A180662 -id:A180662 - cp's OEIS frontend

A099251 Bisection of Motzkin sums (A005043).

1, 1, 3, 15, 91, 603, 4213, 30537, 227475, 1730787, 13393689, 105089229, 834086421, 6684761125, 54022715451, 439742222071, 3602118427251, 29671013856627, 245613376802185, 2042162142208813, 17047255430494497, 142816973618414817

Offset: 0

N. J. A. Sloane, Nov 16 2004

The Kn4 triangle sums of A175136 lead to the sequence given above (n >= 1). For the definition of the Kn4 and other triangle sums see A180662. - Johannes W. Meijer, May 06 2011
Equals the expected value of trace(O)^(2n), where O is a 3 X 3 orthogonal matrix randomly selected according to Haar measure (see MathOverflow link). - Nathaniel Johnston, Sep 05 2014
From Petros Hadjicostas, Jul 23 2020: (Start)
In Smith (1985), we apparently have a(n) = P(2*n), where P(n) is the number of linearly independent three-dimensional n-th order isotropic tensors. In the paper, he refers to Smith (1968) for more details. It is not clear why he does not list the values of P(2*n+1). See also the 1978 letter of D. L. Andrews to N. J. A. Sloane.
Eric Weisstein gives some details on how the material in Smith (1968) about isotropic tensors is related to Motzkin sums. (End)

G. F. Smith, On isotropic tensors and rotation tensors of dimension m and order n, Tensor (N.S.), Vol. 19 (1968), 79-88 (MR0224008).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. L. Andrews, Letter to N. J. A. Sloane, Apr 10 1978.
Georgia Benkart and A. Elduque, Cross products, invariants, and centralizers, arXiv:1606.07588 [math.RT], 2016.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
MathOverflow, Moments of the trace of orthogonal matrices.
G. F. Smith, Lectures on constitutive expressions, Mathematical models and methods in mechanics, pp. 645-678, Banach Center Publ., 15, PWN, Warsaw, 1985 (MR0874855). See p. 653.
Eric Weisstein's World of Mathematics, Isotropic tensor.

Cf. A005043, A099252, A246860, A247304.

G := (1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x)): Gser := series(G,x=0,60):
1, seq(coeff(Gser, x^(2*n)), n=1..25); # Emeric Deutsch
a := n -> hypergeom([1/2, -2*n], [2], 4):
seq(simplify(a(n)), n=0..21); # Peter Luschny, Jul 25 2020

Take[CoefficientList[Series[(1 + x - Sqrt[1 - 2 * x - 3 * x^2])/(2 * x * (1 + x)), {x, 0, 60}], x], {1, -1, 2}] (* Vaclav Kotesovec, Oct 17 2012 *)

a(n):=sum(binomial(2*j,j)*(-1)^(j)*binomial(2*n+1,j+1),j,0,2*n+1)/(2*n+1); /*Vladimir Kruchinin, Apr 02 2017*/

x='x+O('x^66); v=Vec((1+x-sqrt(1-2*x-3*x^2))/(2*x*(1+x))); vector(#v\2,n,v[2*n-1]) \\ Joerg Arndt, May 12 2013

Recurrence: n*(2*n + 1)*a(n) = (2*n - 1)*(13*n - 10)*a(n-1) - 3*(26*n^2 - 87*n + 76)*a(n-2) + 27*(n - 2)*(2*n - 5)*a(n-3). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ 3^(2*n + 3/2)/(16*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
Conjecture: a(n) = (2/Pi)*Integral_{t=0..1} sqrt((1 - t)/t)*(1 - 8*t + 16*t^2)^n. - Benedict W. J. Irwin, Oct 05 2016
a(n) = Sum_{j=0..2*n+1} (C(2*j,j)*(-1)^(j)*C(2*n+1,j+1))/(2*n+1). - Vladimir Kruchinin, Apr 02 2017
a(n) = hypergeom([1/2, -2*n], [2], 4). - Peter Luschny, Jul 25 2020

More terms from Emeric Deutsch, Nov 18 2004

A180664 Golden Triangle sums: a(n) = a(n-1) + A001654(n+1) with a(0)=0.

