A001835 -id:A001835 - cp's OEIS frontend

A001570 Numbers k such that k^2 is centered hexagonal.

1, 13, 181, 2521, 35113, 489061, 6811741, 94875313, 1321442641, 18405321661, 256353060613, 3570537526921, 49731172316281, 692665874901013, 9647591076297901, 134373609193269601, 1871582937629476513, 26067787517619401581, 363077442309042145621

Offset: 1

N. J. A. Sloane

Chebyshev T-sequence with Diophantine property. - Wolfdieter Lang, Nov 29 2002
a(n) = L(n,14), where L is defined as in A108299; see also A028230 for L(n,-14). - Reinhard Zumkeller, Jun 01 2005
Numbers x satisfying x^2 + y^3 = (y+1)^3. Corresponding y given by A001921(n)={A028230(n)-1}/2. - Lekraj Beedassy, Jul 21 2006
Mod[ a(n), 12 ] = 1. (a(n) - 1)/12 = A076139(n) = Triangular numbers that are one-third of another triangular number. (a(n) - 1)/4 = A076140(n) = Triangular numbers T(k) that are three times another triangular number. - Alexander Adamchuk, Apr 06 2007
Also numbers n such that RootMeanSquare(1,3,...,2*n-1) is an integer. - Ctibor O. Zizka, Sep 04 2008
a(n), with n>1, is the length of the cevian of equilateral triangle whose side length is the term b(n) of the sequence A028230. This cevian divides the side (2*x+1) of the triangle in two integer segments x and x+1. - Giacomo Fecondo, Oct 09 2010
For n>=2, a(n) equals the permanent of the (2n-2)X(2n-2) tridiagonal matrix with sqrt(12)'s along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
Beal's conjecture would imply that set intersection of this sequence with the perfect powers (A001597) equals {1}. In other words, existence of a nontrivial perfect power in this sequence would disprove Beal's conjecture. - Max Alekseyev, Mar 15 2015
Numbers n such that there exists positive x with x^2 + x + 1 = 3n^2. - Jeffrey Shallit, Dec 11 2017
Given by the denominators of the continued fractions [1,(1,2)^i,3,(1,2)^{i-1},1]. - Jeffrey Shallit, Dec 11 2017
A near-isosceles integer-sided triangle with an angle of 2*Pi/3 is a triangle whose sides (a, a+1, c) satisfy Diophantine equation (a+1)^3 - a^3 = c^2. For n >= 2, the largest side c is given by a(n) while smallest and middle sides (a, a+1) = (A001921(n-1), A001922(n-1)) (see Julia link). - Bernard Schott, Nov 20 2022

			G.f. = x + 13*x^2 + 181*x^3 + 2521*x^4 + 35113*x^5 + 489061*x^6 + 6811741*x^7 + ...

E.-A. Majol, Note #2228, L'Intermédiaire des Mathématiciens, 9 (1902), pp. 183-185. - N. J. A. Sloane, Mar 03 2022
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).

G. C. Greubel, Table of n, a(n) for n = 1..870 (terms 1..101 from T. D. Noe)
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
G. Julia, Triangles dont un angle mesure 120 degrés, Problème Capes, part C (in French).
Tanya Khovanova, Recursive Sequences
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
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.
Sociedad Magic Penny Patagonia, Leonardo en Patagonia
V. Thebault, Consecutive cubes with difference a square, Amer. Math. Monthly, 56 (1949), 174-175.
Eric Weisstein's World of Mathematics, Hex Number
Wikipedia, Beal's conjecture
Index entries for sequences related to Chebyshev polynomials.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (14,-1).

Bisection of A003500/4. Cf. A006051, A001921, A001922.
One half of odd part of bisection of A001075. First differences of A007655.
Cf. A077417 with companion A077416.
Row 14 of array A094954.
Cf. A076139, A076140, A102871.
A122571 is another version of the same sequence.
Row 2 of array A188646.
Cf. similar sequences listed in A238379.
Cf. A028231, which gives the corresponding values of x in 3n^2 = x^2 + x + 1.
Similar sequences of the type cosh((2*m+1)*arccosh(k))/k are listed in A302329. This is the case k=2.

[((2 + Sqrt(3))^(2*n - 1) + (2 - Sqrt(3))^(2*n - 1))/4: n in [1..50]]; // G. C. Greubel, Nov 04 2017

A001570:=-(-1+z)/(1-14*z+z**2); # Simon Plouffe in his 1992 dissertation.

NestList[3 + 7*#1 + 4*Sqrt[1 + 3*#1 + 3*#1^2] &, 0, 24] (* Zak Seidov, May 06 2007 *)
f[n_] := Simplify[(2 + Sqrt@3)^(2 n - 1) + (2 - Sqrt@3)^(2 n - 1)]/4; Array[f, 19] (* Robert G. Wilson v, Oct 28 2010 *)
a[c_, n_] := Module[{},
   p := Length[ContinuedFraction[ Sqrt[ c]][[2]]];
   d := Denominator[Convergents[Sqrt[c], n p]];
   t := Table[d[[1 + i]], {i, 0, Length[d] - 1, p}];
   Return[t];
  ] (* Complement of A041017 *)
a[12, 20] (* Gerry Martens, Jun 07 2015 *)
LinearRecurrence[{14, -1}, {1, 13}, 19] (* Jean-François Alcover, Sep 26 2017 *)
CoefficientList[Series[x (1-x)/(1-14x+x^2),{x,0,20}],x] (* Harvey P. Dale, Sep 18 2024 *)

{a(n) = real( (2 + quadgen( 12)) ^ (2*n - 1)) / 2}; /* Michael Somos, Feb 15 2011 */

a(n) = ((2 + sqrt(3))^(2*n - 1) + (2 - sqrt(3))^(2*n - 1)) / 4. - Michael Somos, Feb 15 2011
G.f.: x * (1 - x) / (1 -14*x + x^2). - Michael Somos, Feb 15 2011
Let q(n, x) = Sum_{i=0, n} x^(n-i)*binomial(2*n-i, i) then a(n) = q(n, 12). - Benoit Cloitre, Dec 10 2002
a(n) = S(n, 14) - S(n-1, 14) = T(2*n+1, 2)/2 with S(n, x) := U(n, x/2), resp. T(n, x), Chebyshev's polynomials of the second, resp. first, kind. See A049310 and A053120. S(-1, x)=0, S(n, 14)=A007655(n+1) and T(n, 2)=A001075(n). - Wolfdieter Lang, Nov 29 2002
a(n) = A001075(n)*A001075(n+1) - 1 and thus (a(n)+1)^6 has divisors A001075(n)^6 and A001075(n+1)^6 congruent to -1 modulo a(n) (cf. A350916). - Max Alekseyev, Jan 23 2022
4*a(n)^2 - 3*b(n)^2 = 1 with b(n)=A028230(n+1), n>=0.
a(n)*a(n+3) = 168 + a(n+1)*a(n+2). - Ralf Stephan, May 29 2004
a(n) = 14*a(n-1) - a(n-2), a(0) = a(1) = 1. a(1 - n) = a(n) (compare A122571).
a(n) = 12*A076139(n) + 1 = 4*A076140(n) + 1. - Alexander Adamchuk, Apr 06 2007
a(n) = (1/12)*((7-4*sqrt(3))^n*(3-2*sqrt(3))+(3+2*sqrt(3))*(7+4*sqrt(3))^n -6). - Zak Seidov, May 06 2007
a(n) = A102871(n)^2+(A102871(n)-1)^2; sum of consecutive squares. E.g. a(4)=36^2+35^2. - Mason Withers (mwithers(AT)semprautilities.com), Jan 26 2008
a(n) = sqrt((3*A028230(n+1)^2 + 1)/4).
a(n) = A098301(n+1) - A001353(n)*A001835(n).
a(n) = A000217(A001571(n-1)) + A000217(A133161(n)), n>=1. - Ivan N. Ianakiev, Sep 24 2013
a(n)^2 = A001922(n-1)^3 - A001921(n-1)^3, for n >= 1. - Bernard Schott, Nov 20 2022
a(n) = 2^(2*n-3)*Product_{k=1..2*n-1} (2 - sin(2*Pi*k/(2*n-1))). Michael Somos, Dec 18 2022
a(n) = A003154(A101265(n)). - Andrea Pinos, Dec 19 2022

A140480 RMS numbers: numbers n such that root mean square of divisors of n is an integer.

