A033571 -id:A033571 - cp's OEIS frontend

A000566 Heptagonal numbers (or 7-gonal numbers): n(5n-3)/2.

0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, 1918, 2059, 2205, 2356, 2512, 2673, 2839, 3010, 3186, 3367, 3553, 3744, 3940, 4141, 4347, 4558, 4774, 4995, 5221, 5452, 5688

Offset: 0

N. J. A. Sloane

Binomial transform of (0, 1, 5, 0, 0, 0, ...). Binomial transform is A084899. - Paul Barry, Jun 10 2003
Row sums of triangle A131413. - Gary W. Adamson, Jul 08 2007
Sequence starting (1, 7, 18, 34, ...) = binomial transform of (1, 6, 5, 0, 0, 0, ...). Also row sums of triangle A131896. - Gary W. Adamson, Jul 24 2007
Also the partial sums of A016861, a zero added in front; therefore a(n) = n (mod 5). - R. J. Mathar, Mar 19 2008
Also sequence found by reading the line from 0, in the direction 0, 7, ..., and the line from 1, in the direction 1, 18, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. These parallel lines are the semi-axes perpendicular to the main axis A195015 in the same spiral. - Omar E. Pol, Oct 14 2011
Also sequence found by reading the line from 0, in the direction 0, 7, ... and the parallel line from 1, in the direction 1, 18, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 18 2012
Partial sums give A002413. - Omar E. Pol, Jan 12 2013
The n-th heptagonal number equals the sum of the n consecutive integers starting at 2*n-1; for example, 1, 3+4, 5+6+7, 7+8+9+10, etc. In general, the n-th (2k+1)-gonal number is the sum of the n consecutive integers starting at (k-1)*n - (k-2). When k = 1 and 2, this result generates the triangular numbers, A000217, and the pentagonal numbers, A000326, respectively. - Charlie Marion, Mar 02 2022

			G.f. = x + 7*x^2 + 18*x^3 + 34*x^4 + 55*x^5 + 81*x^6 + 112*x^7 + ... - _Michael Somos_, Jan 25 2019

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 38.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
Leonard E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 2.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 123.

Daniel Mondot, Table of n, a(n) for n = 0..10000 (first 1000 terms 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 341.
Bir Kafle, Florian Luca, and Alain Togbé, Pentagonal and heptagonal repdigits, Annales Mathematicae et Informaticae, pp. 137-145.
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.
Omar E. Pol, Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567.
B. Srinivasa Rao, Heptagonal Numbers in the Pell Sequence and Diophantine Equations 2x^2 = y^2(5y - 3)^2 ± 2, Fib. Quarterly, 43 (2005), 194-201.
B. Srinivasa Rao, Heptagonal numbers in the associated Pell sequence and Diophantine equations x^2(5x - 3)^2 = 8y^2 ± 4, Fib. Quarterly, 43 (2005), 302-306.
Leo Tavares, Illustration.
Eric Weisstein's World of Mathematics, Heptagonal Number.
Index to sequences related to polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014637, A014640, A014773, A014792, A069099, A131413, A131896, A134483, A000384.
a(n)= A093562(n+1, 2), (5, 1)-Pascal column.
Cf. A006564, A147875, A244639.
Cf. sequences listed in A254963.

a000566 n = n * (5 * (n - 1) + 2) `div` 2
a000566_list = scanl (+) 0 a016861_list  -- Reinhard Zumkeller, Jun 16 2013

a000566:=func< n | n*(5*n-3) div 2 >; [ a000566(n): n in [0..50] ];

A000566 := proc(n)
        n*(5*n-3)/2 ;
end proc:
seq(A000566(n),n=0..30); # R. J. Mathar, Oct 02 2020

Table[n (5n - 3)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 1, 7}, 50] (* Harvey P. Dale, Oct 13 2011 *)
Join[{0},Accumulate[Range[1,315,5]]] (* Harvey P. Dale, Mar 26 2016 *)
(* For Mathematica 10.4+ *) Table[PolygonalNumber[RegularPolygon[7], n], {n, 0, 48}] (* Arkadiusz Wesolowski, Aug 27 2016 *)
PolygonalNumber[7,Range[0,50]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 23 2021 *)

makelist(n*(5*n-3)/2, n, 0, 20); /* Martin Ettl, Dec 11 2012 */

PARI
```
a(n) = n * (5*n - 3) / 2
```

# Intended to compute the initial segment of the sequence, not isolated terms.
def aList():
     x, y = 1, 1
     yield 0
     while True:
         yield x
         x, y = x + y + 5, y + 5
A000566 = aList()
print([next(A000566) for i in range(49)]) # Peter Luschny, Aug 04 2019