Original entry on oeis.org

0, 2, 8, 23, 63, 167, 440, 1154, 3024, 7919, 20735, 54287, 142128, 372098, 974168, 2550407, 6677055, 17480759, 45765224, 119814914, 313679520, 821223647, 2149991423, 5628750623, 14736260448, 38580030722, 101003831720

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) (terms doubled) are the Kn13 sums of the Golden Triangle A180662. See A180662 for information about these knight and other chess sums.

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

Cf. A000032, A000045, A005248, A064831, A115730, A180662, A180664, A180665, A180666.

[(1/10)*((-1)^n - 15 + 2*Lucas(2*n+4)): n in [0..40]]; // G. C. Greubel, Jan 21 2022

nmax:=26: with(combinat): for n from 0 to nmax+1 do A001654(n):=fibonacci(n) * fibonacci(n+1) od: a(0):=0: for n from 1 to nmax do a(n) := a(n-1)+A001654(n+1) od: seq(a(n),n=0..nmax);

Table[Sum[Fibonacci[i+2]*Fibonacci[i+3], {i,0,n-1}], {n,0,40}] (* Rigoberto Florez, Jul 07 2020 *)
LinearRecurrence[{3,0,-3,1},{0,2,8,23},30] (* Harvey P. Dale, Mar 30 2023 *)

[(1/10)*((-1)^n - 15 + 2*lucas_number2(2*n+4,1,-1)) for n in (0..40)] # G. C. Greubel, Jan 21 2022

a(n+1) = Sum_{k=0..n} A180662(2*n-k+2, k+2).
a(n) = (-15 + (-1)^n + (6-2*A)*A^(-n-1) + (6-2*B)*B^(-n-1))/10 with A=(3+sqrt(5))/2 and B=(3-sqrt(5))/2.
G.f.: (2*x+2*x^2-x^3)/(1-3*x-x^4+3*x^3).
a(n) = Sum_{i=0..n-1} F(i+2)*F(i+3), where F(i) = A000045(i). - Rigoberto Florez, Jul 07 2020
a(n) = (1/10)*((-1)^n - 15 + 2*Lucas(2*n+4)). - G. C. Greubel, Jan 21 2022

A190717 Triplicated tetrahedral numbers A000292.

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 10, 10, 10, 20, 20, 20, 35, 35, 35, 56, 56, 56, 84, 84, 84, 120, 120, 120, 165, 165, 165, 220, 220, 220, 286, 286, 286, 364, 364, 364, 455, 455, 455, 560, 560, 560, 680, 680, 680, 816, 816, 816, 969, 969, 969

Offset: 0

Johannes W. Meijer, May 18 2011

The Ca1 and Ze3 triangle sums, see A180662 for their definitions, of the triangle A159797 are linear sums of shifted versions of the triplicated tetrahedral numbers, e.g. Ca1(n) = a(n-1) + a(n-2) + 2*a(n-3) + a(n-6).

The Ca1, Ca2, Ze3 and Ze4 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above.

Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3,0,-3,3,0,1,-1).

Cf. A000292 (tetrahedral numbers), A058187 (duplicated), this sequence (triplicated), A190718 (quadruplicated), A049347, A144677.

A190717:= proc(n) option remember; A190717(n):= binomial(floor(n/3)+3,3) end: seq(A190717(n),n=0..50);

LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{1,1,1,4,4,4,10,10,10,20},60] (* Harvey P. Dale, Mar 09 2018 *)

a(n) = binomial(floor(n/3)+3,3).
a(n) + a(n-1) + a(n-2) = A144677(n).
a(n) = Sum_{k=0..n} (A144677(n-k)*A049347(k)).
G.f.: 1/((x-1)^4*(x^2+x+1)^3).
Sum_{n>=0} 1/a(n) = 9/2. - Amiram Eldar, Aug 18 2022

A190718 Quadruplicated tetrahedral numbers A000292.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 4, 4, 10, 10, 10, 10, 20, 20, 20, 20, 35, 35, 35, 35, 56, 56, 56, 56, 84, 84, 84, 84, 120, 120, 120, 120, 165, 165, 165, 165, 220, 220, 220, 220, 286, 286, 286, 286, 364, 364, 364, 364, 455, 455, 455, 455

Offset: 0

Johannes W. Meijer, May 18 2011

The Gi1 triangle sums, for the definitions of these and other triangle sums see A180662, of the triangle A159797 are linear sums of shifted versions of the quadruplicated tetrahedral numbers A000292, i.e., Gi1(n) = a(n-1) + a(n-2) + a(n-3) + 2*a(n-4) + a(n-8).

