A001075 -id:A001075 - cp's OEIS frontend

A082841 a(n) = 4*a(n-1) - a(n-2) for n>1, a(0)=3, a(1)=9.

3, 9, 33, 123, 459, 1713, 6393, 23859, 89043, 332313, 1240209, 4628523, 17273883, 64467009, 240594153, 897909603, 3351044259, 12506267433, 46674025473, 174189834459, 650085312363, 2426151414993, 9054520347609, 33791929975443

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 14 2003

y-values in the solutions to 3*x^2+6 = y^2. - Sture Sjöstedt, Nov 25 2011
Positive values of x (or y) satisfying x^2 - 4*x*y + y^2 + 18 = 0. - Colin Barker, Feb 04 2014
Positive values of x (or y) satisfying x^2 - 14*x*y + y^2 + 288 = 0. - Colin Barker, Feb 16 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4,-1).

First differences of A005320.

Cf. A001834.

a:=[3,9];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Feb 25 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (3-6*x+3*x^2)/((1-x)*(1-4*x+x^2)) )); // G. C. Greubel, Feb 25 2019

a:=proc(n) option remember; if n=0 then 3 elif n=1 then 9 else 4*a(n-1)-a(n-2); fi; end: seq(a(n), n=0..40); # Wesley Ivan Hurt, Jan 21 2017

CoefficientList[Series[(3-6x+3x^2)/((1-x)(1-4x+x^2)), {x, 0, 25}], x]
LinearRecurrence[{4,-1},{3,9},30] (* Harvey P. Dale, Aug 28 2019 *)

my(x='x+O('x^30)); Vec((3-6*x+3*x^2)/((1-x)*(1-4*x+x^2))) \\ G. C. Greubel, Feb 25 2019

((3-6*x+3*x^2)/((1-x)*(1-4*x+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 25 2019

G.f.: (3 -6*x +3*x^2)/((1-x)*(1-4*x+x^2)).
a(n) = sqrt(3/2)*(a^(n+1/2) + b^(n+1/2)), with a=2+sqrt(3) and b=2-sqrt(3).
a(n) = sqrt(3*(11 +12*A082840(n) +4*A082840(n)^2)).
a(n) = sqrt((3/2)*(A003500(2n+1) +2)).
a(n) - a(n-1) = 6*A001353(n).
a(n) == 3 (mod 6).
a(n) = 3 * A001835(n+1).
a(n) = 3*x(n) + 3*y(n) for x(n)= A001075(n) and y(n) = A001353(n) the solutions to x^2 - 3*y^2 = 1. - Greg Dresden and his Math 222 Linear Algebra class, Oct 05 2022

A238731 Riordan array ((1-2x)/(1-3x+x^2), x/(1-3*x+x^2)).

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 5, 13, 7, 1, 13, 40, 33, 10, 1, 34, 120, 132, 62, 13, 1, 89, 354, 483, 308, 100, 16, 1, 233, 1031, 1671, 1345, 595, 147, 19, 1, 610, 2972, 5561, 5398, 3030, 1020, 203, 22, 1, 1597, 8495, 17984, 20410, 13893, 5943, 1610, 268, 25, 1, 4181

Offset: 0

Philippe Deléham, Mar 03 2014

Unsigned version of A124037 and A126126.
Subtriangle of the triangle given by (0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Row sums are A001075(n).
Diagonal sums are A133494(n).
Sum_{k=0..n} T(n,k)*x^k = A001519(n), A001075(n), A002320(n), A038723(n), A033889(n) for x = 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Mar 05 2014

			Triangle begins:
1;
1, 1;
2, 4, 1;
5, 13, 7, 1;
13, 40, 33, 10, 1;
34, 120, 132, 62, 13, 1;
89, 354, 483, 308, 100, 16, 1;
233, 1031, 1671, 1345, 595, 147, 19, 1;...
Triangle (0, 1, 1, 1, 0, 0, 0, ...) DELTA (1, 0, 2, -2, 0, 0, ...) begins:
1;
0, 1;
0, 1, 1;
0, 2, 4, 1;
0, 5, 13, 7, 1;
0, 13, 40, 33, 10, 1;
0, 34, 120, 132, 62, 13, 1;
0, 89, 354, 483, 308, 100, 16, 1;
0, 233, 1031, 1671, 1345, 595, 147, 19, 1;...

