A005476 -id:A005476 - cp's OEIS frontend

A294831 Numerators of the partial sums of the reciprocals of the numbers (k + 1)(5k + 4) = 2*A005476(k+1), for k >= 0.

1, 11, 83, 410, 16799, 495151, 8516747, 55850623, 309309419, 1088610631, 6561497681, 777210281963, 12475578306953, 287734917200239, 10671842976127147, 844855135994501953, 846430303832665873, 75457260356268267017, 3551759427031132995079, 711302288219532928235, 712163917143684270659

Offset: 0

Wolfdieter Lang, Nov 18 2017

The corresponding denominators are given in A294832.
For the general case V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits see a comment in A294512. Here [m,r] = [5,4].
The limit of the series is V(5,4) = lim_{n -> oo} V(5,4;n) = ((5/2)*log(5) + (2*phi - 1)*(log(phi) - (Pi/5)*sqrt(3 + 4*phi)))/2, with the golden section phi:= (1 + sqrt(5))/2 = A001622. The value is 0.3877929018046... given in A294833.
In the Koecher reference v_5(4) = (1/5)*V(5,4) = 0.07755858036... given there by (1/4)*log(5) + (1/(2*sqrt(5)))*log((1 + sqrt(5))/2) - (Pi/10)*sqrt((5 + 2*sqrt(5))/5).

			The rationals V(5,4;n), n >= 0, begin:1/4, 11/36, 83/252, 410/1197, 16799/47880, 495151/1388520, 8516747/23604840, 55850623/153431460, 309309419/843873030, 1088610631/2953555605, 6561497681/17721333630, 777210281963/2091117368340, 12475578306953/33457877893440, ...
V(5,4;10^6) = 0.3877927018 (Maple 10 digits) to be compared with 0.3877929018 obtained from A294833 with 10 digits.

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189-193. For v_5(4) see p. 192.

G. C. Greubel, Table of n, a(n) for n = 0..600
Eric Weisstein's World of Mathematics, Digamma Function

Cf. A001622, 2*A005476, A294512, A294828/A294829 (V(5,3;n)), A294832, A294833.

[Numerator((&+[1/((k + 1)*(5*k + 4)): k in [0..n]])): n in [0..50]]; // G. C. Greubel, Aug 30 2018

Table[Numerator[Sum[1/((k + 1)*(5*k + 4)), {k, 0, n}]], {n, 0, 50}] (* G. C. Greubel, Aug 30 2018 *)

a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 4)))); \\ Michel Marcus, Nov 19 2017

a(n) = numerator(V(5,4;n)) with V(5,4;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 4)) = Sum_{k=0..n} 1/(2*A005476(k+1)) = Sum_{k=0..n} (1/(k + 4/5) - 1/(k+1)) = -Psi(4/5) + Psi(n+9/5) - (gamma + Psi(n+2)) with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.

A294833 Decimal expansion of the sum of the reciprocals of the numbers (k+1)(5k+4) = 2*A005476(k+1) for k >= 0.

Original entry on oeis.org

3, 8, 7, 7, 9, 2, 9, 0, 1, 8, 0, 4, 6, 0, 5, 5, 9, 8, 7, 8, 4, 7, 8, 5, 5, 4, 3, 0, 7, 4, 4, 3, 2, 9, 8, 8, 5, 9, 2, 0, 0, 1, 1, 5, 3, 7, 5, 5, 2, 9, 9, 2, 3, 0, 3, 0, 4, 0, 4, 3, 5, 5, 9, 3, 6, 0, 0, 9, 3, 4, 8, 0, 6, 0, 8, 0, 1, 9, 3, 3, 3, 3, 6, 9, 4, 2, 9, 5, 9, 5, 9, 4, 8, 1, 7, 5, 3, 0

Offset: 0

Wolfdieter Lang, Nov 18 2017

In the Koecher reference v_5(4) = (1/5)*(present value V(5,4)) = 0.07755858036092111..., given there as (1/4)*log(5) + (1/(2*sqrt(5)))*log((1 + sqrt(5))/2) - (Pi/10)*sqrt((5 + 2*sqrt(5))/5).

			0.387792901804605598784785543074432988592001153755299230304043559360093480608...

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189-193. For v_5(4) see p. 192.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Digamma Function.

Cf. A001620, A001622, 2*A005476, A294831/A294832.