The Gi1 and Gi2 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,3,-3,0,0,-3,3,0,0,1,-1).

Cf. A000292 (tetrahedral numbers), A058187 (duplicated), A190717 (triplicated).

A190718:= proc(n) binomial(floor(n/4)+3,3) end:
seq(A190718(n),n=0..52);

LinearRecurrence[{1,0,0,3,-3,0,0,-3,3,0,0,1,-1},{1,1,1,1,4,4,4,4,10,10,10,10,20},60] (* Harvey P. Dale, Oct 20 2012 *)

a(n) = binomial(floor(n/4)+3,3).
a(n-3) + a(n-2) + a(n-1) + a(n) = A144678(n).
a(n) = +a(n-1) +3*a(n-4) -3*a(n-5) -3*a(n-8) +3*a(n-9) +a(n-12) -a(n-13).
G.f.: 1 / ( (1+x)^3*(1+x^2)^3*(x-1)^4 ).
Sum_{n>=0} 1/a(n) = 6. - Amiram Eldar, Aug 18 2022

A193147 Expansion of 1/(1 - x - 2*x^3 - x^5).

Original entry on oeis.org

1, 1, 1, 3, 5, 8, 15, 26, 45, 80, 140, 245, 431, 756, 1326, 2328, 4085, 7168, 12580, 22076, 38740, 67985, 119305, 209365, 367411, 644761, 1131476, 1985603, 3484490, 6114853, 10730820, 18831276, 33046585, 57992715, 101770120, 178594110, 313410816, 549997641

Offset: 0

Johannes W. Meijer, Jul 20 2011

The Ze3 sums, see A180662 for the definition of these sums, of the "Races with Ties" triangle A035317 equal this sequence.

Number of tilings of a 5 X 2n rectangle with 5 X 1 pentominoes. - M. Poyraz Torcuk, Dec 18 2021

Index entries for linear recurrences with constant coefficients, signature (1,0,2,0,1).

Bisection of A003520.

Cf. A035317, A180662, A373578.

A193147 := proc(n) option remember: if n>=-4 and n<=-1 then 0 elif n=0 then 1 else procname(n-1) + 2*procname(n-3) + procname(n-5) fi: end: seq(A193147(n), n=0..32);

Series[1/(1 - x - 2*x^3 - x^5), {x, 0, 32}] // CoefficientList[#, x]& (* Jean-François Alcover, Apr 02 2015 *)

a(n):=sum(sum(binomial(j,3*n-5*m+2*j)*binomial(2*m-n,j)*2^(3*n-5*m+2*j), j,0,2*m-n),m,floor((n+1)/2),n); /* Vladimir Kruchinin, Mar 10 2013 */

G.f.: 1/(1-x-2*x^3-x^5) = -1 / ( (1+x+x^2)*(x^3-x^2+2*x-1) ).
a(n) = a(n-1) + 2*a(n-3) + a(n-5) with a(n) = 0 for n= -4, -3, -2, -1 and a(0) = 1.
a(n) = (5*b(n+1) - 4*b(n) + 3*b(n-1) + 2*c(n) + 3*c(n-1))/7 with b(n) = A005314(n) and c(n) = A049347(n).
G.f.: 1 + x/(U(0)-x) where G(k)= 1 - x^2*(k+1)/(1 - 1/(1 + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2012
a(n) = Sum_{m=floor((n+1)/2)..n} Sum_{j=0..2*m-n} C(j,3*n-5*m+2*j) * C(2*m-n,j) * 2^(3*n-5*m+2*j). - Vladimir Kruchinin, Mar 10 2013
With offset 1, the INVERT transform of (1 + 2x^2 + x^4). - Gary W. Adamson, Mar 30 2017
a(n) = Sum_{k=0..floor(2*n/5)} binomial(2*n-4*k,k). - Seiichi Manyama, Jun 14 2024

A001558 Number of hill-free Dyck paths of semilength n+3 and having length of first descent equal to 1 (a hill in a Dyck path is a peak at level 1).