Cf. A001519, A000012, A016777, A062708.

Cf. A001906, A124037, A126126.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1-2#)/(1-3#+#^2)&, x/(1-3#+#^2)&, 10] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.

G.f.: (1-2*x)/(1-(y+3)*x+x^2). - Philippe Deléham, Mar 05 2014

A204517 Square root of floor[A055859(n)/7].

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 17, 48, 96, 271, 765, 1530, 4319, 12192, 24384, 68833, 194307, 388614, 1097009, 3096720, 6193440, 17483311, 49353213, 98706426, 278635967, 786554688, 1573109376, 4440692161, 12535521795, 25071043590, 70772438609, 199781794032, 399563588064

Offset: 1

M. F. Hasler, Jan 15 2012

nonn

M. F. Hasler, Truncated squares, OEIS wiki, Jan 16 2012
Index to sequences related to truncating digits of squares.

See also A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9), A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7), A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5), A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3), A001541=sqrt(A055792) (base 2).

b=7;for(n=1,2e9,issquare(n^2\b) & print1(sqrtint(n^2\b),","))

A204517(n)=polcoeff((x^4 + 3*x^5 + 6*x^6 + x^7)/(1 - 16*x^3 + x^6+O(x^n)),n)

A204517(n) = sqrt(floor(A204516(n)^2/7)).

G.f. = (x^4 + 3*x^5 + 6*x^6 + x^7)/(1 - 16*x^3 + x^6)

A322836 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{n}(x), evaluated at x=k.

Original entry on oeis.org

1, 1, 0, 1, 1, -1, 1, 2, 1, 0, 1, 3, 7, 1, 1, 1, 4, 17, 26, 1, 0, 1, 5, 31, 99, 97, 1, -1, 1, 6, 49, 244, 577, 362, 1, 0, 1, 7, 71, 485, 1921, 3363, 1351, 1, 1, 1, 8, 97, 846, 4801, 15124, 19601, 5042, 1, 0, 1, 9, 127, 1351, 10081, 47525, 119071, 114243, 18817, 1, -1

Offset: 0

Seiichi Manyama, Dec 28 2018

			Square array begins:
   1, 1,    1,     1,      1,      1,       1, ...
   0, 1,    2,     3,      4,      5,       6, ...
  -1, 1,    7,    17,     31,     49,      71, ...
   0, 1,   26,    99,    244,    485,     846, ...
   1, 1,   97,   577,   1921,   4801,   10081, ...
   0, 1,  362,  3363,  15124,  47525,  120126, ...
  -1, 1, 1351, 19601, 119071, 470449, 1431431, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Mirror of A101124.
Columns 0-20 give A056594, A000012, A001075, A001541, A001091, A001079, A023038, A011943(n+1), A001081, A023039, A001085, A077422, A077424, A097308, A097310, A068203, A322888, A056771, A322889, A078986, A322890.
Rows 0-10 give A000012, A001477, A056220, A144129, A144130, A243131, A243132, A243133, A243134, A243135, A243136.
Main diagonal gives A115066.
Cf. A323182 (Chebyshev polynomial of the second kind).

Table[ChebyshevT[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Dec 28 2018 *)

T(n,k) = polchebyshev(n,1,k);
matrix(7, 7, n, k, T(n-1,k-1)) \\ Michel Marcus, Dec 28 2018

T(n, k) = round(cos(n*acos(k)));\\ Seiichi Manyama, Mar 05 2021

T(n, k) = if(n==0, 1, n*sum(j=0, n, (2*k-2)^j*binomial(n+j, 2*j)/(n+j))); \\ Seiichi Manyama, Mar 05 2021

A(0,k) = 1, A(1,k) = k and A(n,k) = 2 * k * A(n-1,k) - A(n-2,k) for n > 1.
A(n,k) = cos(n*arccos(k)). - Seiichi Manyama, Mar 05 2021
A(n,k) = n * Sum_{j=0..n} (2*k-2)^j * binomial(n+j,2*j)/(n+j) for n > 0. - Seiichi Manyama, Mar 05 2021

A011922 a(n) = 15a(n-1) - 15a(n-2) + a(n-3) with a(0)=1, a(1)=3, and a(2)=33.

