A064038 -id:A064038 - cp's OEIS frontend

A001809 a(n) = n! * n(n-1)/4.

0, 0, 1, 9, 72, 600, 5400, 52920, 564480, 6531840, 81648000, 1097712000, 15807052800, 242853811200, 3966612249600, 68652904320000, 1255367393280000, 24186745110528000, 489781588488192000, 10400656084955136000, 231125690776780800000, 5364548928029491200000

Offset: 0

N. J. A. Sloane

a(n) = n!*n*(n-1)/4 gives the total number of inversions in all the permutations of [n]. [Stern, Terquem] Proof: For fixed i,j and for fixed I,J (i < j, I > J, 1 <= i,j,I,J <= n), we have (n-2)! permutations p of [n] for which p(i)=I and p(j)=J (permute {1,2,...,n} \ {I,J} in the positions (1,2,...,n) \ {i,j}). There are n*(n-1)/2 choices for the pair (i,j) with i < j and n*(n-1)/2 choices for the pair (I,J) with I > J. Consequently, the total number of inversions in all the permutations of [n] is (n-2)!*(n*(n-1)/2)^2 = n!*n*(n-1)/4. - Emeric Deutsch, Oct 05 2006
To state this another way, a(n) is the number of occurrences of the pattern 12 in all permutations of [n]. - N. J. A. Sloane, Apr 12 2014
Equivalently, this is the total Denert index of all permutations on n letters (cf. A008302). - N. J. A. Sloane, Jan 20 2014
Also coefficients of Laguerre polynomials. a(n)=A021009(n,2), n >= 2.
a(n) is the number of edges in the Cayley graph of the symmetric group S_n with respect to the generating set consisting of transpositions. - Avi Peretz (njk(AT)netvision.net.il), Feb 20 2001
a(n+1) is the sum of the moments over all permutations of n. E.g. a(4) is [1,2,3].[1,2,3] + [1,3,2].[1,2,3] + [2,1,3].[1,2,3] + [2,3,1].[1,2,3] + [3,1,2].[1,2,3] + [3,2,1].[1,2.3] = 14 + 13 + 13 + 11 + 11 + 10 = 72. - Jon Perry, Feb 20 2004
Derivative of the q-factorial [n]!, evaluated at q=1. Example: a(3)=9 because (d/dq)[3]!=(d/dq)((1+q)(1+q+q^2))=2+4q+3q^2 is equal to 9 at q=1. - Emeric Deutsch, Apr 19 2007
Also the number of maximal cliques in the n-transposition graph for n > 1. - Eric W. Weisstein, Dec 01 2017
a(n-1) is the number of trees on [n], rooted at 1, with exactly two leaves.  A leaf is a non-root vertex of degree 1. - Nikos Apostolakis, Dec 27 2021

			G.f. = x^2 + 9*x^3 + 72*x^4 + 600*x^5 + 5400*x^6 + 52920*x^7 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.
Simon Altmann and Eduardo L. Ortiz, Editors, Mathematical and Social Utopias in France: Olinde Rodrigues and His Times, Amer. Math. Soc., 2005.
David M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 90.
Cornelius Lanczos, Applied Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 519.
Edward M. Reingold, Jurg Nievergelt and Narsingh Deo, Combinatorial Algorithms, Prentice-Hall, 1977, section 7.1, p. 287.
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).
Olry Terquem, Liouville's Journal, 1838.