SetDefaultRealField(RealField(100)); R:= RealField(); ((5/2)*Log(5) + Sqrt(5)*(Log((1+Sqrt(5))/2) - (Pi(R)/5)*Sqrt(5+2*Sqrt(5))))/2; // G. C. Greubel, Sep 05 2018

RealDigits[-PolyGamma[0, 4/5] + PolyGamma[0, 1], 10, 100][[1]] (* G. C. Greubel, Sep 05 2018 *)

default(realprecision, 100); phi=(1+sqrt(5))/2; ((5/2)*log(5) + (2*phi - 1)*(log(phi) - (Pi/5)*sqrt(3 + 4*phi)))/2 \\ G. C. Greubel, Sep 05 2018

Sum_{k>=0} 1/((5*n + 4)*(n + 1)) =: V(5,4) = ((5/2)*log(5) + (2*phi - 1)*(log(phi) - (Pi/5)*sqrt(3 + 4*phi)))/2 = -Psi(4/5) + Psi(1) with the golden section phi =(1 + sqrt(5))/2 = A001622 with the digamma function Psi and Psi(1) = -gamma = A001620.
The partial sums of this series are given in A294831/A294832.
Equals Sum_{k>=2} zeta(k)/5^(k-1). - Amiram Eldar, May 31 2021

A294832 Denominators of the partial sums of the reciprocals of the numbers (k + 1)(5k + 4) = 2*A005476(k+1), for k >= 0.

Original entry on oeis.org

4, 36, 252, 1197, 47880, 1388520, 23604840, 153431460, 843873030, 2953555605, 17721333630, 2091117368340, 33457877893440, 769531191549120, 28472654087317440, 2249339672898077760, 2249339672898077760, 200191230887928920640, 9408987851732659270080, 1881797570346531854016

Offset: 0

Wolfdieter Lang, Nov 18 2017

The corresponding numerators are given in A294831. Details are found there.

			For the rationals V(5,4;n) see A294831.

Cf. 2*A005476, A294831.

a(n) = denominator(sum(k=0, n, 1/((k + 1)*(5*k + 4)))); \\ Michel Marcus, Nov 19 2017

a(n) = denominator(V(5,4;n)) with V(5,4;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 4)) = Sum_{k=0..n} 1/(2*A005476(k+1)) = Sum_{k=0..n} (1/(k + 4/5) - 1/(k+1)).

For this sum in terms of the digamma function Psi see A294831.

A183296 Complement of A005476.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Clark Kimberling, Jan 03 2011

nonn

Cf. A005476.

  a=5/2; b=-1/2;
F[n_]:=a*n^2+b*n;
R[n_]:=(n/a+((b-1)/(2a))^2)^(1/2);
G[n_]:=n-1+Ceiling[R[n]-(b-1)/(2a)];
Table[F[n], {n,60}]
Table[G[n], {n,100}]

(See the Mathematica code.)

A000326 Pentagonal numbers: a(n) = n(3n-1)/2.

