A001106 -id:A001106 - cp's OEIS frontend

A007584 9-gonal (or enneagonal) pyramidal numbers: a(n) = n(n+1)(7*n-4)/6.

0, 1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, 1606, 2080, 2639, 3290, 4040, 4896, 5865, 6954, 8170, 9520, 11011, 12650, 14444, 16400, 18525, 20826, 23310, 25984, 28855, 31930, 35216, 38720, 42449, 46410, 50610, 55056, 59755, 64714, 69940, 75440, 81221

Offset: 0

N. J. A. Sloane, R. K. Guy

For n > 1, the digital roots of this sequence A010888(A007584(n)) form the purely periodic 27-cycle 1, 1, 7, 8, 2, 5, 6, 3, 3, 4, 4, 1, 2, 5, 8, 9, 6, 6, 7, 7, 4, 5, 8, 2, 3, 9, 9. For n > 1, the units digits of this sequence A010879(A007584(n)) form the purely periodic 20-cycle 1, 0, 4, 0, 5, 6, 0, 4, 5, 0, 6, 0, 9, 0, 0, 6, 5, 4, 0, 0. - Ant King, Oct 30 2012

Partial sums of A001106. - Joerg Arndt, Jun 10 2013

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A093564 ((7, 1) Pascal, column m=3).

Cf. similar sequences listed in A237616.

I:=[0, 1, 10, 34, 80]; [n le 5 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 10 2013

a:=n->sum((n+j)^2-(n+j), j=0..n): seq(a(n)/2, n=0..30); # Zerinvary Lajos, May 26 2008

Table[n*(n+1)(7n-4)/6, {n, 0,100}] (* Vladimir Joseph Stephan Orlovsky, Jun 25 2009 *)
LinearRecurrence[{4,-6,4,-1},{1,10,34,80},30] (* Ant King, Oct 27 2012 *)
CoefficientList[Series[x (1 + 6 x) / (1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 10 2013 *)

A007584[n]:=n*(n+1)*(7*n-4)/6$
makelist(A007584[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

a(n) = n*(n+1)*(7*n-4)/6; \\ Michel Marcus, Mar 04 2014

a(n) = (7*n-4)*binomial(n+1, 2)/3.
G.f.: x*(1+6*x)/(1-x)^4.
From Ant King, Oct 27 2012: (Start)
a(n) = a(n-1) + n*(7*n-5)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 7.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = (n+1)*(2*A001106(n)+n)/6.
a(n) = A000292(n) + 6*A000292(n-1).
a(n) = A002414(n) + A000292(n-1).
a(n) = A000217(n) + 7*A000292(n-1).
a(n) = binomial(n+2,3) + 6*binomial(n+1,3). (End)
a(n) = Sum_{i = 0..n-1} (n-i)*(7*i+1) for n>0. - Bruno Berselli, Feb 10 2014
a(n) = A080851(7,n-1). - R. J. Mathar, Jul 28 2016
E.g.f.: (x/6)*(6 + 24*x + 7*x^2)*exp(x). - G. C. Greubel, Oct 29 2017

A186029 a(n) = n(7n+3)/2.

Original entry on oeis.org

0, 5, 17, 36, 62, 95, 135, 182, 236, 297, 365, 440, 522, 611, 707, 810, 920, 1037, 1161, 1292, 1430, 1575, 1727, 1886, 2052, 2225, 2405, 2592, 2786, 2987, 3195, 3410, 3632, 3861, 4097, 4340, 4590, 4847, 5111, 5382, 5660, 5945, 6237, 6536, 6842, 7155, 7475

Offset: 0

Bruno Berselli, Feb 11 2011

This sequence is related to A050409 by A050409(n) = n*a(n) - Sum_{i=0..n-1} a(i).

			From _Ilya Gutkovskiy_, Mar 31 2016: (Start)
.                                           o o o o o o o o o o o o
.                                           o                     o
.         o o o o o o   o  o o o o o o  o   o  o  o o o o o o  o  o
.         o         o   o  o         o  o   o  o  o         o  o  o
. o   o   o  o   o  o   o  o  o   o  o  o   o  o  o  o   o  o  o  o
. o o o   o  o o o  o   o  o  o o o  o  o   o  o  o  o o o  o  o  o
.                       o               o   o  o               o  o
.                       o o o o o o o o o   o  o o o o o o o o o  o
.
.  n=1        n=2              n=3                    n=4
(End)

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

Cf. numbers of the form n*(d*n+10-d)/2 indexed in A140090.
Cf. A017041 (first differences).
Cf. A001106, A022264, A022265, A024966, A050409, A067727, A174738, A179986, A218471.