Original entry on oeis.org

1, 7, 41, 239, 287, 1673, 3055, 6665, 9545, 9799, 9855, 21385, 26095, 34697, 46655, 66815, 68593, 68985, 125255, 155287, 182665, 242879, 273265, 380511, 391345, 404055, 421655, 627215, 730145, 814463, 823537, 876785, 1069895, 1087009, 1166399, 1204281, 1256489

Offset: 1

Ctibor O. Zizka, Jun 29 2008, Jul 11 2008

For any numbers, A and B, both appearing in the sequence, if gcd(A,B)=1, then A*B is also in the sequence. - Andrew Weimholt, Jul 01 2008
The primes in this sequence are the NSW primes (A088165). For the terms less than 2^31, the only powers greater than 1 appearing in the prime factorization of numbers are 3^3 and 13^2. It appears that all terms are +-1 (mod 8). See A224988 for even numbers. - T. D. Noe, Jul 06 2008, Apr 25 2013
A basis for this sequence is given by A002315. This can be considered as the convergents of quasiregular continued fractions or a special 6-ary numeration system (see A. S. Fraenkel) which gives the characterization of positions of some heap or Wythoff game. What is the Sprague-Grundy function of this game?
Sequence generalized: sigma_r-numbers are numbers n for which sigma_r(n)/sigma_0(n) = c^r. Sigma_r(n) denotes sum of r-th powers of divisors of n; c,r positive integers. This sequence are sigma_2-numbers, A003601 are sigma_1-numbers. In a weaker form we have sigma_r(n)/sigma_0(n) = c^t; t is an integer from <1,r>. - Ctibor O. Zizka, Jul 14 2008
The primes in this sequence are prime numerators with an odd index in A001333. The RMS values (A141812) of prime RMS numbers (this sequence) are prime Pell numbers (A000129) with an odd index. - Ctibor O. Zizka, Aug 13 2008
From Ctibor O. Zizka, Aug 30 2008: (Start)
The set of RMS numbers n could be split into subsets according to the number and form of divisors of n. By definition, RMS(n) = sqrt(sigma_2(n) / sigma_0(n)) should be an integer. Now consider some examples. For n prime number, n has 2 divisors [1,n] and we have to solve Pell's equation n^2 = 2*C^2 - 1; C positive integer. The solution is a prime n of the form u(i) = 6*u(i-1) - u(i-2), i >= 2, u(0)=1, u(1)=7, known as an NSW prime (A088165). For n = p_1*p_2, p_1 and p_2 primes, n has 4 divisors {1; p_1; p_2; p_1*p_2}. There are 2 possible cases. Firstly p^2 = (2*C)^2 - 1 which does not hold for any prime p; secondly p_1^2 = 2*C_1^2 - 1 and p_2^2 = 2*C_2^2 - 1; C_1 and C_2 positive integers.
The solution is that p_1 and p_2 are different NSW primes. If n = p^3, divisors of n are {1; p; p^2; p^3} and we have to solve the Diophantine equation (p^8 - 1)/(p - 1) = (2*C)^2. This equation has no solution for any prime p. RMS numbers n with 4 divisors are only of the form n = p_1*p_2, with p_1 and p_2 NSW primes. The general case is n = p_1*...*p_t, n has 2^t divisors, and for t >= 3, NSW primes are not the only solution. If some of the prime divisors are equals p_i = p_j = ... = p_k, the general case n = p_1*...*p_t is "degenerate" because of the multiplicity of prime factors and therefore n has fewer than 2^t divisors. (End)
General recurrence is a(n) = (a(1)-1)*a(n-1) - a(n-2), a(1) >= 4, lim_{n->infinity} a(n) = x*(k*x+1)^n, k = a(1) - 3, x = (1 + sqrt((a(1)+1)/(a(1)-3)))/2. Examples in OEIS: a(1)=4 gives A002878, whose prime terms give A121534. a(1)=5 gives A001834, whose prime terms give A086386. a(1)=6 gives A030221, whose prime terms {29, 139, 3191, ...} are not a sequence on the OEIS. a(1)=7 gives A002315, whose prime terms give A088165. a(1)=8 gives A033890; the OEIS does not have its prime terms as a sequence (do there exist any prime terms?). a(1)=9 gives A057080, whose prime terms {71, 34649, 16908641, ...} are not a sequence in the OEIS. a(1)=10 gives A057081, whose prime terms {389806471, 192097408520951, ...} are not a sequence in the OEIS. - Ctibor O. Zizka, Sep 02 2008
16 of the first 1660 terms are even (the smallest is 2217231104). The first 16 even terms are all divisible by 30976. - Donovan Johnson, Apr 16 2013
All the 83 even terms up to 10^13 (see A224988) are divisible by 30976. - Giovanni Resta, Oct 29 2019

Giovanni Resta, Table of n, a(n) for n = 1..7430 (terms < 10^13, first 455 terms from T. D. Noe, terms 456..1660 from Donovan Johnson)
A. S. Fraenkel, Heap games, numeration systems and sequences, arXiv:math/9809074 [math.CO], 1998; Annals of Combinatorics, 2 (1998), 197-210.
Aviezri S. Fraenkel, On the recurrence f(m+1)= b(m)*f(m)-f(m-1) and applications, Discrete Mathematics 224 (2000), pp. 273-279.
H. W. Lenstra Jr., Solving the Pell Equation, Notices of the AMS, Vol.49, No.2, Feb. 2002, p.182-192. [_Ctibor O. Zizka_, Aug 30 2008]
Eric Weisstein's World of Math, Root Mean Square

Cf. A002315, A001653, A001834, A001835, A001599, A000005, A000040, A003601, A010052, A001157, A020486, A158294, A224988.

a140480 n = a140480_list !! (n-1)
a140480_list = filter
    ((== 1) . a010052 . (\x -> a001157 x `div` a000005 x)) a020486_list
-- Reinhard Zumkeller, Jan 15 2013

rmsQ[n_] := IntegerQ[Sqrt[DivisorSigma[2, n]/DivisorSigma[0, n]]]; m = 160000; sel1 = Select[8*Range[0, m]+1, rmsQ]; sel7 = Select[8*Range[m]-1, rmsQ]; Union[sel1, sel7] (* Jean-François Alcover, Aug 31 2011, after T. D. Noe's comment *)
Select[Range[1300000],IntegerQ[RootMeanSquare[Divisors[#]]]&] (* Harvey P. Dale, Mar 24 2016 *)

More terms from T. D. Noe and Andrew Weimholt, Jul 01 2008

A003480 a(0) = 1, a(1) = 2, for n > 1, a(n) = 4a(n-1) - 2a(n-2).

Original entry on oeis.org

1, 2, 7, 24, 82, 280, 956, 3264, 11144, 38048, 129904, 443520, 1514272, 5170048, 17651648, 60266496, 205762688, 702517760, 2398545664, 8189147136, 27959497216, 95459694592, 325919783936, 1112759746560, 3799199418368, 12971278180352, 44286713884672, 151204299177984

Offset: 0

N. J. A. Sloane

Gives the number of L-convex polyominoes with n cells, that is convex polyominoes where any two cells can be connected by a path internal to the polyomino and which has at most 1 change of direction (i.e., one of the four orientation of the L). - Simone Rinaldi (rinaldi(AT)unisi.it), Feb 19 2007
Joe Keane (jgk(AT)jgk.org) observes that this sequence (beginning at 2) is "size of raises in pot-limit poker, one blind, maximum raising".
Dimensions of the graded components of the Hopf algebra of noncommutative multi-symmetric functions of level 2. For level r, the sequence would be the INVERT transform of binomial(n+r-1,n). - Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008
The sum of the numbers in the n-th row of the summatory Pascal triangle (A059576). - Ron R. King, Jan 22 2009
(1 + 2x + 7x^2 + 24x^3 + ...) = 1 / (1 - 2x - 3x^2 - 4x^3 - ...). - Gary W. Adamson, Jul 27 2009
Let M be a triangle with the odd-indexed Fibonacci numbers (1, 2, 5, 13, ...) in every column, with the leftmost column shifted upwards one row. A003480 = lim_{n->oo} M^n, the left-shifted vector considered as a sequence. The analogous operation using the even-indexed Fibonacci numbers generates A001835 starting with offset 1. - Gary W. Adamson, Jul 27 2010
a(n) is the number of generalized compositions of n when there are i+1 different types of the part i, (i=1,2,...). - Milan Janjic, Sep 24 2010
Let h(t) = (1-t)^2/(2*(1-t)^2-1) = 1/(1-(2*t + 3*t^2 + 4*t^3 + ...)),
  an o.g.f. for A003480, then
  A001003(n) = (1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=1. - Tom Copeland, Sep 06 2011
Excluding the initial 1, a(n) is the 2nd subdiagonal of A228405. - Richard R. Forberg, Sep 02 2013