Original entry on oeis.org

0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151

Offset: 0

N. J. A. Sloane

The average of the first n (n > 0) pentagonal numbers is the n-th triangular number. - Mario Catalani (mario.catalani(AT)unito.it), Apr 10 2003
a(n) is the sum of n integers starting from n, i.e., 1, 2 + 3, 3 + 4 + 5, 4 + 5 + 6 + 7, etc. - Jon Perry, Jan 15 2004
Partial sums of 1, 4, 7, 10, 13, 16, ... (1 mod 3), a(2k) = k(6k-1), a(2k-1) = (2k-1)(3k-2). - Jon Perry, Sep 10 2004
Starting with offset 1 = binomial transform of [1, 4, 3, 0, 0, 0, ...]. Also, A004736 * [1, 3, 3, 3, ...]. - Gary W. Adamson, Oct 25 2007
If Y is a 3-subset of an n-set X then, for n >= 4, a(n-3) is the number of 4-subsets of X having at least two elements in common with Y. - Milan Janjic, Nov 23 2007
Solutions to the duplication formula 2*a(n) = a(k) are given by the index pairs (n, k) = (5,7), (5577, 7887), (6435661, 9101399), etc. The indices are integer solutions to the pair of equations 2(6n-1)^2 = 1 + y^2, k = (1+y)/6, so these n can be generated from the subset of numbers [1+A001653(i)]/6, any i, where these are integers, confined to the cases where the associated k=[1+A002315(i)]/6 are also integers. - R. J. Mathar, Feb 01 2008
a(n) is a binomial coefficient C(n,4) (A000332) if and only if n is a generalized pentagonal number (A001318). Also see A145920. - Matthew Vandermast, Oct 28 2008
Even octagonal numbers divided by 8. - Omar E. Pol, Aug 18 2011
Sequence found by reading the line from 0, in the direction 0, 5, ... and the line from 1, in the direction 1, 12, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011
The hyper-Wiener index of the star-tree with n edges (see A196060, example). - Emeric Deutsch, Sep 30 2011
More generally the n-th k-gonal number is equal to n + (k-2)*A000217(n-1), n >= 1, k >= 3. In this case k = 5. - Omar E. Pol, Apr 06 2013
Note that both Euler's pentagonal theorem for the partition numbers and Euler's pentagonal theorem for the sum of divisors refer more exactly to the generalized pentagonal numbers, not this sequence. For more information see A001318, A175003, A238442. - Omar E. Pol, Mar 01 2014
The Fuss-Catalan numbers are Cat(d,k)= [1/(k*(d-1)+1)]*binomial(k*d,k) and enumerate the number of (d+1)-gon partitions of a (k*(d-1)+2)-gon (cf. Schuetz and Whieldon link). a(n)= Cat(n,3), so enumerates the number of (n+1)-gon partitions of a (3*(n-1)+2)-gon. Analogous sequences are A100157 (k=4) and A234043 (k=5). - Tom Copeland, Oct 05 2014
Binomial transform of (0, 1, 3, 0, 0, 0, ...) (A169585 with offset 1) and second partial sum of (0, 1, 3, 3, 3, ...). - Gary W. Adamson, Oct 05 2015
For n > 0, a(n) is the number of compositions of n+8 into n parts avoiding parts 2 and 3. - Milan Janjic, Jan 07 2016
a(n) is also the number of edges in the Mycielskian of the complete graph K[n]. Indeed, K[n] has n vertices and n(n-1)/2 edges. Then its Mycielskian has n + 3n(n-1)/2 = n(3n-1)/2. See p. 205 of the West reference. - Emeric Deutsch, Nov 04 2016
Sum of the numbers from n to 2n-1. - Wesley Ivan Hurt, Dec 03 2016
Also the number of maximal cliques in the n-Andrásfai graph. - Eric W. Weisstein, Dec 01 2017
Coefficients in the hypergeometric series identity 1 - 5*(x - 1)/(2*x + 1) + 12*(x - 1)*(x - 2)/((2*x + 1)*(2*x + 2)) - 22*(x - 1)*(x - 2)*(x - 3)/((2*x + 1)*(2*x + 2)*(2*x + 3)) + ... = 0, valid for Re(x) > 1. Cf. A002412 and A002418. Column 2 of A103450. - Peter Bala, Mar 14 2019
A generalization of the Comment dated Apr 10 2003 follows. (k-3)*A000292(n-2) plus the average of the first n (2k-1)-gonal numbers is the n-th k-gonal number. - Charlie Marion, Nov 01 2020
a(n+1) is the number of Dyck paths of size (3,3n+1); i.e., the number of NE lattice paths from (0,0) to (3,3n+1) which stay above the line connecting these points. - Harry Richman, Jul 13 2021
a(n) is the largest sum of n positive integers x_1, ..., x_n such that x_i | x_(i+1)+1 for each 1 <= i <= n, where x_(n+1) = x_1. - Yifan Xie, Feb 21 2025

			Illustration of initial terms:
.
.                                       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
.
.  1    5     12       22           35
- _Philippe Deléham_, Mar 30 2013

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pages 2 and 311.
Raymond Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; p. 129.
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 pp. 38, 40.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
L. 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. 1.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.6 Figurate Numbers, p. 291.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 284.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 64.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 52-53, 129-130, 132.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 7-10.
André Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhäuser, Boston, 1984; see p. 186.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 98-100.
Douglas B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001.

Daniel Mondot, Table of n, a(n) for n = 0..10000 (first 1000 terms by T. D. Noe)
George E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
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.
Jordan Bell, Euler and the pentagonal number theorem, arXiv:math/0510054 [math.HO], 2005-2006.
Anicius Manlius Severinus Boethius, De institutione arithmetica libri duo, Book 2, sections 13-14.
Charles K. Cook and Michael R. Bacon, Some polygonal number summation formulas, Fib. Q., 52 (2014), 336-343.
Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 33.
Stephan Eberhart, Letter to N. J. A. Sloane, Jan 06 1978, also scanned copy of Mathematical-Physical Correspondence, No. 22, Christmas 1977.
Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 1
Leonhard Euler, Observatio de summis divisorum p. 8.
Leonhard Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004, 2009. See p. 8.
Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Shyam Sunder Gupta, Beauty of Number 153, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 15, 399-410.
Rodney T. Hansen, Arithmetic of pentagonal numbers, Fib. Quart., 8 (1970), 83-87.
Alfred Hoehn, Illustration of initial terms of A000326, A005449, A045943, A115067
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 339
Milan Janjic and Boris Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
Jangwon Ju and Daejun Kim, The pentagonal theorem of sixty-three and generalizations of Cauchy's lemma, arXiv:2010.16123 [math.NT], 2020.
Bir Kafle, Florian Luca and Alain Togbé, Pentagonal and heptagonal repdigits, Annales Mathematicae et Informaticae. pp. 137-145.
Sameen Ahmed Khan, Sums of the powers of reciprocals of polygonal numbers, Int'l J. of Appl. Math. (2020) Vol. 33, No. 2, 265-282.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
R. P. Loh, A. G. Shannon and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
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
Cliff Reiter, Polygonal Numbers and Fifty One Stars, Lafayette College, Easton, PA (2019).
Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.
W. Sierpiński, Sur les nombres pentagonaux, Bull. Soc. Royale Sciences Liège, 33 (No. 9-10, 1964), 513-517. [Annotated scanned copy]
N. J. A. Sloane, Illustration of initial terms of A000217, A000290, A000326.
Michel Waldschmidt, Continued fractions, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18 - 29 mai 2015: Oujda (Maroc).
Eric Weisstein's World of Mathematics, Andrásfai Graph.
Eric Weisstein's World of Mathematics, Maximal Clique.
Eric Weisstein's World of Mathematics, Pentagonal Number.
Wikipedia, Mycielskian.
Wikipedia, Pentagonal number
Index entries for "core" sequences
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for two-way infinite sequences