[n*(5*n-3)//2 for n in range(50)] # Gennady Eremin, Mar 24 2022

G.f.: x*(1 + 4*x)/(1 - x)^3. Simon Plouffe in his 1992 dissertation.
a(n) = C(n, 1) + 5*C(n, 2). - Paul Barry, Jun 10 2003
a(n) = Sum_{k = 1..n} (4*n - 3*k). - Paul Barry, Sep 06 2005
a(n) = n + 5*A000217(n-1) - Floor van Lamoen, Oct 14 2005
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for a(0) = 0, a(1) = 1, a(2) = 7. -  Jaume Oliver Lafont, Dec 02 2008
a(n+1) = A153126(n) + n mod 2; a(2*n+1) = A033571(n); a(2*(n+1)) = A153127(n) + 1. - Reinhard Zumkeller, Dec 20 2008
40*a(n)+ 9 = A017354(n-1). - Ken Rosenbaum, Dec 02 2009.
a(n) = 2*a(n-1) - a(n-2) + 5, with a(0) = 0 and a(1) = 1. - Mohamed Bouhamida, May 05 2010
a(n) = A000217(n) + 4*A000217(n-1). - Vincenzo Librandi, Nov 20 2010
a(n) = a(n-1) + 5*n - 4, with a(0) = 0. - Vincenzo Librandi, Nov 20 2010
a(n) = A130520(5*n). - Philippe Deléham, Mar 26 2013
a(5*a(n) + 11*n + 1) = a(5*a(n) + 11*n) + a(5*n + 1). - Vladimir Shevelev, Jan 24 2014
Sum_{n>=1} 1/a(n) = sqrt(1 - 2/sqrt(5))*Pi/3 + 5*log(5)/6 - sqrt(5)*log((1 + sqrt(5))/2)/3 = 1.32277925312238885674944226131... . See A244639. - Vaclav Kotesovec, Apr 27 2016
E.g.f.: x*(2 + 5*x)*exp(x)/2. - Ilya Gutkovskiy, Aug 27 2016
From Charlie Marion, May 02 2017: (Start)
a(n+m) = a(n) + 5*n*m + a(m);
a(n-m) = a(n) - 5*n*m + a(m) + 3*m;
a(n) - a(m) = (5*(n + m) - 3)*(n - m)/2.
In general, let P(k,n) be the n-th k-gonal number. Then
P(k,n+m) = P(k,n) + (k - 2)*n*m + P(k,m);
P(k,n-m) = P(k,n) - (k - 2)*n*m + P(k,m) + (k - 4)*m;
P(k,n) - P(k,m) = ((k-2)*(n + m) + 4 - k)*(n - m)/2.
(End)
a(n) = A147875(-n) for all n in Z. - Michael Somos, Jan 25 2019
a(n) = A000217(n-1) + A000217(2*n-1). - Charlie Marion, Dec 19 2019
Product_{n>=2} (1 - 1/a(n)) = 5/7. - Amiram Eldar, Jan 21 2021
a(n) + a(n+1) = (2*n+1)^2 + n^2 - 2*n. In general, if we let P(k,n) = the n-th k-gonal number, then P(k^2-k+1,n)+ P(k^2-k+1,n+1) = ((k-1)*n+1)^2 + (k-2)*(n^2-2*n) = ((k-1)*n+1)^2 + (k-2)*A005563(n-2). When k = 2, this formula reduces to the well-known triangular number formula: T(n) + T(n+1) = (n+1)^2. - Charlie Marion, Jul 01 2021

Partially edited by Joerg Arndt, Mar 11 2010

A008596 Multiples of 14.

Original entry on oeis.org

0, 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 182, 196, 210, 224, 238, 252, 266, 280, 294, 308, 322, 336, 350, 364, 378, 392, 406, 420, 434, 448, 462, 476, 490, 504, 518, 532, 546, 560, 574, 588, 602, 616, 630, 644, 658, 672, 686, 700, 714, 728

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 326.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008589, A008594, A008595, A033571, A135628, A158186.