Magma
```
[n*(7*n+3)/2: n in [0..44]];
```

Table[(n - 1) (7 n - 4)/2, {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)
LinearRecurrence[{3,-3,1},{0,5,17},50] (* Harvey P. Dale, Sep 07 2022 *)

a(n)=n*(7*n+3)/2 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: x*(5+2*x)/(1-x)^3.
a(n) - a(-n) = A008585(n).
a(n) + a(-n) = A033582(n).
n*a(n+1) - (n+1)*a(n) = A024966(n). - Bruno Berselli, May 30 2012
n*a(n+2) - (n+2)*a(n) = A067727(n) for n>0. - Bruno Berselli, May 30 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=5, a(2)=17. - Philippe Deléham, Mar 26 2013
a(n) = A174738(7*n+4). - Philippe Deléham, Mar 26 2013
E.g.f.: (1/2)*(7*x^2 + 10*x)*exp(x). - G. C. Greubel, Jul 17 2017

A022265 a(n) = n(7n + 1)/2.

Original entry on oeis.org

0, 4, 15, 33, 58, 90, 129, 175, 228, 288, 355, 429, 510, 598, 693, 795, 904, 1020, 1143, 1273, 1410, 1554, 1705, 1863, 2028, 2200, 2379, 2565, 2758, 2958, 3165, 3379, 3600, 3828, 4063, 4305, 4554, 4810

Offset: 0

N. J. A. Sloane

For n >= 4, a(n) is the sum of the numbers appearing in the 4th row of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - Wesley Ivan Hurt, May 17 2021

			From _Bruno Berselli_, Oct 27 2017: (Start)
After 0:
4  =       -(1)       +             (2 + 3).
15 =     -(1 + 2)     +         (3 + 4 + 5 + 6).
33 =   -(1 + 2 + 3)   +     (4 + 5 + 6 + 7 + 8 + 9).
58 = -(1 + 2 + 3 + 4) + (5 + 6 + 7 + 8 + 9 + 10 + 11 + 12). (End)

G. C. Greubel, Table of n, a(n) for n = 0..5000
Leo Tavares, Illustration: Square-sided Triangles.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001106, A022264, A024966, A110449, A174738, A179986, A186029, A218471.
Cf. similar sequences listed in A022289.
Cf. A000217, A000290.

seq(binomial(7*n+1,2)/7, n=0..37); # Zerinvary Lajos, Jan 21 2007
seq(binomial(6*n+1,2)/3-binomial(5*n+1,2)/5, n=0..42); # Zerinvary Lajos, Jan 21 2007

Table[n (7 n + 1)/2, {n, 0, 40}] (* Bruno Berselli, Oct 13 2016 *)
LinearRecurrence[{3,-3,1},{0,4,15},40] (* Harvey P. Dale, Oct 09 2018 *)

a(n)=n*(7*n+1)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A110449(n, 3) for n>2.
a(n) = A049453(n) - A005475(n). - Zerinvary Lajos, Jan 21 2007
a(n) = 7*n + a(n-1) - 3 for n>0, a(0)=0. - Vincenzo Librandi, Aug 04 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with a(0)=0, a(1)=4, a(2)=15. - Philippe Deléham, Mar 26 2013
a(n) = A174738(7n+3). - Philippe Deléham, Mar 26 2013
a(n) = A000217(4*n) - A000217(3*n). - Bruno Berselli, Oct 13 2016
G.f.: x*(4 + 3*x)/(1 - x)^3. - Ilya Gutkovskiy, Oct 13 2016
E.g.f.: (x/2)*(7*x + 8)*exp(x). - G. C. Greubel, Aug 23 2017
a(n) = A000217(n) + 3*A000290(n). - Leo Tavares, Mar 15 2025

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

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).

A152759 3 times 9-gonal (or nonagonal) numbers: a(n) = 3n(7*n-5)/2.