The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.
Cf. A001318 (generalized pentagonal numbers), A049452, A033570, A010815, A034856, A051340, A004736, A033568, A049453, A002411 (partial sums), A033579.
See A220083 for a list of numbers of the form n*P(s,n)-(n-1)*P(s,n-1), where P(s,n) is the n-th polygonal number with s sides.
Cf. A240137: sum of n consecutive cubes starting from n^3.
Cf. similar sequences listed in A022288.
Partial sums of A016777.

List([0..50],n->n*(3*n-1)/2); # Muniru A Asiru, Mar 18 2019

a000326 n = n * (3 * n - 1) `div` 2  -- Reinhard Zumkeller, Jul 07 2012

[n*(3*n-1)/2 : n in [0..100]]; // Wesley Ivan Hurt, Oct 15 2015

A000326 := n->n*(3*n-1)/2: seq(A000326(n), n=0..100);
A000326:=-(1+2*z)/(z-1)**3; # Simon Plouffe in his 1992 dissertation
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]-a[n-2]+3 od: seq(a[n], n=0..50); # Miklos Kristof, Zerinvary Lajos, Feb 18 2008

Table[n (3 n - 1)/2, {n, 0, 60}] (* Stefan Steinerberger, Apr 01 2006 *)
Array[# (3 # - 1)/2 &, 47, 0] (* Zerinvary Lajos, Jul 10 2009 *)
LinearRecurrence[{3, -3, 1}, {0, 1, 5}, 61] (* Harvey P. Dale, Dec 27 2011 *)
pentQ[n_] := IntegerQ[(1 + Sqrt[24 n + 1])/6]; pentQ[0] = True; Select[Range[0, 3200], pentQ@# &] (* Robert G. Wilson v, Mar 31 2014 *)
Join[{0}, Accumulate[Range[1, 312, 3]]] (* Harvey P. Dale, Mar 26 2016 *)
(* For Mathematica 10.4+ *) Table[PolygonalNumber[RegularPolygon[5], n], {n, 0, 46}] (* Arkadiusz Wesolowski, Aug 27 2016 *)
CoefficientList[Series[x (-1 - 2 x)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)
PolygonalNumber[5, Range[0, 20]] (* Eric W. Weisstein, Dec 01 2017 *)

PARI
```
a(n)=n*(3*n-1)/2
```

vector(100, n, n--; binomial(3*n, 2)/3) \\ Altug Alkan, Oct 06 2015

is_a000326(n) = my(s); n==0 || (issquare (24*n+1, &s) && s%6==5); \\ Hugo Pfoertner, Aug 03 2023

# 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 + 3, y + 3
A000326 = aList()
print([next(A000326) for i in range(47)]) # Peter Luschny, Aug 04 2019