Range[0, 1000, 14] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
CoefficientList[Series[14 x / (x - 1)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Jun 10 2013 *)

a(n)=14*n \\ Charles R Greathouse IV, Sep 28 2015

a(n) = A033571(n) - A158186(n). - Reinhard Zumkeller, Mar 13 2009
From R. J. Mathar, Jun 23 2009: (Start)
a(n) = 14*n.
a(n) = 2*a(n-1) - a(n-2).
G.f.: 14*x/(x-1)^2. (End)
From Elmo R. Oliveira, Apr 08 2025: (Start)
E.g.f.: 14*x*exp(x).
a(n) = 2*A008589(n) = A135628(n)/2. (End)

A153126 Sums of rows of the triangle in A153125.

Original entry on oeis.org

1, 6, 18, 33, 55, 80, 112, 147, 189, 234, 286, 341, 403, 468, 540, 615, 697, 782, 874, 969, 1071, 1176, 1288, 1403, 1525, 1650, 1782, 1917, 2059, 2204, 2356, 2511, 2673, 2838, 3010, 3185, 3367, 3552, 3744, 3939, 4141, 4346, 4558, 4773, 4995, 5220, 5452

Offset: 0

Reinhard Zumkeller, Dec 20 2008

Sequence found by reading the line from 1, in the direction 1, 6,..., and the same line from 1, in the direction 1, 18,..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. Line perpendicular to the main axis A195015 in the same spiral. - Omar E. Pol, Oct 14 2011

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000566, A001622, A033571, A153125, A153127.

LinearRecurrence[{2,0,-2,1},{1,6,18,33},50] (* Harvey P. Dale, Apr 13 2014 *)

a(n)=n*(5*n+7)/2 + 1 - n%2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = n*(5*n+7)/2 + 1 - n mod 2.
a(n) = Sum_{k=1..n+1} A153125(n+1,k).
a(2*n) = A033571(n); a(2*n+1) = A153127(n).
a(n) = A000566(n+1) - n mod 2.
From Colin Barker, Jul 07 2012: (Start)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: (1+4*x+6*x^2-x^3)/((1-x)^3*(1+x)). (End)
Sum_{n>=0} 1/a(n) = 5/7 + 2*sqrt(1+2/sqrt(5))*Pi/21 + 2*sqrt(5)*log(phi)/21 + 5*log(5)/21 - 8*log(2)/21, where phi is the golden ratio (A001622). - Amiram Eldar, Aug 23 2022

A195015 Main axis of the square spiral whose edges have length A195013 and whose vertices are the numbers A195014.

Original entry on oeis.org

0, 2, 12, 24, 44, 66, 96, 128, 168, 210, 260, 312, 372, 434, 504, 576, 656, 738, 828, 920, 1020, 1122, 1232, 1344, 1464, 1586, 1716, 1848, 1988, 2130, 2280, 2432, 2592, 2754, 2924, 3096, 3276, 3458, 3648, 3840, 4040, 4242, 4452, 4664, 4884

Offset: 0

Omar E. Pol, Sep 26 2011

Sequence found by reading the line from 0, in the direction 0, 2, ..., and the same line from 0, in the direction 0, 12, ..., in the square spiral mentioned above. Axis perpendicular to A195016 in the same spiral.

Also four times A005475 and positives A152965 interleaved.

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

Cf. A000566, A005475, A005476, A028895, A033571, A152965, A153126, A195013, A195014, A195016.

[(2*n*(5*n+2)+3*(-1)^n-3)/4: n in [0..50]]; // Vincenzo Librandi, Oct 28 2011

LinearRecurrence[{2, 0, -2, 1}, {0, 2, 12, 24}, 50] (* Paolo Xausa, Feb 09 2024 *)

From Bruno Berselli, Oct 14 2011: (Start)
G.f.: 2*x*(1+4*x)/((1+x)*(1-x)^3).
a(n) = (2*n*(5*n+2) + 3*(-1)^n-3)/4.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(n) + a(n-1) = A135706(n). (End)

A010010 a(0) = 1, a(n) = 20*n^2 + 2 for n>0.

