A195014 -id:A195014 - 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

A028895 5 times triangular numbers: a(n) = 5n(n+1)/2.

Original entry on oeis.org

0, 5, 15, 30, 50, 75, 105, 140, 180, 225, 275, 330, 390, 455, 525, 600, 680, 765, 855, 950, 1050, 1155, 1265, 1380, 1500, 1625, 1755, 1890, 2030, 2175, 2325, 2480, 2640, 2805, 2975, 3150, 3330, 3515, 3705, 3900, 4100, 4305, 4515, 4730, 4950, 5175, 5405, 5640

Offset: 0

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

Sequence found by reading the line from 0, in the direction 0, 5, ... and the same line from 0, in the direction 0, 15, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. Axis perpendicular to A195142 in the same spiral. - Omar E. Pol, Sep 18 2011
Bisection of A195014. Sequence found by reading the line from 0, in the direction 0, 5, ..., and the same line from 0, in the direction 0, 15, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. This is the main diagonal of the spiral. - Omar E. Pol, Sep 25 2011
a(n) = the Wiener index of the graph obtained by applying Mycielski's construction to the complete graph K(n) (n>=2). - Emeric Deutsch, Aug 29 2013
Sum of the numbers from 2*n to 3*n for n=0,1,2,... - Wesley Ivan Hurt, Nov 27 2015
Numbers k such that the concatenation k625 is a square, where also 625 is a square. - Bruno Berselli, Nov 07 2018
From Paul Curtz, Nov 29 2019: (Start)
Main column of the pentagonal spiral for n (A001477):
                         50
                      49 30 31
                   48 29 15 16 32
                47 28 14  5  6 17 33
             46 27 13  4  0  1  7 18 34
               45 26 12  3  2  8 19 35
                 44 25 11 10  9 20 36
                   43 24 23 22 21 37
                    42 41 40 39 38
(End)

D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 205.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Rangaswami Balakrishnan and S. Francis Raj, The Wiener number of powers of the Mycielskian, Discussiones Math. Graph Theory, Vol. 30, No. 3 (2010), pp. 489-498 (see Theorem 2.1).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A001068, A005891, A028896, A046092, A055998, A085787, A130520, A195013, A195014, A195142.
Cf. index to numbers of the form n*(d*n+10-d)/2 in A140090.
Cf. A000566, A005475, A005476, A033583, A085787, A147875, A192136, A326725 (all in the spiral).

[5*n*(n+1)/2 : n in [0..50]]; // Wesley Ivan Hurt, Jun 09 2014

[seq(5*binomial(n,2), n=1..45)]; # Zerinvary Lajos, Nov 24 2006

Table[Sum[i + 2*n - 1, {i, 2, n}], {n, 45}] (* Zerinvary Lajos, Jul 11 2009 *)
Table[5 n (n + 1)/2, {n, 0, 50}] (* Bruno Berselli, Sep 23 2016 *)
5 Accumulate[Range[0,60]] (* Harvey P. Dale, Jun 17 2025 *)

a(n)=5*n*(n+1)/2 \\ Charles R Greathouse IV, Sep 24 2015

def A028895(n): return 5*n*(n+1)>>1 # Chai Wah Wu, Aug 07 2025

G.f.: 5*x/(1-x)^3.
a(n) = 5*n*(n+1)/2 = 5*A000217(n).
a(n+1) = 5*n+a(n). - Vincenzo Librandi, Aug 05 2010
a(n) = A005891(n) - 1. - Omar E. Pol, Oct 03 2011
a(n) = A130520(5n+4). - Philippe Deléham, Mar 26 2013
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. - Wesley Ivan Hurt, Nov 27 2015
a(n) = Sum_{i=0..n} A001068(4i). - Wesley Ivan Hurt, May 06 2016
E.g.f.: 5*x*(2 + x)*exp(x)/2. - Ilya Gutkovskiy, May 06 2016
a(n) = A055998(3*n) - A055998(2*n). - Bruno Berselli, Sep 23 2016
From Amiram Eldar, Feb 26 2022: (Start)
Sum_{n>=1} 1/a(n) = 2/5.
Sum_{n>=1} (-1)^(n+1)/a(n) = (2/5)*(2*log(2) - 1). (End)
Product_{n>=1} (1 - 1/a(n)) = -(5/(2*Pi))*cos(sqrt(13/5)*Pi/2). - Amiram Eldar, Feb 21 2023

A005475 a(n) = n(5n+1)/2.