Product_{m > 0} (1 - q^m) = Sum_{k} (-1)^k*x^a(k). - Paul Barry, Jul 20 2003
G.f.: x*(1+2*x)/(1-x)^3.
E.g.f.: exp(x)*(x+3*x^2/2).
a(n) = n*(3*n-1)/2.
a(-n) = A005449(n).
a(n) = binomial(3*n, 2)/3. - Paul Barry, Jul 20 2003
a(n) = A000290(n) + A000217(n-1). - Lekraj Beedassy, Jun 07 2004
a(0) = 0, a(1) = 1; for n >= 2, a(n) = 2*a(n-1) - a(n-2) + 3. - Miklos Kristof, Mar 09 2005
a(n) = Sum_{k=1..n} (2*n - k). - Paul Barry, Aug 19 2005
a(n) = 3*A000217(n) - 2*n. - Lekraj Beedassy, Sep 26 2006
a(n) = A126890(n, n-1) for n > 0. - Reinhard Zumkeller, Dec 30 2006
a(n) = A049452(n) - A022266(n) = A033991(n) - A005476(n). - Zerinvary Lajos, Jun 12 2007
Equals A034856(n) + (n - 1)^2. Also equals A051340 * [1,2,3,...]. - Gary W. Adamson, Jul 27 2007
a(n) = binomial(n+1, 2) + 2*binomial(n, 2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0) = 0, a(1) = 1, a(2) = 5. - Jaume Oliver Lafont, Dec 02 2008
a(n) = a(n-1) + 3*n-2 with n > 0, a(0)=0. - Vincenzo Librandi, Nov 20 2010
a(n) = A000217(n) + 2*A000217(n-1). - Vincenzo Librandi, Nov 20 2010
a(n) = A014642(n)/8. - Omar E. Pol, Aug 18 2011
a(n) = A142150(n) + A191967(n). - Reinhard Zumkeller, Jul 07 2012
a(n) = (A000290(n) + A000384(n))/2 = (A000217(n) + A000566(n))/2 = A049450(n)/2. - Omar E. Pol, Jan 11 2013
a(n) = n*A000217(n) - (n-1)*A000217(n-1). - Bruno Berselli, Jan 18 2013
a(n) = A005449(n) - n. - Philippe Deléham, Mar 30 2013
From Oskar Wieland, Apr 10 2013: (Start)
a(n) = a(n+1) - A016777(n),
a(n) = a(n+2) - A016969(n),
a(n) = a(n+3) - A016777(n)*3 = a(n+3) - A017197(n),
a(n) = a(n+4) - A016969(n)*2 = a(n+4) - A017641(n),
a(n) = a(n+5) - A016777(n)*5,
a(n) = a(n+6) - A016969(n)*3,
a(n) = a(n+7) - A016777(n)*7,
a(n) = a(n+8) - A016969(n)*4,
a(n) = a(n+9) - A016777(n)*9. (End)
a(n) = A000217(2n-1) - A000217(n-1), for n > 0. - Ivan N. Ianakiev, Apr 17 2013
a(n) = A002411(n) - A002411(n-1). - J. M. Bergot, Jun 12 2013
Sum_{n>=1} a(n)/n! = 2.5*exp(1). - Richard R. Forberg, Jul 15 2013
a(n) = floor(n/(exp(2/(3*n)) - 1)), for n > 0. - Richard R. Forberg, Jul 27 2013
From Vladimir Shevelev, Jan 24 2014: (Start)
a(3*a(n) + 4*n + 1) = a(3*a(n) + 4*n) + a(3*n+1).
A generalization. Let {G_k(n)}_(n >= 0) be sequence of k-gonal numbers (k >= 3). Then the following identity holds: G_k((k-2)*G_k(n) + c(k-3)*n + 1) = G_k((k-2)*G_k(n) + c(k-3)*n) + G_k((k-2)*n + 1), where c = A000124. (End)
A242357(a(n)) = 1 for n > 0. - Reinhard Zumkeller, May 11 2014
Sum_{n>=1} 1/a(n)= (1/3)*(9*log(3) - sqrt(3)*Pi). - Enrique Pérez Herrero, Dec 02 2014. See the decimal expansion A244641.
a(n) = (A000292(6*n+k-1)-A000292(k))/(6*n-1)-A000217(3*n+k), for any k >= 0. - Manfred Arens, Apr 26 2015 [minor edits from Wolfdieter Lang, May 10 2015]
a(n) = A258708(3*n-1,1) for n > 0. - Reinhard Zumkeller, Jun 23 2015
a(n) = A007584(n) - A245301(n-1), for n > 0. - Manfred Arens, Jan 31 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(sqrt(3)*Pi - 6*log(2))/3 = 0.85501000622865446... - Ilya Gutkovskiy, Jul 28 2016
a(m+n) = a(m) + a(n) + 3*m*n. - Etienne Dupuis, Feb 16 2017
In general, let P(k,n) be the n-th k-gonal number. Then P(k,m+n) = P(k,m) + (k-2)mn + P(k,n). - Charlie Marion, Apr 16 2017
a(n) = A023855(2*n-1) - A023855(2*n-2). - Luc Rousseau, Feb 24 2018
a(n) = binomial(n,2) + n^2. - Pedro Caceres, Jul 28 2019
Product_{n>=2} (1 - 1/a(n)) = 3/5. - Amiram Eldar, Jan 21 2021
(n+1)*(a(n^2) + a(n^2+1) + ... + a(n^2+n)) = n*(a(n^2+n+1) + ... + a(n^2+2n)). - Charlie Marion, Apr 28 2024
a(n) = Sum_{k = 0..3*n} (-1)^(n+k+1) * binomial(k, 2)*binomial(3*n+k-1, 2*k). - Peter Bala, Nov 04 2024

Incorrect example removed by Joerg Arndt, Mar 11 2010

A033991 a(n) = n(4n-1).