T. D. Noe, Table of n, a(n) for n=0..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Eric Babson and Einar Steingrimsson, Generalized Permutation Patterns and Classification of the Mahonian Statistics, Séminaire Lotharingien de Combinatoire, B44b (2000), 18 pp.
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]
FindStat - Combinatorial Statistic Finder, The Denert index of a permutation
Dominique Foata and Marcel-Paul Schützenberger, Major Index and inversion number of permutations , Math. Nachr. 83 (1978), 143-159
Dexter Jane L. Indong and Gilbert R. Peralta, Inversions of permutations in Symmetric, Alternating, and Dihedral Groups, JIS, Vol. 11 (2008), Article 08.4.3.
Warren P. Johnson, Review of Altmann-Ortiz book, Amer. Math. Monthly, Vol. 114, No. 8 (2007), pp. 752-758.
Cornelius Lanczos, Applied Analysis. (Annotated scans of selected pages)
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles. (Annotated scans of some selected pages)
M. Stern, Aufgaben, Journal für die reine und angewandte Mathematik, Vol. 18 (1838), p. 100.
Thotsaporn Thanatipanonda, Inversions and major index for permutations, Math. Mag., Vol. 77, No. 2 (April 2004), pp. 136-140.
Eric Weisstein's World of Mathematics, Maximal Clique.
Eric Weisstein's World of Mathematics, Transposition Graph.
Index entries for sequences related to Laguerre polynomials.

Cf. A034968 (the inversion numbers of permutations listed in alphabetic order). See also A053495 and A064038.

Cf. A001620, A008302, A099285.

[Factorial(n)*n*(n-1)/4: n in [0..20]]; // Vincenzo Librandi, Jun 15 2015

A001809 := n->n!*n*(n-1)/4;
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card=1)}, labeled]: subs(r=1, stack): seq(count(subs(r=2, ZL), size=m), m=0..19); # Zerinvary Lajos, Feb 07 2008
with (combstruct):with (combinat):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(1):seq(count(ZLL, size=n)*binomial(n,2)/2, n=0..21); # Zerinvary Lajos, Jun 11 2008

Table[n! n (n - 1)/4, {n, 0, 18}]
Table[n! Binomial[n, 2]/2, {n, 0, 20}] (* Eric W. Weisstein, Dec 01 2017 *)
Coefficient[Table[n! LaguerreL[n, x], {n, 20}], x, 2] (* Eric W. Weisstein, Dec 01 2017 *)

PARI
```
{a(n) = n! * n * (n-1) / 4};
```

[factorial(m) * binomial(m, 2) / 2 for m in range(19)]  # Zerinvary Lajos, Jul 05 2008

E.g.f.: (1/2)*x^2/(1-x)^3.
a(n) = a(n-1)*n^2/(n-2), n > 2; a(2)=1.
a(n) = n*a(n-1) + (n-1)!*n*(n-1)/2, a(1) = 0, a(2) = 1; a(n) = sum (first n! terms of A034968); a(n) = sum of the rises j of permutations (p(j)If we define f(n,i,x) = Sum_{k=i..n} (Sum_{j=i..k}(C(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j))) then a(n)=(-1)^n*f(n,2,-3), (n>=2). - Milan Janjic, Mar 01 2009
a(n) = Sum_k k*A008302(n,k). - N. J. A. Sloane, Jan 20 2014
a(n+2) = n*n!*(n+1)^2 / 4 = A000142(n) * (A000292(n) + A000330(n))/2 = sum of the cumulative sums of all the permutations of numbers from 1 to n, where A000142(n) = n! and sequences A000292(n) and A000330(n) are sequences of minimal and maximal values of cumulative sums of all the permutations of numbers from 1 to n. - Jaroslav Krizek, Sep 13 2014
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=2} 1/a(n) = 12 - 4*e.
Sum_{n>=2} (-1)^n/a(n) = 8*gamma - 4 - 4/e - 8*Ei(-1), where gamma is Euler's constant (A001620) and -Ei(-1) is the negated exponential integral at -1 (A099285). (End)

More terms and new description from Michael Somos, May 19 2000

Simpler description from Emeric Deutsch, Oct 05 2006

A014695 Poincaré series [or Poincare series] (or Molien series) for mod 2 cohomology of Q_8.