Original entry on oeis.org

0, 3, 11, 24, 42, 65, 93, 126, 164, 207, 255, 308, 366, 429, 497, 570, 648, 731, 819, 912, 1010, 1113, 1221, 1334, 1452, 1575, 1703, 1836, 1974, 2117, 2265, 2418, 2576, 2739, 2907, 3080, 3258, 3441, 3629

Offset: 0

N. J. A. Sloane

Sequence found by reading the line from 0, in the direction 0, 11, ..., and the line from 3, in the direction 3, 24, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. - Omar E. Pol, Sep 26 2011

For n >= 3, a(n) is the sum of the numbers appearing in the 3rd 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

G. C. Greubel, Table of n, a(n) for n = 0..5000
A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory, arXiv:quant-ph/0409152, 2004.
Leo Tavares, Illustration: Hexagonal Trapeziums.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001622, A110449, A130520, A162147, A202803.

Cf. similar sequences listed in A022289.

seq(binomial(5*n+1,2)/5, n=0..34); # Zerinvary Lajos, Jan 21 2007
a:=n->sum(2*n+j, j=1..n): seq(a(n), n=0..38); # Zerinvary Lajos, Apr 29 2007

Table[n (5 n + 1)/2, {n, 0, 40}] (* Bruno Berselli, Oct 13 2016 *)

a(n)=n*(5*n+1)/2; \\ Joerg Arndt, Mar 27 2013

a(n) = A110449(n, 2) for n>1.
a(n) = a(n-1) + 5*n - 2 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010
a(n) = A130520(5*n+2). - Philippe Deléham, Mar 26 2013
a(n) = A202803(n)/2. - Philippe Deléham, Mar 27 2013
a(n) = A162147(n) - A162147(n-1). - J. M. Bergot, Jun 21 2013
a(n) = A000217(3*n) - A000217(2*n). - Bruno Berselli, Oct 13 2016
From G. C. Greubel, Aug 23 2017: (Start)
G.f.: x*(2*x + 3)/(1-x)^3.
E.g.f.: (x/2)*(5*x+6)*exp(x). (End)
Sum_{n>=1} 1/a(n) = 10+2*gamma+2*Psi(1/5) = 0.57635... see A001620 and A200135. - R. J. Mathar, May 30 2022
Sum_{n>=1} 1/a(n) = 10 - sqrt(1+2/sqrt(5))*Pi - sqrt(5)*log(phi) - 5*log(5)/2, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 10 2022

Incorrect comment deleted and minor errors corrected by Johannes W. Meijer, Feb 04 2010

A195013 Multiples of 2 and of 3 interleaved: a(2n-1) = 2n, a(2n) = 3n.

Original entry on oeis.org

2, 3, 4, 6, 6, 9, 8, 12, 10, 15, 12, 18, 14, 21, 16, 24, 18, 27, 20, 30, 22, 33, 24, 36, 26, 39, 28, 42, 30, 45, 32, 48, 34, 51, 36, 54, 38, 57, 40, 60, 42, 63, 44, 66, 46, 69, 48, 72, 50, 75, 52, 78, 54, 81, 56, 84, 58, 87, 60, 90, 62, 93, 64, 96, 66, 99, 68, 102

Offset: 1

Omar E. Pol, Sep 09 2011

First differences of A195014.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. H. Bailey, J. M. Borwein, and J. S. Kimberley, Discovery of large Poisson polynomials using a new arbitrary precision software package, Slides, 2015.
D. H. Bailey, J. M. Borwein, and J. S. Kimberley, Computer discovery and analysis of large Poisson polynomials, 2016.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A005843, A008585, A195014, A195019.

Cf. A111712 (partial sums of this sequence prepended with 1).

import Data.List (transpose)
a195013 n = a195013_list !! (n-1)
a195013_list = concat $ transpose [[2, 4 ..], [3, 6 ..]]
-- Reinhard Zumkeller, Apr 06 2015

&cat[[2*n,3*n]: n in [1..34]]; // Bruno Berselli, Sep 25 2011

With[{r = Range[50]}, Riffle[2*r, 3*r]] (* or *)
LinearRecurrence[{0, 2, 0, -1}, {2, 3, 4, 6}, 100] (* Paolo Xausa, Feb 09 2024 *)

a(n)=(5*n+(n-2)*(-1)^n+2)/4 \\ Charles R Greathouse IV, Sep 24 2015

Pair(2*n, 3*n).
From Bruno Berselli, Sep 26 2011: (Start)
G.f.: x*(2+3*x)/(1-x^2)^2.
a(n) = (5*n+(n-2)*(-1)^n+2)/4.
a(n) = 2*a(n-2) - a(n-4) = a(n-2) + A010693(n-1).
a(n)+a(-n) = A010673(n).
a(n)-a(-n) = A106832(n). (End)

A033571 a(n) = (2n + 1)(5*n + 1).