G. Castiglione and A. Restivo, L-convex polyominoes: a survey, Chapter 2 of K. G. Subranian et al., eds., Formal Models, Languages and Applications, World Scientific, 2015.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe)
Daniela Battaglino, Jean-Marc Fédou, Simone Rinaldi, and Samanta Socci, The number of k-parallelogram polyominoes, FPSAC 2013 Paris, France DMTCS Proc. AS, 2013, 1143-1154.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, (an + b)-color compositions, arXiv:1707.07798 [math.CO], 2017.
Adrien Boussicault, Simone Rinaldi, and Samanta Socci, The number of directed k-convex polyominoes, arXiv preprint arXiv:1501.00872 [math.CO], 2015; Discrete Math., 343 (2020), #111731, 22 pages. See t_n.
Steve Butler, Jeongyoon Choi, Kimyung Kim, and Kyuhyeok Seo, Enumerating multiplex juggling patterns, arXiv:1702.05808 [math.CO], 2017.
Peter J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
Giuseppa Castiglione, Andrea Frosini, Emanuele Munarini, Antonio Restivo, and Simone Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.
Yumin Cho, Jaehyun Kim, Jang Soo Kim, and Nakyung Lee, Enumeration of multiplex juggling card sequences using generalized q-derivatives, arXiv:2402.09903 [math.CO], 2024. See p. 6.
Ana Djurdjevac, Vesa Kaarnioja, Claudia Schillings, and André-Alexander Zepernick, Uncertainty quantification for stationary and time-dependent PDEs subject to Gevrey regular random domain deformations, arXiv:2502.12345 [math.NA], 2025. See p. 34.
Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607.
Enrica Duchi, Simone Rinaldi, and Gilles Schaeffer, The number of Z-convex polyominoes, arXiv:math/0602124 [math.CO], 2006.
Andrea Frosini and Simone Rinaldi, An object grammar for the class of L-convex polyominoes, PU.M.A. Vol. 17 (2006), No. 1-2, pp. 97-110.
Taras Goy and Mark Shattuck, Toeplitz-Hessenberg determinant formulas for the sequence F_n-1, Online J. Anal. Comb. (2025) Vol. 19, Paper 1, 1-26. See pp. 12, 18
Yu-hong. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq (12).
Harri Hakula, Helmut Harbrecht, Vesa Kaarnioja, Frances Y. Kuo, and Ian H. Sloan, Uncertainty quantification for random domains using periodic random variables, arXiv:2210.17329 [math.NA], 2022.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 418
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Jean-Christophe Novelli and Jean-Yves Thibon, Free quasi-symmetric functions and descent algebras for wreath products and noncommutative multi-symmetric functions, arXiv:0806.3682 [math.CO], 2008.
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
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]
Index entries for sequences related to poker
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Row sums of A059576 and of A181289. Second differences of A007070.
Cf. A007052, A126764.
Cf. A000129, A001003, A001835, A006012, A029744, A145839, A145840, A145841, A228405.
Column k=2 of A261780.

a003480 n = a003480_list !! n
a003480_list = 1 : 2 : 7 : (tail $ zipWith (-)
   (tail $ map (* 4) a003480_list) (map (* 2) a003480_list))
-- Reinhard Zumkeller, Jan 16 2012, Oct 03 2011

INVERT([seq(n+1,n=1..20)]); # Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Jun 26 2008

a[0]=1; a[1]=2; a[2]=7; a[n_]:=a[n]=4*a[n-1] - 2*a[n-2]; Table[a[n],{n,0,24}] (* Jean-François Alcover, Mar 22 2011 *)
Join[{1},LinearRecurrence[{4,-2},{2,7},40]] (* Harvey P. Dale, Oct 23 2011 *)

a(n)=polcoeff((1-x)^2/(1-4*x+2*x^2)+x*O(x^n),n)

a(n)=local(x); if(n<1,n==0,x=(2+quadgen(8))^n; imag(x)+real(x)/2)

a(n) = (n+1)*a(0) + n*a(1) + ... + 3*a(n-2) + 2*a(n-1). - Amarnath Murthy, Aug 17 2002
G.f.: (1-x)^2/(1-4*x+2*x^2). - Simon Plouffe in his 1992 dissertation
a(n) = A007070(n)/2, n > 0.
G.f.: 1/( 1 - Sum_{k>=1} (k+1)*x^k ).
a(n+1)*a(n+1) - a(n+2)*a(n) = 2^n, n > 0. - D. G. Rogers, Jul 12 2004
For n > 0, a(n) = ((2+sqrt(2))^(n+1) - (2-sqrt(2))^(n+1))/(4*sqrt(2)). - Rolf Pleisch, Aug 03 2009
If the leading 1 is removed, 2, 7, 24, ... is the binomial transform of 2, 5, 12, 29, ..., which is A000129 without its first 2 terms, and the second binomial transform of 2, 3, 4, 6, ..., which is A029744, again without its leading 1. - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009
a(n) = Sum((1+p_1)*(1+p_2)*...*(1+p_m)), summation being over all compositions (p_1, p_2, ..., p_m) of n. Example: a(3)=24; indeed, the compositions of 3 are (1,1,1), (1,2), (2,1), (3) and we have 2*2*2 + 2*3 + 3*2 + 4 = 24. - Emeric Deutsch, Oct 17 2010
a(n) = Sum_{k>=0} binomial(n+2*k-1,n) / 2^(k+1). - Vaclav Kotesovec, Dec 31 2013
E.g.f.: (1 + exp(2*x)*(cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x)))/2. - Stefano Spezia, May 20 2024

A061278 a(n) = 5a(n-1) - 5a(n-2) + a(n-3) with a(1) = 1 and a(k) = 0 if k <= 0.

Original entry on oeis.org

0, 1, 5, 20, 76, 285, 1065, 3976, 14840, 55385, 206701, 771420, 2878980, 10744501, 40099025, 149651600, 558507376, 2084377905, 7779004245, 29031639076, 108347552060, 404358569165, 1509086724601, 5631988329240, 21018866592360, 78443478040201, 292755045568445

Offset: 0

Henry Bottomley, Jun 04 2001

Indices m of triangular numbers T(m) which are one-third of another triangular number: 3*T(m) = T(k); the k's are given by A001571. - Bruce Corrigan (scentman(AT)myfamily.com), Oct 31 2002
On the previous comment: for m=0 this is actually one third of the same triangular number. - Zak Seidov, Apr 07 2011
Also numbers n such that the n-th centered 24-gonal number 12*n*(n+1)+1 is a perfect square A001834(n)^2, where A001834(n) is defined by the recursion: a(0) = 1, a(1) = 5, a(n) = 4*a(n-1) - a(n-2) + 1. - Alexander Adamchuk, Apr 21 2007
Also numbers n such that RootMeanSquare(5,...,6*n-1) is an integer. - Ctibor O. Zizka, Dec 17 2008 (Corrected by Robert K. Moniot, Jul 22 2020)
Also numbers n such that n*(n+1) = Sum_{i=1..x} n+i for some x. (This does not apply to the first term.). - Gil Broussard, Dec 23 2008
From John P. McSorley, May 26 2020: (Start)
Consecutive terms (a(n-1), a(n)) = (u,v) give all points on the hyperbola u^2 - u + v^2 - v - 4*u*v = 0 in quadrant I with both coordinates an integer.
Also related to the block sizes of small multi-set designs. (End)
If a(n) white balls and a(n+1) black balls are mixed in a bag, and a pair of balls is drawn without replacement, the probability that one ball of each color is drawn is exactly 1/3. These are the only integers for which the probability is 1/3. For example, if there are 20 white balls and 76 black balls, the probability of drawing one of each is (20/96)*(76/95) + (76/96)*(20/95) = 1/3. - Elliott Line, May 13 2022

			a(2)=5 and T(5)=15 which is 1/3 of 45=T(9).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., Vol. 58, No. 2 (2020), 140-142.
Niccolò Castronuovo, On the number of fixed points of the map gamma, arXiv:2102.02739 [math.NT], 2021. Mentions this sequence.
Brian Lawrence and Will Sawin, The Shafarevich conjecture for hypersurfaces in abelian varieties, arXiv:2004.09046 [math.NT], 2020.
Ioana-Claudia Lazăr, Lucas sequences in t-uniform simplicial complexes, arXiv:1904.06555 [math.GR], 2019.
L. A. Medina and A. Straub, On multiple and infinite log-concavity, 2013.
S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons, Annales de l'institut Fourier, 62 no. 3 (2012), 937-987. - From _N. J. A. Sloane_, Dec 26 2012
Robert Phillips, A triangular number result, 2009.
Vladimir Pletser, Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers, arXiv:2101.00998 [math.NT], 2021.
Vladimir Pletser, Closed Form Equations for Triangular Numbers Multiple of Other Triangular Numbers, arXiv:2102.12392 [math.GM], 2021.
Vladimir Pletser, Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations, arXiv:2102.13494 [math.NT], 2021.
Vladimir Pletser, Congruence Properties of Indices of Triangular Numbers Multiple of Other Triangular Numbers, arXiv:2103.03019 [math.GM], 2021.
Vladimir Pletser, Searching for multiple of triangular numbers being triangular numbers, 2021.
Vladimir Pletser, Using Pell equation solutions to find all triangular numbers multiple of other triangular numbers, 2022.
Eric Weisstein, Centered Polygonal Number.
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Cf. A001075, A001353, A001571, A001834, A001835, A079935, A101265. Also cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

I:=[0, 1]; [n le 2 select I[n] else 4*Self(n-1) - Self(n-2) + 1: n in [1..30]]; // Vincenzo Librandi, Dec 23 2012

f:= gfun:-rectoproc({a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3),a(1)=1,a(0)=0,a(-1)=0},a(n),remember):
map(f, [$0..50]); # Robert Israel, Jun 05 2015

CoefficientList[Series[x/(1 - 5*x + 5*x^2 - x^3), {x, 0, nn}], x] (* T. D. Noe, Jun 04 2012 *)
LinearRecurrence[{5,-5,1},{0,1,5},30] (* Harvey P. Dale, Dec 23 2012 *)