Original entry on oeis.org

1, 22, 82, 182, 322, 502, 722, 982, 1282, 1622, 2002, 2422, 2882, 3382, 3922, 4502, 5122, 5782, 6482, 7222, 8002, 8822, 9682, 10582, 11522, 12502, 13522, 14582, 15682, 16822, 18002, 19222, 20482, 21782, 23122, 24502, 25922, 27382, 28882, 30422, 32002, 33622

Offset: 0

N. J. A. Sloane

Bruno Berselli, Table of n, a(n) for n = 0..1000
Volodymyr Mazorchuk and Xiaoyu Zhu, Combinatorics of infinite rank module categories over finite dimensional sl_3-modules in Lie-algebraic context, arXiv:2501.00291 [math.RT], 2024. See p. 2.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A206399.

[1] cat [20*n^2 + 2: n in [1..50]]; // Vincenzo Librandi, Aug 03 2015

Join[{1}, 20 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {22, 82, 182}, 50]] (* Vincenzo Librandi, Aug 03 2015 *)

a(n) = A033571(n)+A158186(n) = A158187(n)*2 for n>0. - Reinhard Zumkeller, Mar 13 2009
G.f.: (1+x)*(1+18*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*20+2)*e^x-1. - Gopinath A. R., Feb 14 2012
Sum_{n>=0} 1/a(n) = 3/4+sqrt(10)/40*Pi*coth(Pi/sqrt(10)) = 1.0772981051444036327... - R. J. Mathar, May 07 2024
a(n) = A069133(n)+A069133(n+1). - R. J. Mathar, May 07 2024

A153127 a(n) = (2n + 1)(5*n + 6).

Original entry on oeis.org

6, 33, 80, 147, 234, 341, 468, 615, 782, 969, 1176, 1403, 1650, 1917, 2204, 2511, 2838, 3185, 3552, 3939, 4346, 4773, 5220, 5687, 6174, 6681, 7208, 7755, 8322, 8909, 9516, 10143, 10790, 11457, 12144, 12851, 13578, 14325, 15092, 15879, 16686, 17513

Offset: 0

Reinhard Zumkeller, Dec 20 2008

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

Cf. A000566, A001622, A005408, A016861, A019952, A033571, A153126.

Table[(2n+1)(5n+6),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{6,33,80},50] (* Harvey P. Dale, Jun 07 2021 *)

a(n)=(2*n+1)*(5*n+6) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A153126(2*n+1) = A000566(2*(n+1)) - 1.
a(n) = a(n-1) + 20*n + 7 (with a(0)=6). - Vincenzo Librandi, Dec 27 2010
G.f.: (-6-15*x+x^2)/(-1+x)^3 - Harvey P. Dale, Jun 07 2021
Sum_{n>=0} 1/a(n) = 5/7 - sqrt(1+2/sqrt(5))*Pi/14 - sqrt(5)*log(phi)/14 - 5*log(5)/28 + 2*log(2)/7, where phi is the golden ratio (A001622). - Amiram Eldar, Aug 23 2022
From Elmo R. Oliveira, Oct 27 2024: (Start)
E.g.f.: exp(x)*(6 + 27*x + 10*x^2).
a(n) = A005408(n)*A016861(n+1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A158186 a(n) = 10n^2 - 7n + 1.

Original entry on oeis.org

1, 4, 27, 70, 133, 216, 319, 442, 585, 748, 931, 1134, 1357, 1600, 1863, 2146, 2449, 2772, 3115, 3478, 3861, 4264, 4687, 5130, 5593, 6076, 6579, 7102, 7645, 8208, 8791, 9394, 10017, 10660, 11323, 12006, 12709, 13432, 14175, 14938, 15721, 16524, 17347

Offset: 0

Reinhard Zumkeller, Mar 13 2009

Sequence found by reading the segment (1, 4) together with the line (one of the diagonal axes) from 4, in the direction 4, 27, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Sep 10 2011

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

Cf. A001622, A008596, A010010, A033571, A085787.

Table[10n^2-7n+1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,4,27},50] (* Harvey P. Dale, Apr 06 2020 *)

a(n)=10*n^2-7*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (2*n-1)*(5*n-1).
a(n) = A033571(n) - A008596(n) = A010010(n) - A033571(n).
G.f.: (1+x+18*x^2)/(1-x)^3. - Jaume Oliver Lafont, Mar 27 2009
a(n) = a(n-1) + 20*n - 17 (with a(0)=1). - Vincenzo Librandi, Dec 03 2010
Sum_{n>=0} 1/a(n) = 1 + (2*sqrt(1+2/sqrt(5))*Pi - 2*sqrt(5)*log(phi) - 5*log(5) + 8*log(2))/12, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 22 2022

Typo in definition corrected by Reinhard Zumkeller, Dec 03 2009

A195016 a(n) = (n(5n+7)-(-1)^n+1)/2.