Original entry on oeis.org

1, 18, 55, 112, 189, 286, 403, 540, 697, 874, 1071, 1288, 1525, 1782, 2059, 2356, 2673, 3010, 3367, 3744, 4141, 4558, 4995, 5452, 5929, 6426, 6943, 7480, 8037, 8614, 9211, 9828, 10465, 11122, 11799, 12496, 13213, 13950, 14707, 15484, 16281, 17098, 17935, 18792, 19669, 20566, 21483

Offset: 0

N. J. A. Sloane

Sequence found by reading the line from 1, in the direction 1, 18, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. This is one of the diagonals in the spiral. - Omar E. Pol, Sep 10 2011

Also sequence found by reading 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. This is a line perpendicular to the main axis A195015 in the same spiral. - Omar E. Pol, Oct 14 2011

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

Cf. A153127. - Reinhard Zumkeller, Dec 20 2008
Cf. A000566, A008596, A010010, A153126, A158186.
Cf. A001622, A003154, A007742, A019952.

List([0..50], n-> (2*n+1)*(5*n+1)); # G. C. Greubel, Oct 12 2019

[(2*n+1)*(5*n+1): n in [0..50]]; // G. C. Greubel, Oct 12 2019

seq((2*n+1)*(5*n+1), n=0..50); # G. C. Greubel, Oct 12 2019

Table[(2*n+1)*(5*n+1), {n,0,50}] (* G. C. Greubel, Oct 12 2019 *)

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

[(2*n+1)*(5*n+1) for n in range(50)] # G. C. Greubel, Oct 12 2019

a(n) = A153126(2*n) = A000566(2*n+1). - Reinhard Zumkeller, Dec 20 2008
From Reinhard Zumkeller, Mar 13 2009: (Start)
a(n) = A008596(n) + A158186(n), for n > 0.
a(n) = A010010(n) - A158186(n). (End)
a(n) = a(n-1) + 20*n - 3 (with a(0)=1). - Vincenzo Librandi, Nov 17 2010
From G. C. Greubel, Oct 12 2019: (Start)
G.f.: (1 + 15*x + 4*x^2)/(1-x)^3.
E.g.f.: (1 + 17*x + 10*x^2)*exp(x). (End)
a(n) = A003154(n+1) + A007742(n). - Leo Tavares, Mar 27 2022
Sum_{n>=0} 1/a(n) = sqrt(1+2/sqrt(5))*Pi/6 + sqrt(5)*log(phi)/6 + 5*log(5)/12 - 2*log(2)/3, where phi is the golden ratio (A001622). - Amiram Eldar, Aug 23 2022

Terms a(36) onward added by G. C. Greubel, Oct 12 2019

A152745 5 times hexagonal numbers: 5n(2*n-1).

Original entry on oeis.org

0, 5, 30, 75, 140, 225, 330, 455, 600, 765, 950, 1155, 1380, 1625, 1890, 2175, 2480, 2805, 3150, 3515, 3900, 4305, 4730, 5175, 5640, 6125, 6630, 7155, 7700, 8265, 8850, 9455, 10080, 10725, 11390, 12075, 12780, 13505, 14250, 15015

Offset: 0

Omar E. Pol, Dec 12 2008

Sequence found by reading the line from 0, in the direction 0, 5, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Sep 18 2011

Also sequence found by reading the line from 0, in the direction 0, 5, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. This is one of the four semi-diagonals of the spiral. - Omar E. Pol, Oct 14 2011

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

Cf. A000384, A085250, A152746.

Bisection of A028895.