M = [1, 1, 0; 1, 3, 1; 0, 1, 1]; for(i=1, 30, print1(([1, 0, 0]*M^i)[3], ",")) \\ Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jan 25 2005

a(n) = 4*a(n-1) - a(n-2) + 1.
a(n) = A001075(n) - a(n-1) - 1.
a(n) = (A001835(n+1) - 1)/2 = (A001353(n+1) - A001353(n) - 1)/2.
a(n) = a(n-1) + A001353(n), i.e., partial sum of A001353.
From Bruce Corrigan (scentman(AT)myfamily.com), Oct 31 2002: (Start)
a(n+2) = 4*a(n+1) - a(n) + 1 for a(0)=0, a(1)=1.
G.f.: x/((1 - x)*(1 - 4*x + x^2)).
a(n) = (1/12)*((3 - sqrt(3))*(2 - sqrt(3))^n + (3 + sqrt(3))*(2 + sqrt(3))^n-6). (End)
a(n) = (1/12)*(A003500(n) + A003500(n+1)-6). - Mario Catalani (mario.catalani(AT)unito.it), Apr 11 2003
a(n+1) = Sum_{k=0..n} U(k, 2) = Sum_{k=0..n} S(k, 4), where U(n,x) and S(n,x) are Chebyshev polynomials. - Paul Barry, Nov 14 2003
G.f.: x/(1 - 5*x + 5*x^2 - x^3).
a(n) = A079935(n+1) + A001571(n) for n>0, a(0)=0. - Gerry Martens, Jun 05 2015
a(n)*a(n-2) = a(n-1)*(a(n-1) - 1) for n>1. - Bruno Berselli, Nov 29 2016
From John P. McSorley, May 25 2020: (Start)
a(n)^2 - a(n) + a(n-1)^2 - a(n-1) - 4*a(n)*a(n-1) = 0.
a(n) = A001834(n-1) + a(n-2). (End)
(T(a(n)-1) + T(a(n+1)-1))/T(a(n) + a(n+1) - 1) = 2/3 where T(i) is the i-th triangular number. - Robert K. Moniot, Oct 11 2020
E.g.f.: exp(x)*(exp(x)*(3*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) - 3)/6. - Stefano Spezia, Feb 05 2021
a(n) = A101265(n) - 1. - Jon E. Schoenfield, Jan 01 2022

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jan 25 2005

A052530 a(n) = 4*a(n-1) - a(n-2), with a(0) = 0, a(1) = 2.

Original entry on oeis.org

0, 2, 8, 30, 112, 418, 1560, 5822, 21728, 81090, 302632, 1129438, 4215120, 15731042, 58709048, 219105150, 817711552, 3051741058, 11389252680, 42505269662, 158631825968, 592022034210, 2209456310872, 8245803209278, 30773756526240

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

a(n-1) and a(n+1) are the solutions for c if b = a(n) in (b^2 + c^2)/(b*c + 1) = 4 and there are no other pairs of solutions apart from consecutive pairs of terms in this sequence. Cf. A061167. - Henry Bottomley, Apr 18 2001
a(n)^2 for n >= 1 gives solutions to A007913(3*x+4) = A007913(x). - Benoit Cloitre, Apr 07 2002
For all terms k of the sequence, 3*k^2 + 4 is a perfect square. Limit_{n->oo} a(n)/a(n-1) = 2 + sqrt(3). - Gregory V. Richardson, Oct 06 2002
a(n) = the number of compositions of the integer 2*n into even parts, where each part 2*i comes in 2*i colors. (Dedrickson, Theorem 3.2.6) An example is given below. Cf. A052529, A095263. - Peter Bala, Sep 17 2013
Except for an initial 1, this is the p-INVERT of (1, 1, 1, 1, 1, ...) for p(S) = 1 - 2*S - 2*S^2; see A291000.  - Clark Kimberling, Aug 24 2017
a(n+1) is the number of spanning trees of the graph P_n, where P_n is a 2 X n grid with two additional vertices, u and v, where u is adjacent to (1,1) and (2,1), and v is adjacent to (1,n) and (2,n). - Kevin Long, May 04 2018
a(n) is also the output of Tesler's formula for the number of perfect matchings of an m X n Mobius band where m is even and n is odd, specialized to m=2. (The twist is on the length-n side.) - Sarah-Marie Belcastro, Feb 15 2022
In general, values of x and y which satisfy (x^2 + y^2)/(x*y + 1) = k^2 are any two adjacent terms of a second-order recurrence with initial terms 0 and k and signature (k^2,-1). This can also be expressed as a first-order recurrence a(n+1) = (k^2*a(n) + sqrt((k^4-4)*a(n)^2 + 4*k^2))/2, n > 1. - Gary Detlefs, Feb 27 2024

			Colored compositions. a(2) = 8: There are two compositions of 4 into even parts, namely 4 and 2 + 2. Using primes to indicate the coloring of parts, the 8 colored compositions are 4, 4', 4'', 4''', 2 + 2, 2 + 2', 2' + 2 and 2' + 2'. - _Peter Bala_, Sep 17 2013

T. D. Noe, Table of n, a(n) for n = 0..200
K. Andersen, L. Carbone, and D. Penta, Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, (an + b)-color compositions, arXiv:1707.07798 [math.CO], 2017.
Niccolò Castronuovo, On the number of fixed points of the map gamma, arXiv:2102.02739 [math.NT], 2021. Mentions this sequence.
C. R. Dedrickson III, Compositions, Bijections, and Enumerations Thesis, 2012, Jack N. Averitt College of Graduate Studies, Georgia Southern University.
J.-P. Ehrmann et al., Problem POLYA002, Integer pairs (x,y) for which (x^2+y^2)/(1+pxy) is an integer.
F. Goebel and A. A. Jagers, On a conjecture of Tutte concerning minimal tree numbers, J. Combin. Theory Ser. B 26 (1979), no. 3, 346-348. MR0535948 (80m:05064). [From _N. J. A. Sloane_, Feb 20 2012]
A. F. Horadam, Basic Properties of a Certain Generalized Sequence of Numbers, Fibonacci Quarterly, Vol. 3, No. 3, 1965, pp. 161-176.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 460
N. J. A. Sloane, Transforms
G. Tesler, Matchings in graphs on non-orientable surfaces, Journal of Combinatorial Theory B, 78(2000), 198-231.
Index entries for linear recurrences with constant coefficients, signature (4,-1).

Cf. A001353, A007913, A003699, A217233.

Cf. A052529, A095263.

a052530 n = a052530_list !! n
a052530_list =
   0 : 2 : zipWith (-) (map (* 4) $ tail a052530_list) a052530_list
-- Reinhard Zumkeller, Sep 29 2011

I:=[0,2]; [n le 2 select I[n] else 4*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Feb 25 2019

spec := [S,{S=Sequence(Prod(Union(Z,Z),Sequence(Z),Sequence(Z)))},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
s := sqrt(3): a := n -> ((2-s)^n-(s+2)^n)/(s*(s-2)*(s+2)):
seq(simplify(a(n)), n=0..24); # Peter Luschny, Apr 28 2020

p=1; c=2; a[0]=0; a[1]=c; a[n_]:=a[n]=p*c^2*a[n-1]-a[n-2]; Table[a[n], {n, 0, 20}]
NestList[2 # + Sqrt[4 + 3 #^2]&, 0, 200] (* Zak Seidov, Mar 31 2011 *)
LinearRecurrence[{4, -1}, {0, 2}, 25] (* T. D. Noe, Jan 09 2012 *)
CoefficientList[Series[2x/(1-4x+x^2),{x,0,30}],x] (* Harvey P. Dale, May 31 2023 *)

{ polya002(p,c,m) = local(v,w,j,a); w=0; print1(w,", "); v=c; print1(v,", "); j=1; while(j<=m,a=p*c^2*v-w; print1(a,", "); w=v; v=a; j++) };
polya002(1,2,25)

my(x='x+O('x^30)); concat([0], Vec(2*x/(1-4*x+x^2))) \\ G. C. Greubel, Feb 25 2019

first(n) = n = max(n, 2); my(res = vector(n)); res[1] = 0; res[2] = 2; for(i = 3, n, res[i] = 4 * res[i-1] - res[i-2]); res \\ David A. Corneth, Apr 28 2020