Original entry on oeis.org

1, 3, 10, 33, 111, 379, 1312, 4596, 16266, 58082, 209010, 757259, 2760123, 10114131, 37239072, 137698584, 511140558, 1904038986, 7115422212, 26668376994, 100221202998, 377570383518, 1425706128480, 5394898197448, 20454676622476

Offset: 0

N. J. A. Sloane

a(n) is also the number of even-length descents to ground level in all Dyck paths of semilength n+2. Example: a(1)=3 because in UDUDUD, UDUU(DD), UU(DD)UD, UUDU(DD) and UUUDDD we have 3 even-length descents to ground level (shown between parentheses). - Emeric Deutsch, Oct 05 2008
Convolution of A000108 with A104629. - Philippe Deléham, Nov 11 2009
The Kn12 triangle sums of A039599 are given by the terms of this sequence. For the definition of this and other triangle sums see A180662. - Johannes W. Meijer, Apr 20 2011

			a(1)=3 because we have uu(d)ududd, uuu(d)uddd and uu(d)uuddd, where u=(1,1), d=(1,-1) (the first descents are shown between parentheses).
G.f. = 1 + 3*x + 10*x^2 + 33*x^3 + 111*x^4 + 379*x^5 + 1312*x^6 + ...

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..1000
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
T. Fine, Extrapolation when very little is known about the source, Information and Control 16 (1970), 331-359.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série. FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.

Cf. A000957, A118972, A118973.

Cf. A111301. - Emeric Deutsch, Oct 05 2008

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-Sqrt(1-4*x))/(x*(3-Sqrt(1-4*x)))*((1-Sqrt(1-4*x))/(2*x))^3 )); // G. C. Greubel, Feb 12 2019

F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: g:=F*C^3: gser:=series(g,z=0,32): seq(coeff(gser,z,n),n=0..27); # Emeric Deutsch, May 08 2006

CoefficientList[Series[(1-Sqrt[1-4*x])/(x*(3-Sqrt[1-4*x]))*((1-Sqrt[1-4*x])/(2*x))^3, {x, 0, 30}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

my(x='x+O('x^30)); Vec((1-sqrt(1-4*x))/(x*(3-sqrt(1-4*x)))*((1-sqrt(1-4*x))/(2*x))^3) \\ G. C. Greubel, Feb 12 2019

((1-sqrt(1-4*x))/(x*(3-sqrt(1-4*x)))*((1-sqrt(1-4*x))/(2*x))^3).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

a(n) = A000957(n+4) - A000957(n+3) - A000957(n+2) (A000957 are the Fine numbers). - Emeric Deutsch, May 08 2006
a(n) = A118972(n+3,1). - Emeric Deutsch, May 08 2006
G.f.: F*C^3, where F = (1-sqrt(1-4z))/(z*(3-sqrt(1-4z))) and C = (1-sqrt(1-4z))/(2z) is the Catalan function. - Emeric Deutsch, May 08 2006
a(n) = Sum_{k>=0} k*A111301(n+2,k). - Emeric Deutsch, Oct 05 2008
(n+3)*a(n) = (-(11/2)*n + 21/2)*a(n-3) + ((9/2)*n + 11/2)*a(n-1) + (-(1/2)*n + 9/2)*a(n-2) + (-2n + 5)*a(n-4). - Simon Plouffe, Feb 09 2012
a(n) ~ 11*2^(2*n+4)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014

Edited by Emeric Deutsch, May 08 2006

A023435 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-5).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 7, 11, 16, 24, 35, 52, 76, 112, 164, 241, 353, 518, 759, 1113, 1631, 2391, 3504, 5136, 7527, 11032, 16168, 23696, 34728, 50897, 74593, 109322, 160219, 234813, 344135, 504355, 739168, 1083304, 1587659, 2326828, 3410132, 4997792, 7324620, 10734753