Original entry on oeis.org

0, 3, 27, 72, 138, 225, 333, 462, 612, 783, 975, 1188, 1422, 1677, 1953, 2250, 2568, 2907, 3267, 3648, 4050, 4473, 4917, 5382, 5868, 6375, 6903, 7452, 8022, 8613, 9225, 9858, 10512, 11187, 11883, 12600, 13338, 14097, 14877, 15678, 16500, 17343, 18207, 19092, 19998

Offset: 0

Omar E. Pol, Dec 14 2008

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

Cf. A001106, A139268, A226491.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152767, A153783, A153448, A153875.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=21: see Comments lines of A226492.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,3,6!,21}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
CoefficientList[Series[3 x (1 + 6 x) / (1 - x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 05 2013 *)
LinearRecurrence[{3,-3,1},{0,3,27},40] (* Harvey P. Dale, May 26 2015 *)

a(n)=3*n*(7*n-5)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (21*n^2 - 15*n)/2 = 3*A001106(n).
a(n) = a(n-1) + 21*n - 18 with n > 0, a(0)=0. - Vincenzo Librandi, Nov 26 2010
G.f.: 3*x*(1+6*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(n) = n + A226491(n). - Bruno Berselli, Jun 11 2013
From Elmo R. Oliveira, Dec 15 2024: (Start)
E.g.f.: 3*exp(x)*x*(2 + 7*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A220212 Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).

Original entry on oeis.org

0, 1, 16, 70, 200, 455, 896, 1596, 2640, 4125, 6160, 8866, 12376, 16835, 22400, 29240, 37536, 47481, 59280, 73150, 89320, 108031, 129536, 154100, 182000, 213525, 248976, 288666, 332920, 382075, 436480, 496496, 562496, 634865, 714000, 800310, 894216, 996151

Offset: 0

Bruno Berselli, Dec 08 2012

Partial sums of A172073.

Apart from 0, all terms are in A135021: a(n) = A135021(A034856(n+1)) with n>0.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Index to sequences related to pyramidal numbers.

Cf. A135021, A172073.
Cf. convolution of the natural numbers (A000027) with the k-gonal numbers (* means "except 0"):
k= 2 (A000027 ): A000292;
k= 3 (A000217 ): A000332 (after the third term);
k= 4 (A000290 ): A002415 (after the first term);
k= 5 (A000326 ): A001296;
k= 6 (A000384*): A002417;
k= 7 (A000566 ): A002418;
k= 8 (A000567*): A002419;
k= 9 (A001106*): A051740;
k=10 (A001107*): A051797;
k=11 (A051682*): A051798;
k=12 (A051624*): A051799;
k=13 (A051865*): A055268.
Cf. similar sequences with formula n*(n+1)*(n+2)*(k*n-k+2)/12 listed in A264850.

A051866:=func; [&+[(n-k+1)*A051866(k): k in [0..n]]: n in [0..37]];

I:=[0,1,16,70,200]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

A051866[k_] := k (6 k - 5); Table[Sum[(n - k + 1) A051866[k], {k, 0, n}], {n, 0, 37}]
CoefficientList[Series[x (1 + 11 x) / (1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 18 2013 *)

G.f.: x*(1+11*x)/(1-x)^5.
a(n) = n*(n+1)*(n+2)*(3*n-2)/6.
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=1} 1/a(n) = 3*(3*sqrt(3)*Pi + 27*log(3) - 17)/80.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*(6*sqrt(3)*Pi - 64*log(2) + 37)/80. (End)

A255184 25-gonal numbers: a(n) = n(23n-21)/2.

Original entry on oeis.org

0, 1, 25, 72, 142, 235, 351, 490, 652, 837, 1045, 1276, 1530, 1807, 2107, 2430, 2776, 3145, 3537, 3952, 4390, 4851, 5335, 5842, 6372, 6925, 7501, 8100, 8722, 9367, 10035, 10726, 11440, 12177, 12937, 13720, 14526, 15355, 16207, 17082, 17980

Offset: 0

Luciano Ancora, Apr 03 2015

If b(n,k) = n*((k-2)*n-(k-4))/2 is n-th k-gonal number, then b(n,k) = A000217(n) + (k-3)* A000217(n-1) (see Deza in References section, page 21, where the formula is attributed to Bachet de Méziriac).
Also, b(n,k) = b(n,k-1) + A000217(n-1) (see Deza and Picutti in References section, page 20 and 137 respectively, where the formula is attributed to Nicomachus). Some examples:
for k=4, A000290(n) = A000217(n) + A000217(n-1);
for k=5, A000326(n) = A000290(n) + A000217(n-1);
for k=6, A000384(n) = A000326(n) + A000217(n-1), etc.
This is the case k=25.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6 (23rd row of the table).
E. Picutti, Sul numero e la sua storia, Feltrinelli Economica (1977), pages 131-147.