(2*x/(1-4*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 25 2019

G.f.: 2*x/(1 - 4*x + x^2).
Invert transform of even numbers: a(n) = 2*Sum_{k=1..n} k*a(n-k). - Vladeta Jovovic, Apr 27 2001
From Gregory V. Richardson, Oct 06 2002: (Start)
a(n) = Sum_{alpha} -(1/3)*(-1 + 2*alpha)*alpha^(-1 - n), alpha = root of (1 - 4*Z + Z^2).
a(n) = (((2+sqrt(3))^(n+1) - (2-sqrt(3))^(n+1)) - ((2+sqrt(3))^n - (2-sqrt(3))^n) + ((2+sqrt(3))^(n-1) - (2-sqrt(3))^(n-1)))/(3*sqrt(3)). (End)
a(n) = A071954(n) - 2. - N. J. A. Sloane, Feb 20 2005
a(n) = (2*sinh(2n*arcsinh(1/sqrt(2))))/sqrt(3). - Herbert Kociemba, Apr 24 2008
a(n) = 2*A001353(n). - R. J. Mathar, Oct 26 2009
a(n) = ((3 - 2*sqrt(3))/3)*(2 - sqrt(3))^(n - 1) + ((3 + 2*sqrt(3))/3)*(2 + sqrt(3))^(n - 1). - Vincenzo Librandi, Nov 20 2010
a(n) = floor((2 + sqrt(3))^n/sqrt(3)). - Zak Seidov, Mar 31 2011
a(n) = ((2 + sqrt(3))^n - (2 - sqrt(3))^n)/sqrt(3). (See Horadam for construction.) - Johannes Boot, Jan 08 2012
a(n) = A217233(n) + A217233(n-1) with A217233(-1) = -1. - Bruno Berselli, Oct 01 2012
a(n) = A001835(n+1) - A001835(n). - Kevin Long, May 04 2018
E.g.f.: (exp((2 + sqrt(3))*x) - exp((2 - sqrt(3))*x))/sqrt(3). - Franck Maminirina Ramaharo, Nov 12 2018
a(n+1) = 2*a(n) + sqrt(3*a(n)^2 + 4), n > 1. - Gary Detlefs, Feb 27 2024

More terms from James Sellers, Jun 06 2000
Edited by N. J. A. Sloane, Nov 11 2006
a(0) changed to 0 and entry revised accordingly by Max Alekseyev, Nov 15 2007
Signs in definition corrected by John W. Layman, Nov 20 2007

A005246 a(n) = (1 + a(n-1)*a(n-2))/a(n-3), a(0) = a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 7, 11, 26, 41, 97, 153, 362, 571, 1351, 2131, 5042, 7953, 18817, 29681, 70226, 110771, 262087, 413403, 978122, 1542841, 3650401, 5757961, 13623482, 21489003, 50843527, 80198051, 189750626, 299303201, 708158977, 1117014753

Offset: 0

N. J. A. Sloane

For n >= 4 we have the linear recurrence a(n) = 4*a(n-2) - a(n-4). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 04 2001
Integer solutions to the equation floor(sqrt(3)*x^2) = x*floor(sqrt(3)*x). - Benoit Cloitre, Mar 18 2004
For n > 2, a(n) is the smallest integer > a(n-1) such that sqrt(3)*a(n) is closer to and greater than an integer than sqrt(3)*a(n-1). I.e., a(n) is the smallest integer > a(n-1) such that frac(sqrt(3)*a(n)) < frac(sqrt(3)*a(n-1)). - Benoit Cloitre, Jan 20 2003
The lower principal and intermediate convergents to 3^(1/2), beginning with 1/1, 3/2, 5/3, 12/7, 19/11, form a strictly increasing sequence; essentially, numerators=A143643 and denominators=A005246. - Clark Kimberling, Aug 27 2008
This sequence is a particular case of the following situation: a(0)=1, a(1)=a, a(2)=b with the recurrence relation a(n+3)=(a(n+2)*a(n+1)+q)/a(n) where q is given in Z to have Q=(a*b^2+q*b+a+q)/(a*b) itself in Z. The g.f. is f: f(z)=(1+a*z+(b-Q)*z^2+(a*b+q-a*Q)*z^3)/(1-Q*z^2+z^4); so we have the linear recurrence: a(n+4)=Q*a(n+2)-a(n). The general form of a(n) is given by: a(2*m) = Sum_{p=0..floor(m/2)} (-1)^p*binomial(m-p,p)*Q^(m-2*p) + (b-Q)*Sum_{p=0..floor((m-1)/2)} (-1)^p*binomial(m-1-p,p)*Q^(m-1-2*p) and a(2*m+1) = a*Sum_{p=0..floor(m/2)} (-1)^p*binomial(m-p,p)*Q^(m-2*p) + (a*b+q-a*Q)*Sum_{p=0..floor((m-1)/2)} (-1)^p*binomial(m-1-p,p)*Q^(m-1-2*p). - Richard Choulet, Feb 24 2010
a(n) for n > 1 are the integer square roots of (floor(m^2/3)+1), where the values of m are given by A143643.  Also see A082630. - Richard R. Forberg, Nov 14 2013
The a(n) = (1 + a(n-1)*a(n-2))/a(n-3) recursion has the Laurent property. If a(0), a(1), a(2) are variables, then a(n) is a Laurent polynomial (a rational function with a monomial denominator). - Michael Somos, Feb 27 2019

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 7*x^5 + 11*x^6 + 26*x^7 + 41*x^8 + ...
From _Richard Choulet_, Feb 24 2010: (Start)
a(4) = 4^2 - 4^0 - 3*4^1 = 3.
a(7) = 4^3 - 4*binomial(2,1) - 2*(4^2-1) = 26. (End)

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
Reid Barton, A combinatorial interpretation of the sequence 1, 1, 2, 6, 21, 77, ...
Reid Barton, A combinatorial interpretation of the sequence 1, 1, 2, 6, 21, 77, ..., [Annotated scanned copy]
Peter Cameron's Blog, The ADE affair, 3, Posted 23/06/2011.
T. Crilly, Double sequences of positive integers, Math. Gaz., 69 (1985), 263-271.
Enrica Duchi, Andrea Frosini, Renzo Pinzani and Simone Rinaldi, A Note on Rational Succession Rules, J. Integer Seqs., Vol. 6, 2003.
R. K. Guy, Letter to N. J. A. Sloane, Feb 1986
Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.
Valentin Ovsienko, Serge Tabachnikov, Dual numbers, weighted quivers, and extended Somos and Gale-Robinson sequences, arXiv:1705.01623 [math.CO], 2017. See p. 10.
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
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Bisections are A001835 and A001075.
Cf. A101265. Row sums of A211956.
Cf. A001353.

a005246 n = a005246_list !! n
a005246_list = 1 : 1 : 1 : map (+ 1) (zipWith div
   (zipWith (*) (drop 2 a005246_list) (tail a005246_list)) a005246_list)
-- Reinhard Zumkeller, Mar 07 2012

A005246:=-(-1-z+2*z**2+z**3)/(1-4*z**2+z**4); # Conjectured by Simon Plouffe in his 1992 dissertation. Gives sequence except for one of the leading 1's.
for q from 1 to 10 do :a:=1:b:=1:Q:=(a*b^2+q*b+a+q)/(a*b): for m from 0 to 15 do U(m):=sum((-1)^p*binomial(m-p,p)*Q^(m-2*p),p=0..floor(m/2))+(b-Q)*sum((-1)^p*binomial(m-1-p,p)*Q^(m-1-2*p),p=0..floor((m-1)/2)):od: for m from 0 to 15 do V(m):=a*sum((-1)^p*binomial(m-p,p)*Q^(m-2*p),p=0..floor(m/2))+(a*b+q-a*Q)*sum((-1)^p*binomial(m-1-p,p)*Q^(m-1-2*p),p=0..floor((m-1)/2)):od:for m from 0 to 15 do W(2*m):=U(m):od:for m from 0 to 14 do W(2*m+1):=V(m):od:seq(W(m),m=0..30):od; # Richard Choulet, Feb 24 2010

RecurrenceTable[{a[0]==a[1]==a[2]==1,a[n]==(1+a[n-1]a[n-2])/a[n-3]},a,{n,40}] (* Harvey P. Dale, May 28 2013 *)
a[n_] := Cosh[(n-1)*ArcSinh[1/Sqrt[2]]]*If[EvenQ[n], Sqrt[2/3], 1]; Table[a[n] // FunctionExpand, {n, 0, 34}] (* Jean-François Alcover, Dec 10 2014, after Peter Bala *)
a[ n_] := With[{m = If[ n < 0, 2 - n, n]}, SeriesCoefficient[ (1 + x - 3 x^2 - 2 x^3) / (1 - 4 x^2 + x^4), {x, 0, m}]]; (* Michael Somos, Feb 10 2017 *)

{a(n) = if( n<0, n = 2 - n); polcoeff((1 + x - 3*x^2 - 2*x^3) / (1 - 4*x^2 + x^4) + x * O(x^n), n)}; /* Michael Somos, Nov 15 2006 */

{a(n) = real( (2 + quadgen(12))^(n\2) * if( n%2, 1, 1 - 1 / quadgen(12)) )}; /* Michael Somos, May 24 2012 */