Original entry on oeis.org

0, 3, 14, 33, 60, 95, 138, 189, 248, 315, 390, 473, 564, 663, 770, 885, 1008, 1139, 1278, 1425, 1580, 1743, 1914, 2093, 2280, 2475, 2678, 2889, 3108, 3335, 3570, 3813, 4064, 4323, 4590, 4865, 5148, 5439, 5738, 6045, 6360, 6683, 7014, 7353, 7700, 8055, 8418

Offset: 0

N. J. A. Sloane

Write 0,1,2,... in a clockwise spiral; sequence gives numbers on negative x axis. (See illustration in Example.)
This sequence is the number of expressions x generated for a given modulus n in finite arithmetic. For example, n=1 (modulus 1) generates 3 expressions: 0+0=0(mod 1), 0-0=0(mod 1), 0*0=0(mod 1). By subtracting n from 4n^2, we eliminate the counting of those expressions that would include division by zero, which would be, of course, undefined. - David Quentin Dauthier, Nov 04 2007
From Emeric Deutsch, Sep 21 2010: (Start)
a(n) is also the Wiener index of the windmill graph D(3,n).
The windmill graph D(m,n) is the graph obtained by taking n copies of the complete graph K_m with a vertex in common (i.e., a bouquet of n pieces of K_m graphs). The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.
Example: a(2)=14; indeed if the triangles are OAB and OCD, then, denoting distance by d, we have d(O,A)=d(O,B)=d(A,B)=d(O,C)=d(O,D)=d(C,D)=1 and d(A,C)=d(A,D)=d(B,C)=d(B,D)=2. The Wiener index of D(m,n) is (1/2)n(m-1)[(m-1)(2n-1)+1]. For the Wiener indices of D(4,n), D(5,n), and D(6,n) see A152743, A028994, and A180577, respectively. (End)
Even hexagonal numbers divided by 2. - Omar E. Pol, Aug 18 2011
For n > 0, a(n) equals the number of length 3*n binary words having exactly two 0's with the n first bits having at most one 0. For example a(2) = 14. Words are 010111, 011011, 011101, 011110, 100111, 101011, 101101, 101110, 110011, 110101, 110110, 111001, 111010, 111100. - Franck Maminirina Ramaharo, Mar 09 2018
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [2n-1; {1, 2, 1, 4n-2}]. For n=1, this collapses to [1; {1, 2}]. - Magus K. Chu, Sep 06 2022

			Clockwise spiral (with sequence terms parenthesized) begins
   16--17--18--19
    |
   15   4---5---6
    |   |       |
  (14) (3) (0)  7
    |   |   |   |
   13   2---1   8
    |           |
   12--11--10---9

S. M. Ellerstein, The square spiral, J. Recreational Mathematics 29 (#3, 1998) 188; 30 (#4, 1999-2000), 246-250.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Emilio Apricena, A version of the Ulam spiral.
Eric W. Weisstein, Windmill Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Sequences from spirals: A001107, A002939, A007742, A033951-A033954, A033989-A033991, A002943, A033996, A033988.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A074378, A014848, A152743, A028994, A000326, A001105, A005476, A014635, A016742, A049452, A118729.

[seq(binomial(4*n, 2)/2, n=0..45)]; # Zerinvary Lajos, Jan 16 2007

Table[n*(4*n - 1), {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)
LinearRecurrence[{3,-3,1},{0,3,14},50] (* Harvey P. Dale, Oct 10 2011 *)

PARI
```
a(n)=4*n^2-n;
```

a(n) = A007742(-n) = A074378(2n-1) = A014848(2n).
G.f.: x*(3+5*x)/(1-x)^3. - Michael Somos, Mar 03 2003
a(n) = A014635(n)/2. - Zerinvary Lajos, Jan 16 2007
From Zerinvary Lajos, Jun 12 2007: (Start)
a(n) = A000326(n) + A005476(n).
a(n) = A049452(n) - A001105(n). (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. - Harvey P. Dale, Oct 10 2011
a(n) = A118729(8n+2). - Philippe Deléham, Mar 26 2013
From Ilya Gutkovskiy, Dec 04 2016: (Start)
E.g.f.: x*(3 + 4*x)*exp(x).
Sum_{n>=1} 1/a(n) = 3*log(2) - Pi/2 = 0.50864521488... (End)
a(n) = Sum_{i=n..3n-1} i. - Wesley Ivan Hurt, Dec 04 2016
From Franck Maminirina Ramaharo, Mar 09 2018: (Start)
a(n) = binomial(2*n, 2) + 2*n^2.
a(n) = A054556(n+1) - 1. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + log(3-2*sqrt(2)))/sqrt(2) - log(2). - Amiram Eldar, Mar 20 2022

Two remarks combined into one by Emeric Deutsch, Oct 03 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

A226488 a(n) = n(13n - 9)/2.

Original entry on oeis.org

0, 2, 17, 45, 86, 140, 207, 287, 380, 486, 605, 737, 882, 1040, 1211, 1395, 1592, 1802, 2025, 2261, 2510, 2772, 3047, 3335, 3636, 3950, 4277, 4617, 4970, 5336, 5715, 6107, 6512, 6930, 7361, 7805, 8262, 8732, 9215, 9711, 10220, 10742, 11277, 11825, 12386, 12960

Offset: 0

Bruno Berselli, Jun 09 2013

Sum of n-th octagonal number and n-th 9-gonal (nonagonal) number.