Original entry on oeis.org

1, 3, 33, 451, 6273, 87363, 1216801, 16947843, 236052993, 3287794051, 45793063713, 637815097923, 8883618307201, 123732841202883, 1723376158533153, 24003533378261251, 334326091137124353, 4656561742541479683, 64857538304443591201, 903348974519668797123, 12582028104970919568513

Offset: 0

Mario Velucchi (mathchess(AT)velucchi.it)

Mario Velucchi, Seeing couples, in Recreational and Educational Computing, to appear 1997.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Christian Aebi and Grant Cairns, Lattice Equable Parallelograms, arXiv:2006.07566 [math.NT], 2020.
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 31, 56.
Z. Franusic, On the Extension of the Diophantine Pair {1,3} in Z[surd d], J. Int. Seq. 13 (2010) # 10.9.6.
Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Cf. A001353, A001075, A007655, A011916, A011918, A011920, A011943, A103974.

I:=[1,3,33]; [n le 3 select I[n] else 15*Self(n-1)-15*Self(n-2)+Self(n-3): n in [1..17]]; // Bruno Berselli, Jul 09 2011

a:= gfun:-rectoproc({a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3), a(0)=1,a(1)=3,a(2)=33},a(n),remember):
map(a,[$0..100]); # Robert Israel, Jul 02 2015

RecurrenceTable[{a[n] == 15 a[n - 1] - 15 a[n - 2] + a[n - 3], a[0] == 1, a[1] == 3, a[2] == 33}, a, {n, 0, 15}] (* Michael De Vlieger, Jul 02 2015 *)
LinearRecurrence[{15,-15,1},{1,3,33},30] (* Harvey P. Dale, Dec 04 2018 *)

a[0]:1$ a[1]:3$ a[2]:33$ a[n]:=15*a[n-1]-15*a[n-2]+a[n-3]$ makelist(a[n], n, 0, 16); /* Bruno Berselli, Jul 09 2011 */

a(n)=([0,1,0; 0,0,1; 1,-15,15]^n*[1;3;33])[1,1] \\ Charles R Greathouse IV, Jul 02 2015

a(n) = (2+sqrt(1+((((2+sqrt(3))^(2*n)-(2-sqrt(3))^(2*n))^2)/4)))/3. [corrected by Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 30 2001]
a(n) = ((7+4*sqrt(3))^n+(7-4*sqrt(3))^n+4)/6. - Bruno Berselli, Jul 09 2011
G.f.: (1-12*x+3*x^2)/ ((1-x) * (x^2-14*x+1)). - R. J. Mathar, Apr 15 2010
Sqrt(3) = 1 + Sum_{n>=1} 2/a(n) = 1 + 2/3 + 2/33 + ... - Gary W. Adamson, Jun 12 2003
a(n)^2 = A103974(n+1)^2 - (4*A007655(n+1))^2. - Paul D. Hanna, Mar 06 2005
a(n) = (A011943(n+1) + 2)/3. - Ralf Stephan, Aug 13 2013
a(n) = A001075(n)^2 - A001353(n)^2. - Richard R. Forberg, Aug 24 2013
E.g.f.: exp(x)*(2 + exp(6*x)*cosh(4*sqrt(3)*x))/3. - Stefano Spezia, Dec 11 2022

Recurrence in definition by R. J. Mathar, Apr 15 2010

A071954 a(n) = 4*a(n-1) - a(n-2) - 4, with a(0) = 2, a(1) = 4.

Original entry on oeis.org

2, 4, 10, 32, 114, 420, 1562, 5824, 21730, 81092, 302634, 1129440, 4215122, 15731044, 58709050, 219105152, 817711554, 3051741060, 11389252682, 42505269664, 158631825970, 592022034212, 2209456310874, 8245803209280, 30773756526242, 114849222895684

Offset: 0

Lekraj Beedassy, Jun 25 2002

a(n) gives the side of a cube having a square number of cubes in its two outermost layers, i.e., solutions p to the equation p^3 - (p - 4)^3 = q^2. The corresponding q is given by 4*A001075(n).