Original entry on oeis.org

1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1

Offset: 0

N. J. A. Sloane

From Klaus Brockhaus, May 14 2010: (Start)
Periodic sequence: Repeat 1, 2, 2, 1.
a(n) = A130658(n+1).
Continued fraction expansion of (5+sqrt(221))/14.
Decimal expansion of 37/303. (End)

A. Adem, Recent developments in the cohomology of finite groups, Notices Amer. Math. Soc., 44 (1997), 806-812.
Eric Weisstein's World of Mathematics, Simple Graph
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Denominators for the sequence whose numerators are A064038.

Cf. A130658, A177841. - Klaus Brockhaus, May 14 2010

Table[Denominator[n*(n + 1)/4], {n, 0, 104}] (* Arkadiusz Wesolowski, Aug 09 2012 *)
LinearRecurrence[{1,-1,1},{1,2,2},120] (* Harvey P. Dale, Jan 19 2020 *)

x='x+O('x^100); Vec((1+2*x+2*x^2+x^3)/(1-x^4)) \\ Altug Alkan, Dec 24 2015

def A014695(n): return (1,2,2,1)[n&3] # Chai Wah Wu, Apr 17 2023

G.f.: (1+x+x^2)/((1-x)*(1+x^2)) =  (1+2*x+2*x^2+x^3)/(1-x^4).
a(n) = (3-sqrt(2)*cos((2*n+1)*Pi/4))/2. - Jaume Oliver Lafont, Nov 28 2009
a(n) = (6-(1+i)*i^n-(1-i)*(-i)^n)/4 where i = sqrt(-1). - Klaus Brockhaus, May 14 2010
a(n) = denominator of Sum_{k=0..n} k/2. - Arkadiusz Wesolowski, Aug 09 2012

More terms from Klaus Brockhaus, May 14 2010

A181318 a(n) = A060819(n)^2.

Original entry on oeis.org

0, 1, 1, 9, 1, 25, 9, 49, 4, 81, 25, 121, 9, 169, 49, 225, 16, 289, 81, 361, 25, 441, 121, 529, 36, 625, 169, 729, 49, 841, 225, 961, 64, 1089, 289, 1225, 81, 1369, 361, 1521, 100, 1681, 441, 1849, 121, 2025, 529, 2209, 144, 2401, 625, 2601, 169, 2809, 729

Offset: 0

Paul Curtz, Jan 26 2011

The first sequence, p=0, of the family A060819(n)*A060819(n+p).
Hence array
p=0:  0, 1, 1,  9,  1, 25,  9,  49,    a(n)=A060819(n)^2,
p=1:  0, 1, 3,  3,  5, 15, 21,  14,     A064038(n),
p=2:  0, 3, 1, 15,  3, 35,  6,  63,     A198148(n),
p=3:  0, 1, 5,  9,  7, 10, 27,  35,     A160050(n),
p=4:  0, 5, 3, 21,  2, 45, 15,  77,     A061037(n),
p=5:  0, 3, 7,  6,  9, 25, 33,  21,     A178242(n),
p=6:  0, 7, 2, 27,  5, 55,  9,  91,     A217366(n),
p=7:  0, 2, 9, 15, 11, 15, 39,  49,     A217367(n),
p=8:  0, 9, 5, 33,  3, 65, 21, 105,     A180082(n).
Compare columns 2, 3 and 5, columns 4 and 7 and columns 6 and 8.
From Peter Bala, Feb 19 2019: (Start)
We make some general remarks about the sequence a(n) = numerator(n^2/(n^2 + k^2)) = (n/gcd(n,k))^2 for k a fixed positive integer (we suppress the dependence of a(n) on k). The present sequence corresponds to the case k = 4.
a(n) is a quasi-polynomial in n. In fact, a(n) = n^2/b(n) where b(n) = gcd(n^2,k^2) is a purely periodic sequence in n.
In addition to being multiplicative these sequences are also strong divisibility sequences, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. In particular, it follows that a(n) is a divisibility sequence: if n divides m then a(n) divides a(m).
By the multiplicativeness and strong divisibility property of the sequence a(n) it follows that if gcd(n,m) = 1 then a( a(n)*a(m) ) = a(a(n)) * a(a(m)), a( a(a(n))*a(a(m)) ) = a(a(a(n))) * a(a(a(m))) and so on.
The sequence a(n) has the rational generating function Sum_{d divides k} f(d)*F(x^d), where F(x) = x*(1 + x)/(1 - x)^3 = x + 4*x^2 + 9*x^3 + 16*x^4 + ... is the o.g.f. for the squares A000290, and where f(n) is the Dirichlet inverse of the Jordan totient function J_2(n) - see A007434. The function f(n) is multiplicative and is defined on prime powers p^k by f(p^k) = (1 - p^2). See A046970. Cf. A060819. (End)
a(n-4) is the constant needed to complete the n-polygonal numbers into squares (see A377851); a(-1) = 1, which completes the triangle numbers, is not shown in the data. - Jonathan Dushoff, Nov 12 2024