G.f.: (1 + x - 3*x^2 - 2*x^3)/(1 - 4*x^2 + x^4).
Limit_{n->oo} a(2n+1)/a(2n) = (3+sqrt(3))/3 = 1.5773502...; lim_{n->oo} a(2n)/a(2n-1) = (3+sqrt(3))/2 = 2.3660254.... - Benoit Cloitre, Aug 07 2002
A101265(n) = a(n)*a(n+1). - Franklin T. Adams-Watters, Apr 24 2006
a(n) = a(2-n) for all n in Z. - Michael Somos, Nov 15 2006
a(2*n + 1) = A001075(n). a(2*n) = A001835(n). a(2*n + 1) - a(2*n) = a(2*n + 2) - a(2*n + 1) = A001353(n). - Michael Somos, May 24 2012
For n > 2: a(n) = a(n-1) + Sum_{k=1..floor((n-1)/2)} a(2*k). - Reinhard Zumkeller, Dec 16 2007
From Richard Choulet, Feb 24 2010: (Start)
a(2*m) = Sum_{p=0..floor(m/2)} (-1)^p*binomial(m-p,p)*4^(m-2*p) - 3*Sum_{p=0..floor((m-1)/2)} (-1)^p*binomial(m-1-p,p)*4^(m-1-2*p).
a(2*m+1) = Sum_{p=0..floor(m/2)} (-1)^p*binomial(m-p,p)*4^(m-2*p) - 2*Sum_{p=0..floor((m-1)/2)} (-1)^p*binomial(m-1-p,p)*4^(m-1-2*p). (End)
From Tim Monahan, Jul 01 2011: (Start)
Closed form without extra leading 1: ((sqrt(6)+3)*(sqrt(2+sqrt(3))^n+(sqrt(2-sqrt(3))^n))+(3-sqrt(6))*((-sqrt(2+sqrt(3)))^n+(-sqrt(2-sqrt(3)))^n))/12.
Closed form with extra leading 1: ((6+3*sqrt(6)-2*sqrt(3)-3*sqrt(2))*(sqrt(2+sqrt(3))^n)+(6+3*sqrt(6)+2*sqrt(3)+3*sqrt(2))*(sqrt(2-sqrt(3))^n)+(6-3*sqrt(6)-2*sqrt(3)+3*sqrt(2))*((-sqrt(2+sqrt(3)))^n)+(6-3*sqrt(6)+2*sqrt(3)-3*sqrt(2))*((-sqrt(2-sqrt(3)))^n))/24. (End)
a(2*n+2) = Sum_{k = 0..n} 2^k*binomial(n+k,2*k); a(2*n+1) = Sum_{k = 0..n} n/(n+k)*2^k*binomial(n+k,2*k) for n >= 1. Row sums of A211956. - Peter Bala, May 01 2012
a(n) = ((sqrt(2)+sqrt(3)+(-1)^n*(sqrt(2)-sqrt(3)))*sqrt(2+(2-sqrt(3))^n*(2+ sqrt(3))-(-2+sqrt(3))*(2+ sqrt(3))^n))/(4*sqrt(3)). - Gerry Martens, Jun 06 2015
0 = a(n) - 2*a(n+1) + a(n+2) if n is even, 0 = a(n) - 3*a(n+1) + a(n+2) if n is odd for all n in Z. - Michael Somos, Feb 10 2017

More terms from Michael Somos, Aug 01 2001

A006253 Number of perfect matchings (or domino tilings) in C_4 X P_n.

Original entry on oeis.org

1, 2, 9, 32, 121, 450, 1681, 6272, 23409, 87362, 326041, 1216800, 4541161, 16947842, 63250209, 236052992, 880961761, 3287794050, 12270214441, 45793063712, 170902040409, 637815097922, 2380358351281, 8883618307200, 33154114877521, 123732841202882

Offset: 0

N. J. A. Sloane

Number of tilings of a box with sides 2 X 2 X n in R^3 by boxes of sides 2 X 1 X 1 (3-dimensional dominoes). - Frans J. Faase
The number of domino tilings in A006253, A004003, A006125 is the number of perfect matchings in the relevant graphs. There are results of Jockusch and Ciucu that if a planar graph has a rotational symmetry then the number of perfect matchings is a square or twice a square - this applies to these 3 sequences. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001
Also stacking bricks.
a(n)*(-1)^n = (1-T(n+1,-2))/3, n>=0, with Chebyshev's polynomials T(n,x) of the first kind, is the r=-2 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found. - Wolfdieter Lang, Oct 18 2004
Partial sums of A217233. - Bruno Berselli, Oct 01 2012
The sequence is the case P1 = 2, P2 = -8, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Apr 03 2014

			G.f. = 1 + 2*x + 9*x^2 + 32*x^3 + 121*x^4 + 450*x^5 + ... - _Michael Somos_, Mar 17 2022

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 360.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Butler and S. Osborne, Counting tilings by taking walks, preprint, 2012; J. Combin. Math. Combin. Comput. 88 2014 17-25. - From _N. J. A. Sloane_, Dec 27 2012
M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97
D. Deford, Seating rearrangements on arbitrary graphs, preprint 2013; involve, Vol. 7 (2014), No. 6, 787-805.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
Charles H. Jepsen, Packing a box with bricks, Math. Mag. 64 (2) (1991) 92-97, Table 1
W. Jockusch, Perfect matchings and perfect squares J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.
R. J. Mathar, Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices, arXiv:1406.7788 (2014), eq. (36).
Valcho Milchev and Tsvetelina Karamfilova, Domino tiling in grid - new dependence, arXiv:1707.09741 [math.HO], 2017.
László Németh, Tilings of (2 X 2 X n)-board with colored cubes and bricks, arXiv:1909.11729 [math.CO], 2019.
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.
Radovan Potůček, The number of fillings a 2 X 2 X n prism with 1 X 1 X 2 prisms, Equations (2024) Vol. 3, 104-114.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for sequences related to dominoes
Index entries for sequences related to bricks
Index entries for linear recurrences with constant coefficients, signature (3,3,-1).

Cf. A002530, A004003, A006125, A217233 (first differences), A109437 (partial sums).
Column k=2 of A181206, A189650, A233308.
Cf. A100047.

a:=[1,2,9];; for n in [4..30] do a[n]:=3*a[n-1]+3*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Nov 16 2018

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)/(1-3*x-3*x^2+x^3))); // G. C. Greubel, Nov 16 2018

CoefficientList[Series[(1 - x)/(1 - 3 x - 3 x^2 + x^3), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 15 2012 *)
RecurrenceTable[{a[1] == 1, a[2] == 2, a[n] == BitXor[1, a[n - 1]]^2/a[n - 2]}, a, {n, 30}] (* Jon Maiga, Nov 16 2018 *)
LinearRecurrence[{3,3,-1}, {1,2,9}, 30] (* G. C. Greubel, Nov 16 2018 *)
a[ n_] := (-1)^n * ChebyshevU[n, Sqrt[-1/2]]^2; (* Michael Somos, Mar 17 2022 *)

a(n)=(sqrt(3)+2)^(n+1) \/ 6 \\ Charles R Greathouse IV, Aug 18 2016

a(n)=([0,1,0; 0,0,1; -1,3,3]^n*[1;2;9])[1,1] \\ Charles R Greathouse IV, Aug 18 2016

Vec((1 - x) / ((1 + x)*(1 - 4*x + x^2)) + O(x^40)) \\ Colin Barker, Dec 16 2017

{a(n) = simplify((-1)^n * polchebyshev(n, 2, quadgen(-8)/2)^2)}; /* Michael Somos, Mar 17 2022 */

s=((1-x)/(1-3*x-3*x^2+x^3)).series(x,30); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 16 2018

G.f.: (1-x)/((1+x)*(1-4*x+x^2)) = (1-x)/(1-3*x-3*x^2+x^3). - Simon Plouffe in his 1992 dissertation; typo corrected by Vincenzo Librandi, Oct 15 2012
Nearest integer to (1/6)*(2+sqrt(3))^(n+1). - Don Knuth, Jul 15 1995
For n >= 4, a(n) = 3a(n-1) + 3a(n-2) - a(n-3). - Avi Peretz (njk(AT)netvision.net.il), Mar 30 2001
For n >= 3, a(n) = 4a(n-1) - a(n-2) + 2*(-1)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 14 2001
From Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 11 2001: The values are a(1) = 2 * 1^2, a(2) = 3^2, a(3) = 2 * 4^2, a(4) = 11^2, a(5) = 2 * 15^2, ... and in general for odd n a(n) is twice a square, for even n a(n) is a square. If we define b(n) by b(n) = sqrt(a(n)) for even n, b(n) = sqrt(a(n)/2) for odd n then apart from the first 2 elements b(n) is A002530(n+1).
a(n) + a(n+1) = A001835(n+2). - R. J. Mathar, Dec 06 2013
From Peter Bala, Apr 03 2014: (Start)
a(n) = |U(n,i/sqrt(2))|^2 where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n-1) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 2; 1, 1] and T(n,x) denotes the Chebyshev polynomial of the first kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)
a(n) = (2*(-1)^n + (2-sqrt(3))^(1+n) + (2+sqrt(3))^(1+n)) / 6. - Colin Barker, Dec 16 2017
a(n) = (1 XOR a(n-1))^2/a(n-2). - Jon Maiga, Nov 16 2018
a(n) = a(-2-n) for all n in Z. - Michael Somos, Mar 17 2022
INVERT transform of sequence p(n), n > 0, where p is the number of nonreducible tilings by height of 2 X 2 X n using dicubes; p is (2, 5, 4, 4, 4, 4...). - Nicolas Bělohoubek, Jun 04 2024

A094954 Array T(k,n) read by antidiagonals. G.f.: x(1-x)/(1-kx+x^2), k>1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 13, 1, 1, 5, 19, 41, 34, 1, 1, 6, 29, 91, 153, 89, 1, 1, 7, 41, 169, 436, 571, 233, 1, 1, 8, 55, 281, 985, 2089, 2131, 610, 1, 1, 9, 71, 433, 1926, 5741, 10009, 7953, 1597, 1, 1, 10, 89, 631, 3409, 13201, 33461, 47956, 29681

Offset: 1

Ralf Stephan, May 31 2004

Also, values of polynomials with coefficients in A098493 (see Fink et al.). See A098495 for negative k.