Sum of reciprocals of a(n), for n>0: 0.629618994194109711163742089971688...

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

Cf. A000567, A001106, A153080 (first differences).

Cf. numbers of the form n*(n*k-k+4)/2 listed in A005843 (k=0), A000096 (k=1), A002378 (k=2), A005449 (k=3), A001105 (k=4), A005476 (k=5), A049450 (k=6), A218471 (k=7), A002939 (k=8), A062708 (k=9), A135706 (k=10), A180223 (k=11), A139267 (n=12), this sequence (k=13), A139268 (k=14), A226489 (k=15), A139271 (k=16), A180232 (k=17), A152995 (k=18), A226490 (k=19), A152965 (k=20), A226491 (k=21), A152997 (k=22).

List([0..50], n-> n*(13*n-9)/2); # G. C. Greubel, Aug 30 2019

Magma
```
[n*(13*n-9)/2: n in [0..50]];
```

I:=[0,2,17]; [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

A226488:=n->n*(13*n - 9)/2; seq(A226488(n), n=0..50); # Wesley Ivan Hurt, Feb 25 2014

Table[n(13n-9)/2, {n, 0, 50}]
LinearRecurrence[{3, -3, 1}, {0, 2, 17}, 50] (* Harvey P. Dale, Jun 19 2013 *)
CoefficientList[Series[x(2+11x)/(1-x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)

a(n)=n*(13*n-9)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(13*n-9)/2 for n in (0..50)] # G. C. Greubel, Aug 30 2019

G.f.: x*(2+11*x)/(1-x)^3.
a(n) + a(-n) = A152742(n).
a(0)=0, a(1)=2, a(2)=17; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jun 19 2013
E.g.f.: x*(4 + 13*x)*exp(x)/2. - G. C. Greubel, Aug 30 2019
a(n) = A000567(n) + A001106(n). - Michel Marcus, Aug 31 2019

A130520 a(n) = Sum_{k=0..n} floor(k/5). (Partial sums of A002266.)

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 18, 21, 24, 27, 30, 34, 38, 42, 46, 50, 55, 60, 65, 70, 75, 81, 87, 93, 99, 105, 112, 119, 126, 133, 140, 148, 156, 164, 172, 180, 189, 198, 207, 216, 225, 235, 245, 255, 265, 275, 286, 297, 308, 319, 330, 342, 354, 366

Offset: 0

Hieronymus Fischer, Jun 01 2007

Complementary with A130483 regarding triangular numbers, in that A130483(n) + 5*a(n) = n*(n+1)/2 = A000217(n).

Given a sequence b(n) defined by variables b(0) to b(5) and recursion b(n) = -(b(n-6) * a(n-2) * (b(n-4) * b(n-2)^3 - b(n-3)^3 * b(n-1)) - b(n-5) * b(n-3) * b(n-1) * (b(n-5) * b(n-2)^2 - b(n-4)^2 * b(n-1)))/(b(n-4) * (b(n-5) * b(n-3)^3 - b(n-4)^3 * b(n-2))). The denominator of b(n+1) has a factor of (b(1) * b(3)^3 - b(2)^3 * b(4))^a(n+1). For example, if b(0) = 2, b(1) = b(2) = b(3) = 1, b(4) = 1+x, b(5) = 4, then the denominator of b(n+1) is x^a(n+1). - Michael Somos, Nov 15 2023

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

Cf. A002264, A002265, A004526, A010872, A010873, A010874, A130481, A130482.

List([0..70], n-> Int((n-1)*(n-2)/10)); # G. C. Greubel, Aug 31 2019

[Round(n*(n-3)/10): n in [0..70]]; // Vincenzo Librandi, Jun 25 2011

seq(floor((n-1)*(n-2)/10), n=0..70); # G. C. Greubel, Aug 31 2019

Accumulate[Floor[Range[0,70]/5]] (* Harvey P. Dale, May 25 2016 *)