			G.f. = 2 + 4*x + 10*x^2 + 32*x^3 + 114*x^4 + 420*x^5 + 1562*x^6 + ...

M. E. Larsen, "Four Cubes" in Puzzler's Tribute, Ed. D. Wolfe & T. Rodgers, pp. 69-70, A. K. Peters, MA, 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Reinhard Zumkeller)
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Equals A052530(n) + 2, n > 0.

Cf. A001075, A003699, A072110, A263942.

a:=[2,4,10];; for n in [4..30] do a[n]:=5*a[n-1]-5*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Feb 25 2019

a071954 n = a071954_list !! n
a071954_list = 2 : 4 : zipWith (-)
               (map ((4 *) . pred) (tail a071954_list)) a071954_list
-- Reinhard Zumkeller, Aug 11 2011

I:=[2,4,10]; [n le 3 select I[n] else 5*Self(n-1) -5*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Feb 25 2019

a[n_]:= a[n] = 4*a[n-1] -a[n-2] -4; a[0]=2; a[1]=4; Table[a[n], {n,0,30}]
LinearRecurrence[{5,-5,1},{2,4,10},30] (* Harvey P. Dale, May 05 2011 *)

Vec((2-6*x)/(1-5*x+5*x^2-x^3)+O(x^30)) \\ Charles R Greathouse IV, Feb 09 2012

{a(n) = my(w=quadgen(12)); simplify( 2 + ((2+w)^n - (2-w)^n) / w)}; /* Michael Somos, Nov 03 2016 */

(2*(1-3*x)/((1-x)*(1-4*x+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 25 2019

a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) for n > 2, with a(0) = 2, a(1) = 4, a(2) = 10.
G.f.: 2*(1 - 3*x)/((1-x)*(1 -4*x +x^2)). - Harvey P. Dale, May 05 2011
a(n) = (2 + (-(2 - sqrt(3))^n + (2 + sqrt(3))^n)/sqrt(3)). - Colin Barker, Nov 03 2016
A263942(n) = -a(-1-n) for all n in Z. - Michael Somos, Nov 03 2016
E.g.f.: (2/3)*(3*exp(x) + sqrt(3)*exp(2*x)*sinh(sqrt(3)*x)). - Franck Maminirina Ramaharo, Nov 14 2018
From G. C. Greubel, Feb 25 2019: (Start)
a(n) = 2*A072110(n).
a(n) = 2*(1 - (-i)^(n+1)*F(n, 4*i)), where i=sqrt(-1) and F(n,x) is the Fibonacci polynomial. (End)

Edited by Robert G. Wilson v, Jun 27 2002

A077236 a(n) = 4*a(n-1) - a(n-2) with a(0) = 4 and a(1) = 11.

Original entry on oeis.org

4, 11, 40, 149, 556, 2075, 7744, 28901, 107860, 402539, 1502296, 5606645, 20924284, 78090491, 291437680, 1087660229, 4059203236, 15149152715, 56537407624, 211000477781, 787464503500, 2938857536219, 10967965641376

Offset: 0

Wolfdieter Lang, Nov 08 2002

a(n)^2 - 3*b(n)^2 = 13, with the companion sequence b(n)= A054491(n).
Bisection (even part) of Chebyshev sequence with Diophantine property.
The odd part is A077235(n) with Diophantine companion A077234(n).

			11 = a(1) = sqrt(3*A054491(1)^2 + 13) = sqrt(3*6^2 + 13)= sqrt(121) = 11.

Luigi Cerlienco, Maurice Mignotte, and F. Piras, Suites récurrentes linéaires: Propriétés algébriques et arithmétiques, L'Enseignement Math., 33 (1987), 67-108. See Example 2, page 93.
Tanya Khovanova, Recursive Sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (4,-1).