Number of dimer tilings of the graph S_{k-1} X P_{2n-2}.

			1,1,1,1,1,1,1,1,1,1,1,1,1,1, ...
1,2,5,13,34,89,233,610,1597, ...
1,3,11,41,153,571,2131,7953, ...
1,4,19,91,436,2089,10009,47956, ...
1,5,29,169,985,5741,33461,195025, ...
1,6,41,281,1926,13201,90481,620166, ...

Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114.
Elizabeth Wilmer, A note on Stephan's conjecture 87
Elizabeth Wilmer, A note on Stephan's conjecture 87 [cached copy]

Rows are first differences of rows in array A073134.
Rows 2-14 are A000012, A001519, A079935/A001835, A004253, A001653, A049685, A070997, A070998, A072256, A078922, A077417, A085260, A001570. Other rows: A007805 (k=18), A075839 (k=20), A077420 (k=34), A078988 (k=66).
Columns include A028387. Diagonals include A094955, A094956. Antidiagonal sums are A094957.

max = 14; row[k_] := Rest[ CoefficientList[ Series[ x*(1-x)/(1-k*x+x^2), {x, 0, max}], x]]; t = Table[ row[k], {k, 2, max+1}]; Flatten[ Table[ t[[k-n+1, n]], {k, 1, max}, {n, 1, k}]] (* Jean-François Alcover, Dec 27 2011 *)

PARI
```
T(k,n)=polcoeff(x*(1-x)/(1-k*x+x*x),n)
```

Recurrence: T(k, 1) = 1, T(k, 2) = k-1, T(k, n) = kT(k, n-1) - T(k, n-2).
For n>3, T(k, n) = [k(k-2) + T(k, n-1)T(k, n-2)] / T(k, n-3).
T(k, n+1) = S(n, k) - S(n-1, k) = U(n, k/2) - U(n-1, k/2), with S, U = Chebyshev polynomials of second kind.
T(k+2, n+1) = Sum[i=0..n, k^(n-i) * C(2n-i, i)] (from comments by Benoit Cloitre).

A079935 a(n) = 4*a(n-1) - a(n-2) with a(1) = 1, a(2) = 3.

Original entry on oeis.org

1, 3, 11, 41, 153, 571, 2131, 7953, 29681, 110771, 413403, 1542841, 5757961, 21489003, 80198051, 299303201, 1117014753, 4168755811, 15558008491, 58063278153, 216695104121, 808717138331, 3018173449203, 11263976658481, 42037733184721, 156886956080403

Offset: 1

Benoit Cloitre and Paul D. Hanna, Jan 20 2003

See A001835 for another version.
Greedy frac multiples of sqrt(3): a(1)=1, Sum_{n>0} frac(a(n)*x) < 1 at x=sqrt(3).
The n-th greedy frac multiple of x is the smallest integer that does not cause Sum_{k=1..n} frac(a(k)*x) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of x.
Binomial transform of A002605. - Paul Barry, Sep 17 2003
In general, Sum_{k=0..n} binomial(2n-k,k)*j^(n-k) = (-1)^n* U(2n, i*sqrt(j)/2), i=sqrt(-1). - Paul Barry, Mar 13 2005
The Hankel transform of this sequence is [1,2,0,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007
From Richard Choulet, May 09 2010: (Start)
This sequence is a particular case of the following situation:
a(0)=1, a(1)=a, a(2)=b with the recurrence relation a(n+3) = (a(n+2)*a(n+1)+q)/a(n)
where q is given in Z to have Q = (a*b^2 + q*b + a + q)/(a*b) itself in Z.
The g.f is f: f(z) = (1 + a*z + (b-Q)*z^2 + (a*b + q - a*Q)*z^3)/(1 - Q*z^2 + z^4);
so we have the linear recurrence: a(n+4) = Q*a(n+2) - a(n).
The general form of a(n) is given by:
a(2*m) = Sum_{p=0..floor(m/2)} (-1)^p*binomial(m-p,p)*Q^(m-2*p) + (b-Q)*Sum_{p=0..floor((m-1)/2)} (-1)^p*binomial(m-1-p,p)*Q^(m-1-2*p) and
a(2*m+1) = a*Sum_{p=0..floor(m/2)} (-1)^p*binomial(m-p,p)*Q^(m-2*p) + (a*b+q-a*Q)*Sum_{p=0..floor((m-1)/2)} (-1)^p*binomial(m-1-p,p)*Q^(m-1-2*p).
(End)
x-values in the solution to 3*x^2 - 2 = y^2. - Sture Sjöstedt, Nov 25 2011
From Wolfdieter Lang, Oct 12 2020: (Start)
[X(n) = S(n, 4) - S(n-1, 4), Y(n) = X(n-1)] gives all positive solutions of X^2 + Y^2 - 4*X*Y = -2, for n = -oo..+oo, where the Chebyshev S-polynomials are given in A049310, with S(-1, 0) = 0, and S(-|n|, x) = - S(|n|-2, x), for |n| >= 2.
This binary indefinite quadratic form has discriminant D = +12. There is only this family representing -2 properly with X and Y positive, and there are no improper solutions.
See also the preceding comment by Sture Sjöstedt.
See the formula for a(n) = X(n-1), for n >= 1, in terms of S-polynomials below.
This comment is inspired by a paper by Robert K. Moniot (private communication). See his Oct 04 2020 comment in A027941 related to the case of x^2 + y^2 - 3*x*y = -1 (special Markov solutions). (End)
a(n) is also the output of Tesler's formula for the number of perfect matchings of an m x n Mobius band where m and n are both even, specialized to m=2. (The twist is on the length-n side.) - Sarah-Marie Belcastro, Feb 15 2022

			a(4) = 41 since frac(1*x) + frac(3*x) + frac(11*x) + frac(41*x) < 1, while frac(1*x) + frac(3*x) + frac(11*x) + frac(k*x) > 1 for all k > 11 and k < 41.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Dalen Dockery, Marie Jameson, and Samuel Wilson, d-Fold Partition Diamonds, arXiv:2307.02579 [math.NT], 2023.
Tanya Khovanova, Recursive Sequences
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), pp. 609-640.
G. Tesler, Matchings in graphs on non-orientable surfaces, Journal of Combinatorial Theory B, 78(2000), 198-231.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory, Vol. 7, No. 5 (2011), pp. 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012), The John Selfridge Memorial Volume.
Index entries for linear recurrences with constant coefficients, signature (4,-1).

Cf. A002530 (denominators of convergents to sqrt(3)), A079934, A079936, A001353.
Cf. A001835 (same except for the first term).
Row 4 of array A094954.
Cf. similar sequences listed in A238379.

a079935 n = a079935_list !! (n-1)
a079935_list =
   1 : 3 : zipWith (-) (map (4 *) $ tail a079935_list) a079935_list
-- Reinhard Zumkeller, Aug 14 2011

I:=[1,3]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jun 06 2015

f:= gfun:-rectoproc({a(n) = 4*a(n-1) - a(n-2),a(1)=1,a(2)=3}, a(n), remember):
seq(f(n),n=1..30); # Robert Israel, Jun 05 2015

a[n_] := (MatrixPower[{{1, 2}, {1, 3}}, n].{{1}, {1}})[[1, 1]]; Table[ a[n], {n, 0, 23}] (* Robert G. Wilson v, Jan 13 2005 *)
LinearRecurrence[{4,-1},{1,3},30] (* or *) CoefficientList[Series[ (1-x)/(1-4x+x^2),{x,0,30}],x]  (* Harvey P. Dale, Apr 26 2011 *)
a[n_] := Sqrt[2/3] Cosh[(-1 - 2 n) ArcCsch[Sqrt[2]]];
Table[Simplify[a[n-1]], {n, 1, 12}] (* Peter Luschny, Oct 13 2020 *)

a(n)=([0,1; -1,4]^(n-1)*[1;3])[1,1] \\ Charles R Greathouse IV, Mar 18 2017

my(x='x+O('x^30)); Vec((1-x)/(1-4*x+x^2)) \\ G. C. Greubel, Feb 25 2019

[lucas_number1(n,4,1)-lucas_number1(n-1,4,1) for n in range(1, 25)] # Zerinvary Lajos, Apr 29 2009