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Completing the square.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A181829, A046970, A007434, A060819, A168077, A377851.

[n^2/GCD(n,4)^2: n in [0..100]]; // G. C. Greubel, Sep 19 2018

a:=n->n^2/gcd(n,4)^2: seq(a(n),n=0..60); # Muniru A Asiru, Feb 20 2019

Table[n^2/GCD[n,4]^2, {n,0,100}] (* G. C. Greubel, Sep 19 2018 *)
LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1},{0,1,1,9,1,25,9,49,4,81,25,121},60] (* Harvey P. Dale, Jan 18 2025 *)

a(n)=n^2/gcd(n,4)^2 \\ Charles R Greathouse IV, Dec 21 2011

[n^2/gcd(n, 4)^2 for n in (0..100)] # G. C. Greubel, Feb 20 2019

a(2*n) = A168077(n), a(2*n+1) = A016754(n).
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
G.f.: x*(1 + x + 9*x^2 + x^3 + 22*x^4 + 6*x^5 + 22*x^6 + x^7 + 9*x^8 + x^9 + x^10)/(1-x^4)^3. - R. J. Mathar, Mar 10 2011
From Peter Bala, Feb 19 2019: (Start)
a(n) = numerator(n^2/(n^2 + 16)) = n^2/(gcd(n^2,16)) = (n/gcd(n,4))^2.
a(n) = n^2/b(n), where b(n) = [1, 4, 1, 16, 1, 4, 1, 16, ...] is a purely periodic sequence of period 4.
a(n) is a quasi-polynomial in n: a(4*n) = n^2; a(4*n + 1) = (4*n + 1)^2; a(4*n + 2) = (2*n + 1)^2; a(4*n + 3) = (4*n + 3)^2.
O.g.f.: Sum_{d divides 4} A046970(d)*x^d*(1 + x^d)/(1 - x^d)^3 = x*(1 + x)/(1 - x)^3 - 3*x^2*(1 + x^2)/(1 - x^2)^3 - 3*x^4*(1 + x^4)/(1 - x^4)^3. (End)
Sum_{n>=1} 1/a(n) = 5*Pi^2/12. - Amiram Eldar, Aug 12 2022
From Amiram Eldar, Nov 25 2022: (Start)
Multiplicative with a(2^e) = 4^max(0, e-2), and a(p^e) = p^(2*e) for p > 2.
Dirichlet g.f.: zeta(s-2)*(1 - 3/2^s - 3/4^s).
Sum_{k=1..n} a(k) ~ (37/192) * n^3. (End)
a(n) = (37 - 27*(-1)^n - 3*(-1)^(n*(n-1)/2) - 3*(-1)^(n*(n+1)/2)) * n^2/64. - Vaclav Kotesovec, Nov 14 2024

Edited by Jean-François Alcover, Oct 01 2012 and Jan 15 2013

More terms from Michel Marcus, Jun 09 2014

A160050 Numerator of the Harary number for the star graph s_n.