Offset: 0

N. J. A. Sloane

nonn

Diagonal sums of Riordan array (1/(1-x), x(1+x+x^2)) yield a(n+1). - Paul Barry, Feb 16 2005
The Ca2 sums, see A180662 for the definition of these sums, of the "Races with Ties" triangle A035317 lead to this sequence. - Johannes W. Meijer, Jul 20 2011
Number of ordered partitions of (n-1) into parts less than or equal to 3, where the order of the 2's is unimportant. (see example). - David Neil McGrath, Apr 26 2015
Number of ordered partitions of (n-1) into parts less than or equal to 4, where the order of the 1's is unimportant.(see example). - David Neil McGrath, May 05 2015
List the partitions of n in nonincreasing order. Freeze the 1's and 2's in place and allow the other summands to vary their order without disturbing the 1's and 2's. The result is a(n+1). - Gregory L. Simay (based on correspondence with George E. Andrews), Jul 11 2016
Number of ordered partitions of n-1 where the order of the 1's and the 2's are unimportant. - Gregory L. Simay, Jul 18 2016

			There are 11 partitions of 6 into parts less than or equal to 3, where the order of 2's is unimportant, a(7)=11. These are (33),(321=231=312),(132=123=213),(3111),(1311),(1131),(1113),(222),(2211=1122=1221=2112=2121=1212),(21111=12111=11211=11121=11112),(111111). - _David Neil McGrath_, Apr 26 2015
There are 11 partitions of 6 into parts less than equal to 4, where the order of 1's is unimportant. These are (42),(24),(411=141=114),(33),(321=312=132),(231=213=123),(3111=1311=1131=1113),(222),(2211=1122=2112=1221=1212=2121),(21111=12111=11211=11121=11112),(111111). - _David Neil McGrath_, May 05 2015
There are a(9)=24 partitions of 8 where the 1's and 2's are frozen []: (8), (7[1]), (6[2]), (53), (35) (44), (6[1][1]), (5,[2][1]), (43[1]), (34[1]), (4[2][2]), (33[2][2]) (5[1][1][1]), (4[2][1][1]), (33[1][1]), (3[2][2][1]), ([2][2][2][2]), (4[1][1][1][1]), (3[2][1][1][1]), ([2][2][2][1][1]), (3[1][1][1][1][1]), ([2][2][1][1][1][1]), ([2][1][1][1][1][1][1]),([1][1][1][1][1][1][1][1]). - _Gregory L. Simay_, Jul 11 2016

Michael De Vlieger, Table of n, a(n) for n = 0..6024
John H. E. Cohn, Letter to the editor, Fib. Quart. 2 (1964), 108.
Verner E. Hoggatt, Jr. and Douglas A. Lind, The dying rabbit problem, Fib. Quart. 7 (1969), 482-487.
Zoltán Kása, On scattered subword complexity, arXiv:1104.4425 [cs.DM], 2011.
William J. Keith, Robert Schneider, and Andrew V. Sills, Composition-theoretic series and false theta functions, Integers (2024) Vol. 24A, Art. No. A11. See p. 11.
Anthony Shannon, François Dubeau, Mine Uysal, and Engin Özkan, A Difference Equation Model of Infectious Disease, Int. J. Bioautomation (2022) Vol. 26, No. 4, 339-352.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1).

First differences are in A013979.

Cf. A077864 (bisection).

I:=[0,1,1,2,3]; [n le 5 select I[n] else Self(n-1)+Self(n-2)-Self(n-5): n in [1..45]]; // Vincenzo Librandi, Apr 27 2015

LinearRecurrence[{1, 1, 0, 0, -1}, {0, 1, 1, 2, 3}, 50] (* Vincenzo Librandi, Apr 27 2015 *)

x='x+O('x^99); concat(0, Vec(x/((x-1)*(1+x)*(x^3+x-1)))) \\ Altug Alkan, Apr 09 2018

G.f.: x / ( (x-1)*(1+x)*(x^3+x-1) ). - R. J. Mathar, Nov 28 2011

More terms from Vincenzo Librandi, Apr 27 2015

A112970 A generalized Stern sequence.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 2, 2, 1, 5, 3, 3, 2, 5, 2, 3, 1, 6, 4, 3, 2, 6, 2, 3, 1, 7, 5, 4, 3, 8, 3, 5, 2, 8, 5, 4, 2, 8, 3, 3, 1, 9, 6, 5, 4, 9, 3, 6, 2, 9, 6, 4, 2, 9, 3, 3, 1, 10, 7, 6, 5, 11, 4, 8, 3, 12, 8, 6, 3, 13, 5, 5, 2, 13, 8, 7, 5, 12, 4, 7, 2, 12, 8, 5, 3, 11, 3, 4, 1, 12, 9, 7, 6

Offset: 0

Paul Barry, Oct 07 2005

Conjectures: a(2^n)=a(2^(n+1)+1)=A033638(n); a(2^n-1)=a(3*2^n-1)=1.