For n > 0, a(n) = ceiling( (2+sqrt(3))^n/(3+sqrt(3)) ).
From Paul Barry, Sep 17 2003: (Start)
G.f.: (1-x)/(1-4*x+x^2).
E.g.f.: exp(2*x)*(sinh(sqrt(3)*x)/sqrt(3) + cosh(sqrt(3)*x)).
a(n) = ( (3+sqrt(3))*(2+sqrt(3))^n + (3-sqrt(3))*(2-sqrt(3))^n )/6 (offset 0). (End)
a(n) = Sum_{k=0..n} binomial(2*n-k, k)*2^(n-k). - Paul Barry, Jan 22 2005 [offset 0]
a(n) = (-1)^n*U(2*n, i*sqrt(2)/2), U(n, x) Chebyshev polynomial of second kind, i=sqrt(-1). - Paul Barry, Mar 13 2005 [offset 0]
a(n) = Jacobi_P(n,-1/2,1/2,2)/Jacobi_P(n,-1/2,1/2,1). - Paul Barry, Feb 03 2006 [offset 0]
a(n) = sqrt(2+(2-sqrt(3))^(2*n-1) + (2+sqrt(3))^(2*n-1))/sqrt(6). - Gerry Martens, Jun 05 2015
a(n) = (1/2 + sqrt(3)/6)*(2-sqrt(3))^n + (1/2 - sqrt(3)/6)*(2+sqrt(3))^n. - Robert Israel, Jun 05 2015
a(n) = S(n-1,4) - S(n-2,4) = (-1)^(n-1)*S(2*(n-1), i*sqrt(2)), with Chebyshev S-polynomials (A049310), the imaginary unit i, S(-1, x) = 0, for n >= 1. See also the formula above by Paul Barry (with offset 0). - Wolfdieter Lang, Oct 12 2020
a(n) = sqrt(2/3)*cosh((-1 - 2*n) arccsch(sqrt(2))), where arccsch is the inverse hyperbolic cosecant function (with offset 0). - Peter Luschny, Oct 13 2020
From Peter Bala, May 04 2025: (Start)
a(n) = (1/sqrt(3)) * sqrt(1 - T(2*n-1, -2)), where T(k, x) denotes the k-th Chebyshev polynomial of the first kind.
a(n) divides a(3*n-1); a(n) divides a(5*n-2); in general, for k >= 0, a(n) divides a((2*k+1)*n - k).
The aerated sequence [b(n)]n>=1 = [1, 0, 3, 0, 11, 0, 41, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -6, Q = 1 of the 3-parameter family of divisibility sequences found by Williams and Guy.
Sum_{n >= 2} 1/(a(n) - 1/a(n)) = 1/2 (telescoping series: for n >= 1, 1/(a(n) - 1/a(n)) = 1/A052530(n-1) - 1/A052530(n).) (End)

A028230 Bisection of A001353. Indices of square numbers which are also octagonal.

Original entry on oeis.org

1, 15, 209, 2911, 40545, 564719, 7865521, 109552575, 1525870529, 21252634831, 296011017105, 4122901604639, 57424611447841, 799821658665135, 11140078609864049, 155161278879431551, 2161117825702177665, 30100488280951055759, 419245718107612602961, 5839339565225625385695

Offset: 1

N. J. A. Sloane

Chebyshev S-sequence with Diophantine property.
4*b(n)^2 - 3*a(n)^2 = 1 with b(n) = A001570(n), n>=0.
y satisfying the Pellian x^2 - 3*y^2 = 1, for even x given by A094347(n). - Lekraj Beedassy, Jun 03 2004
a(n) = L(n,-14)*(-1)^n, where L is defined as in A108299; see also A001570 for L(n,+14). - Reinhard Zumkeller, Jun 01 2005
Product x*y, where the pair (x, y) solves for x^2 - 3y^2 = -2, i.e., a(n) = A001834(n)*A001835(n). - Lekraj Beedassy, Jul 13 2006
Numbers n such that RootMeanSquare(1,3,...,2*A001570(k)-1) = n. - Ctibor O. Zizka, Sep 04 2008
As n increases, this sequence is approximately geometric with common ratio r = lim(n -> oo, a(n)/a(n-1)) = (2 + sqrt(3))^2 = 7 + 4 * sqrt(3). - Ant King, Nov 15 2011

R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 329.
J. D. E. Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996, p. 104.

Vincenzo Librandi, Table of n, a(n) for n = 1..890
K. Andersen, L. Carbone, and D. Penta, Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
T. N. E. Greville, Table for third-degree spline interpolations with equally spaced arguments, Math. Comp., 24 (1970), 179-183.
W. D. Hoskins, Table for third-degree spline interpolation using equi-spaced knots, Math. Comp., 25 (1971), 797-801.
Tanya Khovanova, Recursive Sequences
E. Kilic, Y. T. Ulutas, and N. Omur, A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters, J. Int. Seq. 14 (2011) #11.5.6, table 4, k=1, t=2.
Dino Lorenzini, and Z. Xiang, Integral points on variable separated curves, Preprint 2016.
F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag., 40 (1967), 74-83.
Eric Weisstein's World of Mathematics, Octagonal Square Number.
Index entries for linear recurrences with constant coefficients, signature (14,-1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A001353, A001570, A001834, A001835, A001921, A007655, A036428, A046184, A049310, A094347.

Cf. A077416 with companion A077417.

a:=[1,15];; for n in [3..30] do a[n]:=14*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 06 2019

I:=[1,15]; [n le 2 select I[n] else 14*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 06 2019

seq(coeff(series((1+x)/(1-14*x+x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Dec 06 2019

LinearRecurrence[{14, - 1}, {1, 15}, 17] (* Ant King, Nov 15 2011 *)
CoefficientList[Series[(1+x)/(1-14x+x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 17 2014 *)

Vec((1+x)/(1-14*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 16 2014

isok(n) = ispolygonal(n^2, 8); \\ Michel Marcus, Jul 09 2017

[(lucas_number2(n,14,1)-lucas_number2(n-1,14,1))/12 for n in range(1, 18)] # Zerinvary Lajos, Nov 10 2009

a(n) = 2*A001921(n)+1.
a(n) = 14*a(n-1) - a(n-2) for n>1.
a(n) = S(n, 14) + S(n-1, 14) = S(2*n, 4) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(-1, x) = 0, S(n, 14) = A007655(n+1) and S(n, 4) = A001353(n+1).
G.f.: x*(1+x)/(1-14*x+x^2).
a(n) = (ap^(2*n+1) - am^(2*n+1))/(ap - am) with ap := 2+sqrt(3) and am := 2-sqrt(3).
a(n+1) = Sum_{k=0..n} (-1)^k*binomial(2*n-k, k)*16^(n-k), n>=0.
a(n) = sqrt((4*A001570(n-1)^2 - 1)/3).
a(n) ~ 1/6*sqrt(3)*(2 + sqrt(3))^(2*n-1). - Joe Keane (jgk(AT)jgk.org), May 15 2002
4*a(n+1) = (A001834(n))^2 + 4*(A001835(n+1))^2 - (A001835(n))^2. E.g. 4*a(3) = 4*209 = 19^2 + 4*11^2 - 3^2 = (A001834(2))^2 + 4*(A001835(3))^2 - (A001835(2))^2. Generating floretion: 'i + 2'j + 3'k + i' + 2j' + 3k' + 4'ii' + 3'jj' + 4'kk' + 3'ij' + 3'ji' + 'jk' + 'kj' + 4e. - Creighton Dement, Dec 04 2004
a(n) = f(a(n-1),7) + f(a(n-2),7), where f(x,s) = s*x + sqrt((s^2-1)*x^2+1); f(0,s)=0. - Marcos Carreira, Dec 27 2006
From Ant King, Nov 15 2011: (Start)
a(n) = 1/6 * sqrt(3) * ( (tan(5*Pi/12)) ^ (2n-1) - (tan(Pi/12)) ^ (2n-1) ).
a(n) = floor (1/6 * sqrt(3) * (tan(5*Pi/12)) ^ (2n-1)). (End)
a(n) = A001353(n)^2-A001353(n-1)^2. - Antonio Alberto Olivares, Apr 06 2020
E.g.f.: 1 - exp(7*x)*(3*cosh(4*sqrt(3)*x) - 2*sqrt(3)*sinh(4*sqrt(3)*x))/3. - Stefano Spezia, Dec 12 2022
a(n) = sqrt(A036428(n)). - Bernard Schott, Dec 19 2022

Additional comments from Wolfdieter Lang, Nov 29 2002
Incorrect recurrence relation deleted by Ant King, Nov 15 2011
Minor edits by Vaclav Kotesovec, Jan 28 2015

cp's OEIS Frontend

A001570 Numbers k such that k^2 is centered hexagonal.

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A140480 RMS numbers: numbers n such that root mean square of divisors of n is an integer.

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A003480 a(0) = 1, a(1) = 2, for n > 1, a(n) = 4*a(n-1) - 2*a(n-2).

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A061278 a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) with a(1) = 1 and a(k) = 0 if k <= 0.

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A052530 a(n) = 4*a(n-1) - a(n-2), with a(0) = 0, a(1) = 2.

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A005246 a(n) = (1 + a(n-1)*a(n-2))/a(n-3), a(0) = a(1) = a(2) = 1.

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A003480 a(0) = 1, a(1) = 2, for n > 1, a(n) = 4a(n-1) - 2a(n-2).

A061278 a(n) = 5a(n-1) - 5a(n-2) + a(n-3) with a(1) = 1 and a(k) = 0 if k <= 0.