Original entry on oeis.org

0, 1, 5, 9, 7, 10, 27, 35, 22, 27, 65, 77, 45, 52, 119, 135, 76, 85, 189, 209, 115, 126, 275, 299, 162, 175, 377, 405, 217, 232, 495, 527, 280, 297, 629, 665, 351, 370, 779, 819, 430, 451, 945, 989, 517, 540, 1127, 1175, 612, 637, 1325, 1377, 715, 742, 1539

Offset: 1

Eric W. Weisstein, Apr 30 2009

			0, 1, 5/2, 9/2, 7, 10, 27/2, 35/2, 22, 27, ...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Harary Index.
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A130658 (denominators), A033954 (quadrisection), A001107 (quadrisection), A181890 (quadrisection).

Cf. A000217, A060819, A064038.

m:=50; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x^2*(-1 - 2*x - 5*x^4 + 3*x^5 - 2*x^6 + x^7)/((x - 1)^3*(x^2 + 1)^3))); // G. C. Greubel, Sep 21 2018

f[n_] := n/GCD[n, 4]; Array[f[#] f[# + 3] &, 58]
Rest[CoefficientList[Series[x^2*(-1 - 2*x - 5*x^4 + 3*x^5 - 2*x^6 + x^7)/((x - 1)^3*(x^2 + 1)^3), {x, 0, 50}], x]] (* G. C. Greubel, Sep 21 2018 *)

s=vector(40,n,1/4*(n+2)*(n-1)) /* fractions */
vector(#s,n,numerator(s[n])) /* this sequence */ \\ Joerg Arndt, Jan 04 2011

x='x+O('x^50); concat([0], Vec(x^2*(-1 - 2*x - 5*x^4 + 3*x^5 - 2*x^6 + x^7)/((x - 1)^3*(x^2 + 1)^3))) \\ G. C. Greubel, Sep 21 2018

Numerator of (1/4)*(n+2)*(n-1). - Joerg Arndt, Jan 04 2011
It appears that a(n + 1) = A060819(n-1) * A060819(n + 2). - Paul Curtz, Dec 23 2010 [Corrected by Joerg Arndt, Jan 04 2011]
G.f.: x^2*(-1-2*x-5*x^4+3*x^5-2*x^6+x^7) / ( (x-1)^3*(x^2+1)^3 ). - R. J. Mathar, Jan 04 2011
a(1+4*n) = (A000217(n+1)-1)/2, a(2+4*n) = (A000217(n+2)-1)/2, a(3+4*n) = A000217(n+3)-1, a(4+4*n) = A000217(n+4)-1. - Paul Curtz, Dec 23 2010.
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12). This is not the shortest recurrence. -Paul Curtz, Mar 27 2011
a(1+3*n) = numerator of 9*n*(n+1)/4 = 9*A064038(1+n). - Paul Curtz, Apr 06 2011
a(n) = (n-1)*(n+2)*(3-i^((n-2)*(n-1)))/8, where i=sqrt(-1).  - Bruno Berselli, Apr 07 2011, corrected by Vaclav Kotesovec, Aug 09 2022
Sum_{n>=2} 1/a(n) = 13/9 + Pi/6. - Amiram Eldar, Aug 09 2022

Edited by N. J. A. Sloane, Dec 23 2010

A176126 Numerator of -A127276(n)/A001788(n).

Original entry on oeis.org

-1, -1, 1, 2, 4, 13, 19, 13, 17, 43, 53, 32, 38, 89, 103, 59, 67, 151, 169, 94, 104, 229, 251, 137, 149, 323, 349, 188, 202, 433, 463, 247, 263, 559, 593, 314, 332, 701, 739, 389, 409, 859, 901, 472, 494, 1033, 1079, 563, 587, 1223, 1273, 662, 688, 1429, 1483, 769, 797, 1651, 1709, 884, 914, 1889, 1951, 1007, 1039, 2143, 2209, 1138, 1172, 2413, 2483, 1277, 1313, 2699, 2773, 1424, 1462, 3001, 3079, 1579

Offset: 0

Paul Curtz, Dec 07 2010

The sequence of fractions starts -1/0, -1/1, 1/3, 2/3, 4/5, 13/15, 19/21, 13/14, 17/18, 43/45, 53/55, 32/33, 38/39, ...
The denominators are apparently A064038(n+1) = A061041(4+8*n) (i.e., specified as numerators in A061041).
The difference between denominator and numerator is A014695(n), n > 0.