Cf. A077238 (even and odd parts), A077235, A053120.

a:=[4,11];; for n in [3..30] do a[n]:=4*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Apr 28 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (4-5*x)/(1-4*x+x^2) )); // G. C. Greubel, Apr 28 2019

CoefficientList[Series[(4-5*x)/(1-4*x+x^2), {x,0,20}], x] (* or *) LinearRecurrence[{4,-1}, {4,11}, 30] (* G. C. Greubel, Apr 28 2019 *)

my(x='x+O('x^30)); Vec((4-5*x)/(1-4*x+x^2)) \\ G. C. Greubel, Apr 28 2019

a(n) = polchebyshev(n+1, 1, 2) + 2*polchebyshev(n, 1, 2); \\ Michel Marcus, Oct 13 2021

((4-5*x)/(1-4*x+x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Apr 28 2019

a(n) = T(n+1,2) + 2*T(n,2), with T(n,x) Chebyshev's polynomials of the first kind, A053120. T(n,2) = A001075(n).
G.f.: (4-5*x)/(1-4*x+x^2).
From Al Hakanson (hawkuu(AT)gmail.com), Jul 06 2009: (Start)
a(n) = ((4+sqrt(3))*(2+sqrt(3))^n + (4-sqrt(3))*(2-sqrt(3))^n)/2. Offset 0.
a(n) = second binomial transform of 4,3,12,9,36. (End)
a(n) = (A054491(n+1) - A054491(n-1))/2 = sqrt(3*A054491(n-1)*A054491(n+1) + 52), n >= 1. - Klaus Purath, Oct 12 2021

Edited by N. J. A. Sloane, Sep 07 2018, replacing old definition with simple formula from Philippe Deléham, Nov 16 2008

A204512 Square roots of [A055872/8]: Their square written in base 8, with some digit appended, is again a square.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 12, 35, 70, 204, 408, 1189, 2378, 6930, 13860, 40391, 80782, 235416, 470832, 1372105, 2744210, 7997214, 15994428, 46611179, 93222358, 271669860, 543339720, 1583407981, 3166815962, 9228778026, 18457556052, 53789260175, 107578520350

Offset: 1

M. F. Hasler, Jan 15 2012

Base-8 analog of A031150. The square of the terms (= truncated squares A055872) are listed in A204504.

See also A031149=sqrt(A023110) (base 10), A204502=sqrt(A204503) (base 9), A204514=sqrt(A055872) (base 8), A204516=sqrt(A055859) (base 7), A204518=sqrt(A055851) (base 6), A204520=sqrt(A055812) (base 5), A004275=sqrt(A055808) (base 4), A001075=sqrt(A055793) (base 3), A001541=sqrt(A055792) (base 2).

CoefficientList[Series[(x^4 (1+2x))/(1-6x^2+x^4),{x,0,40}],x] (* Harvey P. Dale, Nov 30 2020 *)

b=8;for(n=1,1e7,issquare(n^2\b) & print1(sqrtint(n^2\b)","))

a(n)=polcoeff((2*x^5 + x^4)/(x^4 - 6*x^2 + 1+O(x^n)),n)

G.f. = x^4*(1 + 2*x)/(1 - 6*x^2 + x^4)

A001922 Numbers k such that 3k^2 - 3k + 1 is both a square (A000290) and a centered hexagonal number (A003215).

Original entry on oeis.org

1, 8, 105, 1456, 20273, 282360, 3932761, 54776288, 762935265, 10626317416, 148005508553, 2061450802320, 28712305723921, 399910829332568, 5570039304932025, 77580639439715776, 1080558912851088833, 15050244140475527880, 209622859053806301481

Offset: 0

N. J. A. Sloane

Also larger of two consecutive integers whose cubes differ by a square. Defined by a(n)^3 - (a(n) - 1)^3 = square.
Let m be the n-th ratio 2/1, 7/4, 26/15, 97/56, 362/209, ... Then a(n) = m*(2-m)/(m^2-3). The numerators 2, 7, 26, ... of m are A001075. The denominators 1, 4, 15, ... of m are A001353.
From Colin Barker, Jan 06 2015: (Start)
Also indices of centered triangular numbers (A005448) which are also centered square numbers (A001844).
Also indices of centered hexagonal numbers (A003215) which are also centered octagonal numbers (A016754).
Also positive integers x in the solutions to 3*x^2 - 4*y^2 - 3*x + 4*y = 0, the corresponding values of y being A156712.
(End)

			8 is in the sequence because 3*8^2 - 3*8 + 1 = 169 is a square and also a centered hexagonal number. - _Colin Barker_, Jan 07 2015

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).