a(n) = sum(k=0, n, k\5); \\ Michel Marcus, May 13 2016

[floor((n-1)*(n-2)/10) for n in (0..70)] # G. C. Greubel, Aug 31 2019

a(n) = floor(n/5)*(2*n - 3 - 5*floor(n/5))/2.
a(n) = A002266(n)*(2*n - 3 - 5*A002266(n))/2.
a(n) = A002266(n)*(n -3 +A010874(n))/2.
G.f.: x^5/((1-x^5)*(1-x)^2) =  x^5/( (1+x+x^2+x^3+x^4)*(1-x)^3 ).
a(n) = floor((n-1)*(n-2)/10). - Mitch Harris, Sep 08 2008
a(n) = round(n*(n-3)/10) = ceiling((n+1)*(n-4)/10) = round((n^2 - 3*n - 1)/10). - Mircea Merca, Nov 28 2010
a(n) = A008732(n-5), n > 4. - R. J. Mathar, Nov 22 2008
a(n) = a(n-5) + n - 4, n > 4. - Mircea Merca, Nov 28 2010
a(5n) = A000566(n), a(5n+1) = A005476(n), a(5n+2) = A005475(n), a(5n+3) = A147875(n), a(5n+4) = A028895(n). - Philippe Deléham, Mar 26 2013
From Amiram Eldar, Sep 17 2022: (Start)
Sum_{n>=5} 1/a(n) = 518/45 - 2*sqrt(2*(sqrt(5)+5))*Pi/3.
Sum_{n>=5} (-1)^(n+1)/a(n) = 8*sqrt(5)*arccoth(3/sqrt(5))/3 + 92*log(2)/15 - 418/45. (End)

A131242 Partial sums of A059995: a(n) = sum_{k=0..n} floor(k/10).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 156, 162, 168, 174, 180, 186, 192, 198

Offset: 0

Hieronymus Fischer, Jun 21 2007

Complementary with A130488 regarding triangular numbers, in that A130488(n)+10*a(n)=n(n+1)/2=A000217(n).

			As square array :
    0,   0,   0,   0,   0,   0,   0,   0,   0,    0
    1,   2,   3,   4,   5,   6,   7,   8,   9,   10
   12,  14,  16,  18,  20,  22,  24,  26,  28,   30
   33,  36,  39,  42,  45,  48,  51,  54,  57,   60
   64,  68,  72,  76,  80,  84,  88,  92,  96,  100
  105, 110, 115, 120, 125, 130, 135, 140, 145,  150
  156, 162, 168, 174, 180, 186, 192, 198, 204,  210
... - _Philippe Deléham_, Mar 27 2013

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,1,-2,1).

Cf. A008728, A059995, A010879, A002266, A130488, A000217, A002620, A130518, A130519, A130520, A174709, A174738, A118729, A218470.

Table[(1/2)*Floor[n/10]*(2*n - 8 - 10*Floor[n/10]), {n,0,50}] (* G. C. Greubel, Dec 13 2016 *)
Accumulate[Table[FromDigits[Most[IntegerDigits[n]]],{n,0,110}]] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,0,0,1,2},120] (* Harvey P. Dale, Apr 06 2017 *)

for(n=0,50, print1((1/2)*floor(n/10)*(2n-8-10*floor(n/10)), ", ")) \\ G. C. Greubel, Dec 13 2016

a(n)=my(k=n\10); k*(n-5*k-4) \\ Charles R Greathouse IV, Dec 13 2016

a(n) = (1/2)*floor(n/10)*(2n-8-10*floor(n/10)).
a(n) = A059995(n)*(2n-8-10*A059995(n))/2.
a(n) = (1/2)*A059995(n)*(n-8+A010879(n)).
a(n) = (n-A010879(n))*(n+A010879(n)-8)/20.
G.f.: x^10/((1-x^10)(1-x)^2).
From Philippe Deléham, Mar 27 2013: (Start)
a(10n)   = A051624(n).
a(10n+1) = A135706(n).
a(10n+2) = A147874(n+1).
a(10n+3) = 2*A005476(n).
a(10n+4) = A033429(n).
a(10n+5) = A202803(n).
a(10n+6) = A168668(n).
a(10n+7) = 2*A147875(n).
a(10n+8) = A135705(n).
a(10n+9) = A124080(n). (End)
a(n) = A008728(n-10) for n>= 10. - Georg Fischer, Nov 03 2018

cp's OEIS Frontend

A294831 Numerators of the partial sums of the reciprocals of the numbers (k + 1)*(5*k + 4) = 2*A005476(k+1), for k >= 0.

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A294833 Decimal expansion of the sum of the reciprocals of the numbers (k+1)*(5*k+4) = 2*A005476(k+1) for k >= 0.

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A294832 Denominators of the partial sums of the reciprocals of the numbers (k + 1)*(5*k + 4) = 2*A005476(k+1), for k >= 0.

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A183296 Complement of A005476.

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A000326 Pentagonal numbers: a(n) = n*(3*n-1)/2.

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A033991 a(n) = n*(4*n-1).

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A294831 Numerators of the partial sums of the reciprocals of the numbers (k + 1)(5k + 4) = 2*A005476(k+1), for k >= 0.

A294833 Decimal expansion of the sum of the reciprocals of the numbers (k+1)(5k+4) = 2*A005476(k+1) for k >= 0.

A294832 Denominators of the partial sums of the reciprocals of the numbers (k + 1)(5k + 4) = 2*A005476(k+1), for k >= 0.

A000326 Pentagonal numbers: a(n) = n(3n-1)/2.

A033991 a(n) = n(4n-1).

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

A226488 a(n) = n(13n - 9)/2.