Cf. A001788, A127276.

Cf. A061041, A064038, A014695.

A001788 := proc(n) n*(n+1)*2^(n-2) ; end proc:
A127276 := proc(n) 2^n-A001788(n) ; end proc:
A176126 := proc(n) if n = 0 then -1 else 2^n/A001788(n)-1 ; numer(-%) ; end if; end proc:
seq(A176126(n),n=0..40) ;

Conjecture: a(n) = +3*a(n-1) -6*a(n-2) +10*a(n-3) -12*a(n-4) +12*a(n-5) -10*a(n-6) +6*a(n-7) -3*a(n-8) +a(n-9) with g.f. (x^4-x^3+3*x^2-x+1)*(x^4-x^3-2*x^2-x+1) / ( (x-1)^3*(x^2+1)^3 ). - R. J. Mathar, Dec 12 2010

a(n) = 3*a(n-4) -3*a(n-8) +a(n-12).

A181890 a(n) = 8n^2 + 14n + 5.

Original entry on oeis.org

5, 27, 65, 119, 189, 275, 377, 495, 629, 779, 945, 1127, 1325, 1539, 1769, 2015, 2277, 2555, 2849, 3159, 3485, 3827, 4185, 4559, 4949, 5355, 5777, 6215, 6669, 7139, 7625, 8127, 8645, 9179, 9729, 10295, 10877, 11475, 12089, 12719, 13365, 14027, 14705, 15399, 16109, 16835, 17577

Offset: 0

Paul Curtz, Feb 01 2011

A160050(4*n+1) = A033954(n); A160050(4*n+2) = A001107(n); the third quadrisection is a(n).
First 16 terms of clockwise spiral for odd numbers are as follows:
.
  13--15--17--19
   |           |
  11   1---3  21
   |       |   |
   9---7---5  23
               |
  31--29--27--25
.
a(n) comes from the third vertical.
Sequence found by reading the line from 5, in the direction 5, 27, in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Dec 25 2011

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

Cf. A000217, A001107, A014106, A014635, A033954, A064038, A160050.

[8*n^2+14*n+5: n in [0..700]]; // Vincenzo Librandi, Feb 01 2011

Table[(2n+1)(4n+5),{n,0,40}]  (* Harvey P. Dale, Feb 06 2011 *)
CoefficientList[Series[(5 + 12 x - x^2)/(1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, May 23 2014 *)

a(n)=8*n^2+14*n+5 \\ Charles R Greathouse IV, Dec 21 2011

a(n) = A160050(4*n+3).
a(n) = (2*n+1)*(4*n+5).
a(n) = a(n-1) + 16*n + 6.
a(n) = 2*a(n-1) - a(n-2) + 16.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (5 + 12*x - x^2)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 25 2011
a(n) = A014635(n+1) - 1. - Omar E. Pol, Dec 25 2011
From Vaclav Kotesovec, Aug 18 2018: (Start)
Sum_{n>=0} 1/a(n) = 2/3 - Pi/12 - log(2)/6 = 0.289342748774193011891907697817...
Sum_{n>=0} (-1)^n / a(n) = (1 + sqrt(2))*Pi/12 - 2/3 - sqrt(2)*log(tan(Pi/8))/6 = 0.173114712692423461587883724528539... (End)
a(n) = A014106(2*n+1). - Rick L. Shepherd, Aug 06 2019
E.g.f.: (5 + 22*x + 8*x^2)*exp(x). - Elmo R. Oliveira, Oct 19 2024

A185438 a(n) = 8n^2 - 2n + 1.

