] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)

a(n) = 8*n+2; \\ Michel Marcus, Sep 17 2015

a(n) = 8*n+2; a(n) = 2*a(n-1)-a(n-2). - Vincenzo Librandi, May 28 2011
Sum_{n>=0} (-1)^n/a(n) = (Pi + 2*log(cot(Pi/8)))/(8*sqrt(2)). - Amiram Eldar, Dec 11 2021
From Elmo R. Oliveira, Mar 17 2024: (Start)
G.f.: 2*(1+3*x)/(1-x)^2.
E.g.f.: 2*exp(x)*(1 + 4*x).
a(n) = 2*A016813(n) = A008590(n) + 2. (End)

A017029 a(n) = 7*n + 4.

Original entry on oeis.org

4, 11, 18, 25, 32, 39, 46, 53, 60, 67, 74, 81, 88, 95, 102, 109, 116, 123, 130, 137, 144, 151, 158, 165, 172, 179, 186, 193, 200, 207, 214, 221, 228, 235, 242, 249, 256, 263, 270, 277, 284, 291, 298, 305, 312, 319, 326, 333, 340, 347, 354, 361, 368, 375, 382

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008589, A016993, A017005, A017017.

Cf. similar sequences with closed form (2*k-1)*n+k listed in A269044.

[7*n + 4: n in [0..60]]; // Vincenzo Librandi, Jun 18 2011

Range[4,1000,7] (* Vladimir Joseph Stephan Orlovsky, Jun 25 2009 *)
CoefficientList[Series[(3*x + 4)/(1 - x)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jan 27 2013 *)
LinearRecurrence[{2,-1},{4,11},60] (* Harvey P. Dale, Mar 27 2025 *)

a(n)=7*n+4 \\ Charles R Greathouse IV, Jul 10 2016

G.f.: (3*x + 4)/(1-x)^2. - Vincenzo Librandi, Jan 27 2013
From Elmo R. Oliveira, Apr 12 2025: (Start)
E.g.f.: exp(x)*(4 + 7*x).
a(n) = 2*a(n-1) - a(n-2). (End)

Extended by Ray Chandler, Jan 25 2005

A017017 a(n) = 7*n + 3.

Original entry on oeis.org

3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, 80, 87, 94, 101, 108, 115, 122, 129, 136, 143, 150, 157, 164, 171, 178, 185, 192, 199, 206, 213, 220, 227, 234, 241, 248, 255, 262, 269, 276, 283, 290, 297, 304, 311, 318, 325, 332, 339, 346, 353, 360, 367, 374, 381

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008589, A016993, A017005.

[7*n+3: n in [0..60]]; // Vincenzo Librandi, May 28 2011

A017017:=n->7*n + 3; seq(A017017(n), n=0..100); # Wesley Ivan Hurt, Feb 24 2014

Range[3, 600, 7] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)

a(n)=7*n+3 \\ Charles R Greathouse IV, Jul 02 2013

[7*n+3 for n in range(80)] # G. C. Greubel, Oct 17 2023

From G. C. Greubel, Oct 17 2023: (Start)
G.f.: (3 + 4*x)/(1 - x)^2.
E.g.f.: (3 + 7*x)*exp(x). (End)

A004126 a(n) = n(7n^2 - 1)/6.

Original entry on oeis.org

0, 1, 9, 31, 74, 145, 251, 399, 596, 849, 1165, 1551, 2014, 2561, 3199, 3935, 4776, 5729, 6801, 7999, 9330, 10801, 12419, 14191, 16124, 18225, 20501, 22959, 25606, 28449, 31495, 34751, 38224, 41921, 45849, 50015, 54426, 59089, 64011

Offset: 0

Albert D. Rich (Albert_Rich(AT)msn.com)

3-dimensional analog of centered polygonal numbers.
Sum of n triangular numbers starting from T(n), where T = A000217. E.g., a(4) = T(4) + T(5) + T(6) + T(7) = 10 + 15 + 21 + 28 = 74. - Amarnath Murthy, Jul 16 2004
Also as a(n) = (1/6)*(7*n^3-n), n>0: structured heptagonal diamond numbers (vertex structure 8). Cf. A100179 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Partial sums of A069099, centered heptagonal numbers (A000566). - Jonathan Vos Post, Mar 16 2006
Binomial transform of (0, 1, 7, 7, 0, 0, 0, ...) and third partial sum of (0, 1, 6, 7, 7, 7, ...). - Gary W. Adamson, Oct 05 2015

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
Cf. A000217, A000566, A016993, A069099.
Cf. A000447, A000292.