Vincenzo Librandi, Table of n, a(n) for n = 0..800
J. Brenner and E. P. Starke, Problem E702, Amer. Math. Monthly, 53 (1946), 465.
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
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Cf. A001075, A001353, A001570, A001844, A001921, A003215.

Cf. A005448, A006051, A016754, A076139, A156712.

I:=[1, 8, 105]; [n le 3 select I[n] else 15*Self(n-1)-15*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Apr 16 2012

seq(simplify((1 +ChebyshevU(n,7) +ChebyshevU(n-1,7))/2), n=0..30); # G. C. Greubel, Oct 07 2022

With[{s1=3+2Sqrt[3],s2=3-2Sqrt[3],t1=7+4Sqrt[3],t2=7-4Sqrt[3]}, Simplify[ Table[(s1 t1^n+s2 t2^n+6)/12,{n,0,20}]]] (* or *) LinearRecurrence[ {15,-15,1},{1,8,105},21] (* Harvey P. Dale, Aug 14 2011 *)
CoefficientList[Series[(1-7*x)/(1-15*x+15*x^2-x^3),{x,0,30}],x] (* Vincenzo Librandi, Apr 16 2012 *)

Vec((1-7*x)/(1-15*x+15*x^2-x^3) + O(x^100)) \\ Colin Barker, Jan 06 2015

[(1+chebyshev_U(n,7) +chebyshev_U(n-1,7))/2 for n in range(30)] # G. C. Greubel, Oct 07 2022

a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3).
a(n) = (s1*t1^n + s2*t2^n + 6)/12 where s1 = 3 + 2*sqrt(3), s2 = 3 - 2*sqrt(3), t1 = 7 + 4*sqrt(3), t2 = 7 - 4*sqrt(3).
a(n) = A001075(n)*A001353(n+1).
G.f.: (1-7*x)/((1-x)*(1-14*x+x^2)). - Simon Plouffe (in his 1992 dissertation) and Colin Barker, Jan 01 2012
a(n) = A076139(n+1) - 7*A076139(n). - R. J. Mathar, Jul 14 2015
a(n) = (1/2)*(1 + ChebyshevU(n, 7) + ChebyshevU(n-1, 7)). G. C. Greubel, Oct 07 2022
a(n) = 1 - a(-1-n) = 1 + A001921(n) for all integers n. - Michael Somos, Jul 10 2025

Additional comments from James R. Buddenhagen, Mar 04 2001
Name improved by Colin Barker, Jan 07 2015
Edited by Robert Israel, Feb 20 2017

A006235 Complexity of doubled cycle (regarding case n = 2 as a multigraph).

Original entry on oeis.org

1, 12, 75, 384, 1805, 8100, 35287, 150528, 632025, 2620860, 10759331, 43804800, 177105253, 711809364, 2846259375, 11330543616, 44929049777, 177540878700, 699402223099, 2747583822720, 10766828545725, 42095796462852, 164244726238343, 639620518118400, 2486558615814025

Offset: 1

N. J. A. Sloane

In plain English, a(n) is the number of spanning trees of the n-prism graph Y_n. - Eric W. Weisstein, Jul 15 2011
Also the number of spanning trees of the n-web graph. - Eric W. Weisstein, Jul 15 2011
Also the number of spanning trees of the n-dipyramidal graph. - Eric W. Weisstein, Jun 14 2018
Determinants of the spiral knots S(4,k,(1,-1,1)). a(k) = det(S(4,k,(1,-1,1))). These knots are also the weaving knots W(k,4) and the Turk's Head Links THK(4,k). - Ryan Stees, Dec 14 2014

			For k=3, b(3)=sqrt(6)b(2)-b(1)=6-1=5, so det(S(4,3,(1,-1,1)))=3*5^2=75.