Original entry on oeis.org

1, 7, 29, 67, 121, 191, 277, 379, 497, 631, 781, 947, 1129, 1327, 1541, 1771, 2017, 2279, 2557, 2851, 3161, 3487, 3829, 4187, 4561, 4951, 5357, 5779, 6217, 6671, 7141, 7627, 8129, 8647, 9181, 9731, 10297, 10879, 11477, 12091, 12721, 13367, 14029, 14707, 15401, 16111, 16837, 17579

Offset: 0

Paul Curtz, Feb 03 2011

Odd numbers (A005408) written clockwise as a square spiral:
.
  41--43--45--47--49--51
   |                   |
  39  13--15--17--19  53
   |   |           |   |
  37  11   1---3  21  55
   |   |       |   |   |
  35   9---7---5  23  57
   |               |   |
  33--31--29--27--25  59
                       |
  71--69--67--65--63--61
.
Walking in straight lines away from the center:
  1, 17, 49, ... = A069129(n+1) =  1 - 8*n + 8*n^2,
  1,  3, 21, ... = A033567(n)   =  1 - 6*n + 8*n^2,
  1, 15, 45, ... = A014634(n)   =  1 + 6*n + 8*n^2,
  1,  5, 25, ... = A080856(n)   =  1 - 4*n + 8*n^2,
  1, 13, 41, ... = A102083(n)   =  1 + 4*n + 8*n^2,
  1,  7, 29, ... = a(n)         =  1 - 2*n + 8*n^2,
  1, 11, 37, ... = A188135(n)   =  1 + 2*n + 8*n^2,
  1,  9, 33, ... = A081585(n)   =  1       + 8*n^2,
  5, 29, 69, ... = A108928(n+1) = -3       + 8*n^2,
  7, 31, 71, ... = A157914(n+1) = -1       + 8*n^2,
  9, 35, 77, ... = A033566(n+1) = -1 + 2*n + 8*n^2.
All are quadrisections of sequences in A181407(n) (example: A014634(n) and A033567(n) in A064038(n+1)) or of this family (?): a(n) is a quadrisection of f(n) = 1,1,1,1,2,7,11,8,11,29,37,23,28,67,79,46,... f(n) is just before A064038(n+1) (fifth vertical) in A181407(n). The companion to a(n) is A188135(n), another quadrisection of f(n). Two last quadrisections of f(n) are A054552(n) and A033951(n).
For n >= 1, bisection of A193867. - Omar E. Pol, Aug 16 2011
Also the sequence may be obtained by starting with the segment (1, 7) followed by the line from 7 in the direction 7, 29, ... in the square spiral whose vertices are the generalized hexagonal numbers (A000217). - Omar E. Pol, Aug 01 2016

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

Cf. A000217, A005408, A014634, A014635, A033566, A033567, A033951, A054552, A064038.

Cf. A069129, A080856, A081585, A102083, A108928, A157914, A181407, A188135, A193867.

[1-2*n+8*n^2: n in [0..50]]; // Vincenzo Librandi, Feb 03 2011

Table[1 - 2n + 8n^2, {n, 0, 39}] (* Alonso del Arte, Feb 03 2011 *)
CoefficientList[Series[(-1 - 4 x - 11 x^2)/(x - 1)^3, {x, 0, 47}], x] (* Michael De Vlieger, Aug 01 2016 *)

a(n)=8*n^2-2*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 16*n - 10 (n > 0).
a(n) = 2*a(n-1) - a(n-2) + 16 (n > 1).
a(n) = 3*(n-1) - 3*a(n-2) + a(n-3) (n > 2).
G.f.: (-1 - 4*x - 11*x^2)/(x-1)^3. - R. J. Mathar, Feb 03 2011
a(n) = A014635(n) + 1. - Bruno Berselli, Apr 09 2011
E.g.f.: exp(x)*(1 + 6*x + 8*x^2). - Elmo R. Oliveira, Nov 17 2024

A198148 a(n) = n(n+2)(9 - 7*(-1)^n)/16.