[n*(7*n^2-1)/6: n in [0..50]]; // Vincenzo Librandi, May 15 2011

seq(binomial(2*n+1,3)-binomial(n+1,3), n=0..38); # Zerinvary Lajos, Jan 21 2007

Table[n (7 n^2 - 1)/6, {n, 0, 80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

makelist(n*(7*n^2-1)/6,n,0,30); /* Martin Ettl, Jan 08 2013 */

vector(100, n, n--; n*(7*n^2 - 1)/6) \\ Altug Alkan, Oct 06 2015

a(n) = C(2*n+1,3)-C(n+1,3), n>=0. - Zerinvary Lajos, Jan 21 2007
a(n) = A000447(n) - A000292(n). - Zerinvary Lajos, Jan 21 2007
G.f.: x*(1+5*x+x^2)/(1-x)^4. - Colin Barker, Mar 02 2012
E.g.f.: (x/6)*(7*x^2 + 21*x + 6)*exp(x). - G. C. Greubel, Oct 05 2015
a(n) = Sum_{i = n..2*n-1} A000217(i). - Bruno Berselli, Sep 06 2017
a(n) = n^3 + Sum_{k=0..n-1} k*(k+1)/2. Alternately, a(n) = A000578(n) + A000292(n-1) for n>0. - Bruno Berselli, May 23 2018

A017005 a(n) = 7n + 2.

Original entry on oeis.org

2, 9, 16, 23, 30, 37, 44, 51, 58, 65, 72, 79, 86, 93, 100, 107, 114, 121, 128, 135, 142, 149, 156, 163, 170, 177, 184, 191, 198, 205, 212, 219, 226, 233, 240, 247, 254, 261, 268, 275, 282, 289, 296, 303, 310, 317, 324, 331, 338, 345, 352, 359, 366, 373, 380

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1120
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008589, A016993, A342758.

Magma
```
[ 7*n+2: n in [0..60] ];
```

Range[2, 1000, 7] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)
LinearRecurrence[{2,-1},{2,9},60] (* Harvey P. Dale, Sep 07 2015 *)

G.f.: (2+5*x)/(1-x)^2.
E.g.f.: exp(x)*(2 + 7*x). - Stefano Spezia, Mar 21 2021
a(n) = 2*a(n-1) - a(n-2). - Wesley Ivan Hurt, Mar 22 2021

A017053 a(n) = 7*n + 6.

Original entry on oeis.org

6, 13, 20, 27, 34, 41, 48, 55, 62, 69, 76, 83, 90, 97, 104, 111, 118, 125, 132, 139, 146, 153, 160, 167, 174, 181, 188, 195, 202, 209, 216, 223, 230, 237, 244, 251, 258, 265, 272, 279, 286, 293, 300, 307, 314, 321, 328, 335, 342, 349, 356, 363, 370, 377, 384

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 955
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008589, A016993, A017005, A017017.

[7*n + 6: n in [0..60]]; // Vincenzo Librandi, Jun 18 2011

Array[7*#+6&,100,0] (* Vladimir Joseph Stephan Orlovsky, Dec 14 2009 *)
LinearRecurrence[{2,-1},{6,13},60] (* Harvey P. Dale, Apr 13 2022 *)

a(n)=7*n+6 \\ Charles R Greathouse IV, Jul 10 2016

a(n) = 2*a(n-1) - a(n-2). - Wesley Ivan Hurt, Mar 17 2023

G.f.: (6+x)/(1-x)^2. - Wesley Ivan Hurt, Dec 28 2023

A093564 (7,1) Pascal triangle.

Original entry on oeis.org

1, 7, 1, 7, 8, 1, 7, 15, 9, 1, 7, 22, 24, 10, 1, 7, 29, 46, 34, 11, 1, 7, 36, 75, 80, 45, 12, 1, 7, 43, 111, 155, 125, 57, 13, 1, 7, 50, 154, 266, 280, 182, 70, 14, 1, 7, 57, 204, 420, 546, 462, 252, 84, 15, 1, 7, 64, 261, 624, 966, 1008, 714, 336, 99, 16, 1, 7, 71, 325, 885

Offset: 0

Wolfdieter Lang, Apr 22 2004

The array F(7;n,m) gives in the columns m>=1 the figurate numbers based on A016993, including the 9-gonal numbers A001106, (see the W. Lang link).
This is the seventh member, d=7, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-3, for d=1..6.
This is an example of a Riordan triangle (see A093560 for a comment and A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group). Therefore the o.g.f. for the row polynomials p(n,x):=Sum_{m=0..n} a(n,m)*x^m is G(z,x)=(1+6*z)/(1-(1+x)*z).
The SW-NE diagonals give A022097(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 6. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.