The Gi1 and Gi2 triangle sums, see A180662 for their definitions, of Sierpinski's triangle A047999 equal this sequence. The Gi1 and Gi2 sums can also be interpreted as (i + 4*j = n) and (4*i + j = n) sums, see the Northshield reference. Some A112970(2^n-p) sequences, 0<=p<=32, lead to known sequences, see the crossrefs. - Johannes W. Meijer, Jun 05 2011

Sam Northshield, Sums across Pascal's triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010. - _Johannes W. Meijer_, Jun 05 2011

Cf. A002487, A112971.
Cf. A120562 (Northshield).
Cf. A033638 (p=0), A000012 (p=1), A004526 (p=2, p=3, p=5, p=9, p=17), A002620 (p=4, p=7, p=13, p=25), A000027 (p=6, p=11, p=21), A004116 (p=8, p=15, p=29), A035106 (p=10, p=19), A024206 (p=14, p=27), A007494 (p=18), A014616 (p=22), A179207 (p=26). - Johannes W. Meijer, Jun 05 2011

A112970:=proc(n) option remember; if n <0 then A112970(n):=0 fi: if (n=0 or n=1) then 1 elif n mod 2 = 0 then A112970(n/2) + A112970((n/2)-2) else A112970((n-1)/2); fi; end: seq(A112970(n),n=0..99); # Johannes W. Meijer, Jun 05 2011

a[n_] := a[n] = Which[n<0, 0, n==0 || n==1, 1, Mod[n, 2]==0, a[n/2] + a[n/2-2], True, a[(n-1)/2]];
Table[a[n], {n, 0, 99}] (* Jean-François Alcover, Aug 02 2022 *)

a(n) = Sum_{k=0..n} mod(sum{j=0..n, (-1)^(n-k)*C(j, n-j)*C(k, j-k)}, 2).
From Johannes W. Meijer, Jun 05 2011: (Start)
a(2*n+1) = a(n) and a(2*n) = a(n) + a(n-2) with a(0) = 1, a(1) = 1 and a(n)=0 for n<=-1.
G.f.: Product_{n>=0} (1 + x^(2^n) + x^(4*2^n)). (End)
G.f. A(x) satisfies: A(x) = (1 + x + x^4) * A(x^2). - Ilya Gutkovskiy, Jul 09 2019

A124380 O.g.f.: A(x) = Sum_{n>=0} x^nProduct_{k=0..n} (1 + kx).

Original entry on oeis.org

1, 1, 2, 4, 9, 22, 57, 157, 453, 1368, 4296, 13995, 47138, 163779, 585741, 2152349, 8113188, 31326760, 123748871, 499539900, 2058542819, 8651755865, 37054078481, 161591063250, 717032333816, 3235298221401, 14834735654080, 69085973044125

Offset: 0

Paul D. Hanna, Oct 28 2006

nonn

The Kn11 triangle sums of A094638 are given by the terms of this sequence. For the definitions of this and other triangle sums see A180662. [Johannes W. Meijer, Apr 20 2011]

			A(x) = 1 + x*(1+x) + x^2*(1+x)*(1+2x) + x^3*(1+x)*(1+2x)*(1+3x) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..868
Giulio Cerbai, Anders Claesson, and Bruce E. Sagan, Self-modified difference ascent sequences, arXiv:2408.06959 [math.CO], 2024. See p. 15.

Cf. A024427, A171367.

nmax = 30; CoefficientList[Series[Sum[x^(2*k)*Pochhammer[1 + 1/x, k], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 14 2024 *)
Table[Sum[(-1)^k * StirlingS1[n+1-k, n+1-2*k], {k, 0, (n+1)/2}], {n, 0, 30}] (* Vaclav Kotesovec, Sep 18 2024 *)

a(n)=polcoeff(sum(k=0,n,x^k*prod(j=0,k,1+j*x+x*O(x^n))),n)

O.g.f.: A(x) = 1 + x*(1+x)/(G(0) - x*(1+x)) ; G(k) = 1+x*(k*x+x+1) - x*(k*x + 2*x + 1)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 02 2011
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - (1+x*k)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
G.f.: 1/(x*Q(0)-1)/x^4 + (1+x-x^3)/x^4, where Q(k)= 1 - x/(1 - (k+1)*x - x*(k+1)/(x - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 19 2013
Conjecture: log(a(n)) ~ n*log(n)/2 - n*(1 + log(2))/2. - Vaclav Kotesovec, Sep 18 2024

A144678 Related to enumeration of quantum states (see reference for precise definition).