Original entry on oeis.org

0, 3, 1, 15, 3, 35, 6, 63, 10, 99, 15, 143, 21, 195, 28, 255, 36, 323, 45, 399, 55, 483, 66, 575, 78, 675, 91, 783, 105, 899, 120, 1023, 136, 1155, 153, 1295, 171, 1443, 190, 1599, 210, 1763, 231, 1935, 253, 2115, 276, 2303, 300, 2499, 325

Offset: 0

Paul Curtz, Oct 21 2011

See, in A181318(n), A060819(n)*A060819(n+p): A060819(n)^2, A064038(n), a(n), A160050(n), A061037(n), A178242(n). The second differences a(n+2)-2*a(n+1)+a(n) = -5, 16, -26, 44, -61, 86, -110, 142, -173, 212, -250, 296, -341, 394, -446, 506, taken modulo 9 are periodic with the palindromic period 4, 7, 1, 8, 2, 5, 7, 7, 7, 5, 2, 8, 1, 7, 4.

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

Cf. quadrisections A014105, A001539, A000384, A141759.

Cf. A060819, A000217, A000466, A142705, A000034, A005563, A010689, A028552, A050534.

List([0..60], n -> n*(n+2)*(9-7*(-1)^n)/16); # G. C. Greubel, Feb 21 2019

[n*(n+2)*(9-7*(-1)^n)/16: n in [0..60]]; // Vincenzo Librandi, Nov 25 2011

A198148:=n->n*(n+2)*(9-7*(-1)^n)/16; seq(A198148(k), k=0..100); # Wesley Ivan Hurt, Oct 16 2013

LinearRecurrence[{0,3,0,-3,0,1},{0,3,1,15,3,35},60] (* Vincenzo Librandi, Nov 25 2011 *)

a(n)=n*(n+2)*(9-7*(-1)^n)/16 \\ Charles R Greathouse IV, Oct 16 2015

[n*(n+2)*(9-7*(-1)^n)/16 for n in (0..60)] # G. C. Greubel, Feb 21 2019

a(n) = A060819(n)*A060819(n+2).
a(2n) = n*(n+1)/2 = A000217(n).
a(2n+1) = (2*n+1)*(2*n+3) = A000466(n+1).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6), n>5.
a(n+1) - a(n) = (7*(-1)^n *(2*n^2+6*n+3) +18*n +27)/16.
a(n) = A142705(n) / A000034(n+1).
a(n) = A005563(n) / A010689(n+1). - Franklin T. Adams-Watters, Oct 21 2011
G.f. x*(3 +x +6*x^2 -x^4)/(1-x^2)^3. - R. J. Mathar, Oct 25 2011
a(n)*a(n+1) = a(A028552(n)) = A050534(n+2). - Bruno Berselli, Oct 26 2011
a(n) = numerator( binomial((n+2)/2,2) ). - Wesley Ivan Hurt, Oct 16 2013
E.g.f.: x*((24+x)*cosh(x) + (3+8*x)*sinh(x))/8. - G. C. Greubel, Sep 20 2018
Sum_{n>=1} 1/a(n) = 5/2. - Amiram Eldar, Aug 12 2022

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cp's OEIS Frontend

A330082 a(n) = 5*A064038(n+1).

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A061041 Numerator of 1/16 - 1/n^2.

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A001809 a(n) = n! * n(n-1)/4.

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A014695 Poincaré series [or Poincare series] (or Molien series) for mod 2 cohomology of Q_8.

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A181318 a(n) = A060819(n)^2.

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A160050 Numerator of the Harary number for the star graph s_n.

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A181890 a(n) = 8n^2 + 14n + 5.

A185438 a(n) = 8n^2 - 2n + 1.

A198148 a(n) = n(n+2)(9 - 7*(-1)^n)/16.