[5*n*(2*n-1): n in [0..50]]; // G. C. Greubel, Sep 01 2018

LinearRecurrence[{3,-3,1}, {0, 5, 30}, 50] (* or *) Table[5*n*(2*n-1), {n,0,50}] (* G. C. Greubel, Sep 01 2018 *)

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

a(n) = 10*n^2 - 5*n = A000384(n)*5.
a(n) = a(n-1) + 20*n-15 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
From G. C. Greubel, Sep 01 2018: (Start)
G.f.: 5*x*(1+ 3*x)/(1-x)^3.
E.g.f.: 5*x*(1+2*x)*exp(x). (End)
From Vaclav Kotesovec, Sep 02 2018: (Start)
Sum_{n>=1} 1/a(n) = 2*log(2)/5.
Sum_{n>=1} (-1)^n/a(n) = log(2)/5 - Pi/10. (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

A225622 A(n,k) is the total number of parts in the set of partitions of an n X k rectangle into integer-sided squares, considering only the list of parts; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 5, 3, 4, 9, 9, 4, 5, 15, 16, 15, 5, 6, 21, 31, 31, 21, 6, 7, 30, 47, 59, 47, 30, 7, 8, 38, 73, 102, 102, 73, 38, 8, 9, 50, 101, 170, 156, 170, 101, 50, 9, 10, 60, 142, 250, 307, 307, 250, 142, 60, 10, 11, 75, 185, 375, 460, 529, 460, 375, 185, 75, 11

Offset: 1

Christopher Hunt Gribble, Aug 04 2013

			The square array starts:
1    2    3    4    5    6    7    8    9   10   11   12 ...
2    5    9   15   21   30   38   50   60   75   87  105 ...
3    9   16   31   47   73  101  142  185  244  305  386 ...
4   15   31   59  102  170  250  375  523  726  962 ...
5   21   47  102  156  307  460  711 1040 1517 ...
6   30   73  170  307  529  907 1474 2204 ...
7   38  101  250  460  907 1351 2484 ...
8   50  142  375  711 1474 2484 ...
9   60  185  523 1040 2204 ...
...
A(3,2) = 9 because there are 9 parts overall in the 2 partitions of a 3 X 2 rectangle into squares with integer sides.  One partition comprises 6 1 X 1 squares and the other 2 1 X 1 squares and 1 2 X 2 square giving 9 parts in total.

Alois P. Heinz, Antidiagonals n = 1..23, flattened (Antidiagonals n = 1..14 from Christopher Hunt Gribble)
Christopher Hunt Gribble, C++ program

Diagonal = A226897.

Cf. A034295, A224697, A225542.

b:= proc(n, l) option remember; local i, k, s, t;
      if max(l[])>n then {} elif n=0 or l=[] then {0}
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; s:={};
         for i from k to nops(l) while l[i]=0 do s:=s union
             map(v->v+x^(1+i-k), b(n, [l[j]$j=1..k-1,
                 1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
         od; s
      fi
    end:
A:= (n, k)-> add(coeff(add(j, j=b(max(n, k),
            [0$min(n, k)])), x, i), i=1..n):
seq(seq(A(n, 1+d-n), n=1..d), d=1..15); # Alois P. Heinz, Aug 04 2013

$RecursionLimit = 1000; b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which[ Max[l]>n, {}, n == 0 || l == {}, {0}, Min[l]>0, t = Min[l]; b[n-t, l-t], True, For[k = 1, True, k++, If[l[[k]] == 0, Break[]]]; s = {}; For[i = k, i <= Length[l] && l[[i]] == 0, i++, s = s ~Union~ Map[Function[{v}, v + x^(1+i-k) ], b[n, Join[l[[1 ;; k-1]], Array[1+i-k&, i-k+1], l[[i+1 ;; -1]]]]]]; s]];A[n_, k_] := Sum[Coefficient[Sum[j, {j, b[Max[n, k], Array[0&, Min[n, k]]]}], x, i], {i, 1, n}]; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 15}] // Flatten (* Jean-François Alcover, Mar 06 2015, after Alois P. Heinz *)

A(n,1) = A000027(n) = n.
A(n,2) = A195014(n) = (n+1)(5n+3)/8 when n is odd
                      and 5n(n+2)/8 when n is even.

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

cp's OEIS Frontend

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

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

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A028895 5 times triangular numbers: a(n) = 5*n*(n+1)/2.

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A005475 a(n) = n*(5*n+1)/2.

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A195013 Multiples of 2 and of 3 interleaved: a(2n-1) = 2n, a(2n) = 3n.

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A033571 a(n) = (2*n + 1)*(5*n + 1).

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

A028895 5 times triangular numbers: a(n) = 5n(n+1)/2.

A005475 a(n) = n(5n+1)/2.

A033571 a(n) = (2n + 1)(5*n + 1).

A152745 5 times hexagonal numbers: 5n(2*n-1).

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