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 13, 16, 22, 28, 34, 40, 50, 60, 70, 80, 95, 110, 125, 140, 161, 182, 203, 224, 252, 280, 308, 336, 372, 408, 444, 480, 525, 570, 615, 660, 715, 770, 825, 880, 946, 1012, 1078, 1144, 1222, 1300, 1378, 1456, 1547, 1638, 1729, 1820, 1925, 2030, 2135

Offset: 0

N. J. A. Sloane, Feb 06 2009

The Gi2 triangle sums of the triangle A159797 are linear sums of shifted versions of the sequence given above, i.e., Gi2(n) = a(n-1) + 2*a(n-2) + 2*a(n-3) + 3*a(n-4) + a(n-5). For the definitions of the Gi2 and other triangle sums see A180662. [Johannes W. Meijer, May 20 2011]
Partial sums of 1,1,1,1, 3,3,3,3, 6,6,6,6,..., the quadruplicated A000217. - R. J. Mathar, Aug 25 2013
Number of partitions of n into two different parts of size 4 and two different parts of size 1. a(4) = 7: 4, 4', 1111, 1111', 111'1', 11'1'1', 1'1'1'1'. - Alois P. Heinz, Dec 22 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10b, lambda=4]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,2,-4,2,0,-1,2,-1).

Cf. A000292, A006918, A144677, A144679, A190718.

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)*(1-x^4))^2 )); // G. C. Greubel, Oct 18 2021

n:=80; lambda:=4; S10b:=[];
for ii from 0 to n do
x:=floor(ii/lambda);
snc:=1/6*(x+1)*(x+2)*(3*ii-2*x*lambda+3);
S10b:=[op(S10b),snc];
od:
S10b;
A144678 := proc(n) option remember;
   local k;
   sum(A190718(n-k),k=0..3)
end:
A190718:= proc(n)
   binomial(floor(n/4)+3,3)
end:
seq(A144678(n),n=0..54); # Johannes W. Meijer, May 20 2011

a[n_] = (r = Mod[n, 4]; (4+n-r)(8+n-r)(3+n+2r)/96); Table[a[n], {n, 0, 54}] (* Jean-François Alcover, Sep 02 2011 *)
LinearRecurrence[{2,-1,0,2,-4,2,0,-1,2,-1}, {1,2,3,4,7,10,13,16,22,28}, 60] (* G. C. Greubel, Oct 18 2021 *)

Vec(1/(x-1)^4/(x^3+x^2+x+1)^2+O(x^99)) \\ Charles R Greathouse IV, Jun 20 2013

def A144678_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)*(1-x^4))^2 ).list()
A144678_list(60) # G. C. Greubel, Oct 18 2021

From Johannes W. Meijer, May 20 2011: (Start)
a(n) = A190718(n-3) + A190718(n-2) + A190718(n-1) + A190718(n).
a(n-3) + a(n-2) + a(n-1) + a(n) = A122046(n+3).
G.f.: 1/((x-1)^4*(x^3+x^2+x+1)^2). (End)
a(n) = A009531(n+5)/16 + (n+5)*(2*n^2+20*n+33+3*(-1)^n)/192 . - R. J. Mathar, Jun 20 2013
a(n) = Sum_{i=1..n+8} floor(i/4) * floor((n+8-i)/4). - Wesley Ivan Hurt, Jul 21 2014
From Alois P. Heinz, Dec 22 2021: (Start)
G.f.: 1/((1-x)*(1-x^4))^2.
a(n) = Sum_{j=0..floor(n/4)} (j+1)*(n-4*j+1). (End)

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A124380 O.g.f.: A(x) = Sum_{n>=0} x^nProduct_{k=0..n} (1 + kx).