			Triangle begins
  [1];
  [7,  1];
  [7,  8,  1];
  [7, 15,  9,  1];
  ...

Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch. 5, pp. 109-122.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
W. Lang, First 10 rows and array of figurate numbers .

Row sums: A000079(n+2), n>=1, 1 for n=0, alternating row sums are 1 for n=0, 6 for n=2 and 0 otherwise.
The column sequences give for m=1..9: A016993, A001106 (9-gonal), A007584, A051740, A051877, A050403, A027818, A034266, A055994.
Cf. A093565 (d=8).

a093564 n k = a093564_tabl !! n !! k
a093564_row n = a093564_tabl !! n
a093564_tabl = [1] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [7, 1]
-- Reinhard Zumkeller, Sep 01 2014

N:= 20: # to get the first N rows
T:=Matrix(N,N):
T[1,1]:= 1:
for m from 2 to N do
T[m,1]:= 7:
T[m,2..m]:= T[m-1,1..m-1] + T[m-1,2..m];
od:
for m from 1 to N do
convert(T[m,1..m],list)
od; # Robert Israel, Dec 28 2014

a(n, m)=F(7;n-m, m) for 0<= m <= n, otherwise 0, with F(7;0, 0)=1, F(7;n, 0)=7 if n>=1 and F(7;n, m):=(7*n+m)*binomial(n+m-1, m-1)/m if m>=1.
Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=7 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).
G.f. column m (without leading zeros): (1+6*x)/(1-x)^(m+1), m>=0.
T(n, k) = C(n, k) + 6*C(n-1, k). - Philippe Deléham, Aug 28 2005
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(7 + 15*x + 9*x^2/2! + x^3/3!) = 7 + 22*x + 46*x^2/2! + 80*x^3/3! + 125*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

A280457 Expansion of Product_{k>=0} (1 + x^(7*k+1)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 1, 4, 5, 2, 0, 0, 0, 1, 5, 7, 3, 0, 0, 0, 1, 5, 8, 5, 1, 0, 0, 1, 6, 10, 6, 1, 0, 0, 1, 6, 12, 9, 2, 0, 0, 1, 7, 14, 11, 3, 0, 0, 1, 7, 16, 15, 5

Offset: 0

Ilya Gutkovskiy, Jan 03 2017

nonn

Number of partitions of n into distinct parts congruent to 1 mod 7.

			a(37) = 3 because we have [36, 1], [29, 8] and [22, 15].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.
Index entries for related partition-counting sequences

Cf. A000700, A016993, A169975, A261612, A280454.
Cf. A262928, A147599, A281243, A281244.
Cf. A109703, A281245, A281455, A281456, A281457, A281458, A281459.

nmax = 105; CoefficientList[Series[Product[(1 + x^(7 k + 1)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[If[Mod[k, 7] == 1, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 18 2017 *)

G.f.: Product_{k>=0} (1 + x^(7*k+1)).

a(n) ~ exp(Pi*sqrt(n)/sqrt(21))/(2*2^(1/7)*21^(1/4)*n^(3/4)) * (1 + (13*Pi/(336*sqrt(21)) - 3*sqrt(21)/(8*Pi)) / sqrt(n)). - Ilya Gutkovskiy, Jan 03 2017, extended by Vaclav Kotesovec, Jan 24 2017

cp's OEIS Frontend

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

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Mathematica

PARI

Python

Formula

A008589 Multiples of 7.

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Haskell

Magma

Maple

Mathematica

Maxima

PARI

Formula

A017089 a(n) = 8*n + 2.

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Haskell

Magma

Maple

Mathematica

PARI

Formula

A017029 a(n) = 7*n + 4.

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Magma

Mathematica

PARI

Formula

Extensions

A017017 a(n) = 7*n + 3.

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Magma

Maple

Mathematica

PARI

SageMath

Formula

A004126 a(n) = n*(7*n^2 - 1)/6.

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

A004126 a(n) = n(7n^2 - 1)/6.