Original entry on oeis.org

0, 7, 17, 34, 54, 81, 111, 148, 188, 235, 285, 342, 402, 469, 539, 616, 696, 783, 873, 970, 1070, 1177, 1287, 1404, 1524, 1651, 1781, 1918, 2058, 2205, 2355, 2512, 2672, 2839, 3009, 3186, 3366, 3553, 3743, 3940, 4140, 4347, 4557, 4774, 4994

Offset: 0

Omar E. Pol, Sep 26 2011

Sequence found by reading the line from 0, in the direction 0, 7,..., and the same line from 0, in the direction 0, 17,..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. Axis perpendicular to the main axis A195015 in the same spiral.

Also sequence found by reading the line from 0, in the direction 0, 7,..., and the same line from 0, in the direction 0, 17,..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. This line is parallel to A153126 in the same spiral.

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

Cf. A000566, A005476, A028895, A033571, A085787, A152965, A153126, A195013, A195014, A195015.

&cat[[n*t,(n+1)*t] where t is 10*n+7: n in [0..22]]; // Bruno Berselli, Oct 14 2011

LinearRecurrence[{2, 0, -2, 1}, {0, 7, 17, 34}, 50] (* Paolo Xausa, Feb 09 2024 *)

n*(10*n-3), if n >= 1, and (2*n+1)*(5*n+1)-1, if n >= 0, interleaved.

G.f.: x*(7+3*x)/((1+x)*(1-x)^3). - Bruno Berselli, Oct 14 2011

Concise definition by Bruno Berselli, Oct 14 2011

A195018 a(n) = n(10n-3).

Original entry on oeis.org

0, 7, 34, 81, 148, 235, 342, 469, 616, 783, 970, 1177, 1404, 1651, 1918, 2205, 2512, 2839, 3186, 3553, 3940, 4347, 4774, 5221, 5688, 6175, 6682, 7209, 7756, 8323, 8910, 9517, 10144, 10791, 11458, 12145, 12852, 13579, 14326, 15093, 15880, 16687, 17514, 18361, 19228

Offset: 0

Omar E. Pol, Oct 14 2011

Bisection of heptagonal numbers A000566.
Sequence found by reading the line from 0, in the direction 0, 7, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. This is one of the four semi-diagonals of the square spiral.
Also sequence found by reading the line from 0, in the direction 0, 7, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. Semi-axis perpendicular to the main axis A195015 in the same spiral.

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

Also bisection of A195016.

Cf. A000566, A033571, A085787, A195013, A195014, A195015, A153127.

[n*(10*n-3): n in [0..50]]; // Vincenzo Librandi, Oct 28 2011

Table[n (10 n-3),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,7,34},50] (* Harvey P. Dale, May 27 2012 *)

a(n)=n*(10*n-3) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A153127(n-1) + 1, if n >= 1.
G.f.: -x*(7+13*x)/(x-1)^3. - R. J. Mathar, Oct 15 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=7, a(2)=34. - Harvey P. Dale, May 27 2012
From Elmo R. Oliveira, Dec 15 2024: (Start)
E.g.f.: exp(x)*x*(7 + 10*x).
a(n) = A000566(2*n). (End)

cp's OEIS Frontend

A000566 Heptagonal numbers (or 7-gonal numbers): n*(5*n-3)/2.

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A008596 Multiples of 14.

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A153126 Sums of rows of the triangle in A153125.

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A195015 Main axis of the square spiral whose edges have length A195013 and whose vertices are the numbers A195014.

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A010010 a(0) = 1, a(n) = 20*n^2 + 2 for n>0.

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A153127 a(n) = (2*n + 1)*(5*n + 6).

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A158186 a(n) = 10*n^2 - 7*n + 1.

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A000566 Heptagonal numbers (or 7-gonal numbers): n(5n-3)/2.

A153127 a(n) = (2n + 1)(5*n + 6).

A158186 a(n) = 10n^2 - 7n + 1.

A195016 a(n) = (n(5n+7)-(-1)^n+1)/2.

A195018 a(n) = n(10n-3).