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

Cf. k-gonal numbers: A000217 (k=3), A000290 (k=4), A000326 (k=5), A000384 (k=6), A000566 (k=7), A000567 (k=8), A001106 (k=9), A001107 (k=10), A051682 (k=11), A051624 (k=12), A051865 (k=13), A051866 (k=14), A051867 (k=15), A051868 (k=16), A051869 (k=17), A051870 (k=18), A051871 (k=19), A051872 (k=20), A051873 (k=21), A051874 (k=22), A051875 (k=23), A051876 (k=24), this sequence (k=25), A255185 (k=26), A255186 (k=27), A161935 (k=28), A255187 (k=29), A254474 (k=30).

k:=25; [n*((k-2)*n-(k-4))/2: n in [0..40]]; // Bruno Berselli, Apr 10 2015

Mathematica
```
Table[n (23 n - 21)/2, {n, 40}]
```

a(n)=n*(23*n-21)/2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(-1 - 22*x)/(-1 + x)^3.
a(n) = A000217(n) + 22*A000217(n-1) = A051876(n) + A000217(n-1), see comments.
Product_{n>=2} (1 - 1/a(n)) = 23/25. - Amiram Eldar, Jan 22 2021
E.g.f.: exp(x)*(x + 23*x^2/2). - Nikolaos Pantelidis, Feb 05 2023

A047383 Numbers that are congruent to {1, 5} mod 7.

Original entry on oeis.org

1, 5, 8, 12, 15, 19, 22, 26, 29, 33, 36, 40, 43, 47, 50, 54, 57, 61, 64, 68, 71, 75, 78, 82, 85, 89, 92, 96, 99, 103, 106, 110, 113, 117, 120, 124, 127, 131, 134, 138, 141, 145, 148, 152, 155, 159, 162, 166, 169

Offset: 1

N. J. A. Sloane

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001106.

A047383:=n->((-1)^n+14*n-9)/4; seq(A047383(n), n=1..100); # Wesley Ivan Hurt, Nov 10 2013

Flatten[(#+{1,5})&/@(7Range[0,25])] (* or *) LinearRecurrence[ {1,1,-1},{1,5,8},80] (* Harvey P. Dale, Dec 24 2012 *)

a(n)=7*n\2-2 \\ Charles R Greathouse IV, Jun 11 2015

a(n) = ceiling((7*n+2)/2).
a(n) = 7*n - a(n-1) - 8 (with a(1)=1). - Vincenzo Librandi, Aug 05 2010
G.f.: x*(1+4*x+2*x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
a(1)=1, a(2)=5, a(3)=8; for n>3, a(n) = a(n-1) + a(n-2) - a(n-3). - Harvey P. Dale, Dec 24 2012
From Wesley Ivan Hurt, Nov 10 2013: (Start)
a(n) = 4*n - floor((n-1)/2) - 3.
a(2*k-1) = 7*k-6, a(2*k) = 7*k-2. (End)
E.g.f.: 2 + ((14*x - 9)*exp(x) + exp(-x))/4. - David Lovler, Sep 01 2022

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

A007584 9-gonal (or enneagonal) pyramidal numbers: a(n) = n*(n+1)*(7*n-4)/6.

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A186029 a(n) = n*(7*n+3)/2.

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A022265 a(n) = n*(7*n + 1)/2.

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A093564 (7,1) Pascal triangle.

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Haskell

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

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A152759 3 times 9-gonal (or nonagonal) numbers: a(n) = 3*n*(7*n-5)/2.

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A220212 Convolution of natural numbers (A000027) with tetradecagonal numbers (A051866).

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A007584 9-gonal (or enneagonal) pyramidal numbers: a(n) = n(n+1)(7*n-4)/6.

A186029 a(n) = n(7n+3)/2.

A022265 a(n) = n(7n + 1)/2.

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

A152759 3 times 9-gonal (or nonagonal) numbers: a(n) = 3n(7*n-5)/2.

A255184 25-gonal numbers: a(n) = n(23n-21)/2.

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