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 = 1..200
Zbigniew R. Bogdanowicz, The number of spanning trees in a superprism, Discrete Math. Lett. 13 (2024) 66-73. See p. 66.
N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, Spiral knots, Missouri J. of Math. Sci., 22 (2010).
M. DeLong, M. Russell, and J. Schrock, Colorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/-1(mod m), Involve, Vol. 8 (2015), No. 3, 361-384.
N. Dowdall, T. Mattman, K. Meek, and P. Solis, On the Harary-Kauffman conjecture and turk's head knots, arxiv 0811.0044 [math.GT], 2008.
A. A. Jagers, A note on the number of spanning trees in a prism graph, Int. J. Comput. Math., Vol. 24, 1988 (Issue 2), pp. 151-154.
Seong Ju Kim, R. Stees, and L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
D. E. Knuth, Letter to N. J. A. Sloane, Oct. 1994
Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212.
L. Oesper, p-Colorings of Weaving Knots, Undergraduate Thesis, Pomona College, 2005.
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
Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.
Eric Weisstein's World of Mathematics, Dipyramidal Graph
Eric Weisstein's World of Mathematics, Prism Graph
Eric Weisstein's World of Mathematics, Spanning Tree
Eric Weisstein's World of Mathematics, Web Graph
Index entries for linear recurrences with constant coefficients, signature (10,-35,52,-35,10,-1).

Cf. A006237. Apart from a(2) coincides with A072373. A row or column of A173958.

Cf. A001075, A129743.

A006235:=(1+2*z-10*z**2+2*z**3+z**4)/(z-1)**2/(z**2-4*z+1)**2; # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

LinearRecurrence[{10, -35, 52, -35, 10, -1}, {0, 1, 12, 75, 384, 1805}, 20]
Table[1/2 (-2 + (2 - Sqrt[3])^n + (2 + Sqrt[3])^n) n, {n, 0, 20}] // Expand
Table[n (ChebyshevT[n, 2] - 1), {n, 20}] (* Eric W. Weisstein, Mar 30 2017 *)

a(n)=if(n<0,0,polcoeff(x*(1+2*x-10*x^2+2*x^3+x^4)/((1-x)*(1-4*x+x^2))^2+x*O(x^n),n))

a(n) = (1/2)*n*(-2 + (2 - sqrt(3))^n + (2 + Sqrt(3))^n) (Kreweras). - Eric W. Weisstein, Jul 15 2011
G.f.: x*(1+2*x-10*x^2+2*x^3+x^4)/((1-x)*(1-4*x+x^2))^2.
a(n) = 10*a(n-1)-35*a(n-2)+52*a(n-3)-35*a(n-4)+10*a(n-5)-a(n-6), n>5.
a(n) = (n/2)*A129743(n). - Woong Kook and Seung Kyoon Shin (andrewk(AT)math.uri.edu), Jan 13 2009
a(k) = det(S(4,k,(1,-1,1))) = k*b(k)^2, where b(1)=1, b(2)=sqrt(6), b(k)=sqrt(6)*b(k-1) - b(k-2) = b(2)*b(k-1) - b(k-2). - Ryan Stees, Dec 14 2014
a(n) = n*(A001075(n) - 1). - Eric W. Weisstein, Mar 30 2017
E.g.f.: exp(x)*x*(exp(x)*(2*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) - 1). - Stefano Spezia, May 05 2024

More terms from Michael Somos, Jul 19 2002

Minor edits by N. J. A. Sloane, May 27 2012

cp's OEIS Frontend

A082841 a(n) = 4*a(n-1) - a(n-2) for n>1, a(0)=3, a(1)=9.

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A238731 Riordan array ((1-2*x)/(1-3*x+x^2), x/(1-3*x+x^2)).

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A204517 Square root of floor[A055859(n)/7].

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A322836 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where A(n,k) is Chebyshev polynomial of the first kind T_{n}(x), evaluated at x=k.

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A011922 a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3) with a(0)=1, a(1)=3, and a(2)=33.

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A071954 a(n) = 4*a(n-1) - a(n-2) - 4, with a(0) = 2, a(1) = 4.

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A238731 Riordan array ((1-2x)/(1-3x+x^2), x/(1-3*x+x^2)).

A011922 a(n) = 15a(n-1) - 15a(n-2) + a(n-3) with a(0)=1, a(1)=3, and a(2)=33.

A001922 Numbers k such that 3k^2 - 3k + 1 is both a square (A000290) and a centered hexagonal number (A003215).