A027656 -id:A027656 - cp's OEIS frontend

A158943 INVERT transform of A027656: (1, 0, 2, 0, 3, 0, 4, 0, 5, ...).

1, 1, 3, 5, 10, 19, 36, 69, 131, 250, 476, 907, 1728, 3292, 6272, 11949, 22765, 43371, 82629, 157422, 299915, 571388, 1088589, 2073943, 3951206, 7527704, 14341527, 27322992, 52054840, 99173120, 188941273, 359964521, 685792227, 1306548149

Offset: 1

Gary W. Adamson, Mar 31 2009

nonn

Equals row sums of triangle A158945.

Number of compositions of n into odd parts where there is 1 sort of part 1, 2 sorts of part 3, 3 sorts of part 5, ..., k sorts of part 2*k-1. - Joerg Arndt, Aug 04 2014

			The INVERT transform of (1, N, ...) begins (1, (N+1), ...) so that we have (1, 1, ...) placed in ascending magnitude in the bottom row. In the top row we place an equal number of descending terms: (..., 0, 3, 0, 2, 0, 1). Take the dot product of terms in top and bottom rows, adding the result to the next term A027656: (1, 0, 2, 0, 3, ...). a(6) = 19 given: 3, 0, 2, 0, 1 1, 1, 3, 5, 10 Dot product of top row terms * bottom row terms = (1, 0, 2, 0, 3) dot (1, 1, 3, 5, 10) = (3 + 0 + 6 + 0 + 10) = 19, which is added to the next term in (1, 0, 2, 0, 3, ...); i.e., (an 0) = 19.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024. See p. 18.
Jia Huang, A coin flip game and generalizations of Fibonacci numbers, arXiv:2501.07463 [math.CO], 2025. See p. 9.
Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1).

Cf. A027656, A052535, A158944, A158945.

a:=[1,1,3,5];; for n in [5..40] do a[n]:=a[n-1]+2*a[n-2]-a[n-4]; od; a; # G. C. Greubel, Jul 12 2019

I:=[1,1,3,5]; [n le 4 select I[n] else Self(n-1)+2*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 09 2019

A027656 := proc(n) if type(n,odd) then 0; else n/2+1 ; fi; end: L := [seq(A027656(n), n=0..100)] ; read("transforms"); INVERT(L) ; # R. J. Mathar, Apr 02 2009

LinearRecurrence[{1, 2, 0, -1}, {1, 1, 3, 5}, 40] (* Vincenzo Librandi, Jul 09 2019 *)

my(x='x+O('x^40)); Vec(x/(1-x-2*x^2+x^4)) \\ G. C. Greubel, Jul 12 2019

a=(x/(1-x-2*x^2+x^4)).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jul 12 2019

INVERT transform of (1, 0, 2, 0, 3, 0, 4, ...); i.e., the natural numbers interleaved with zeros.
From R. J. Mathar, Apr 02 2009: (Start)
a(n) = a(n-1) + 2*a(n-2) - a(n-4).
G.f.: x/(1 - x - 2*x^2 + x^4). (End)
The sequence is the second INVERT transform of (1, -1, 3, -5, 10, -19, ...). - Gary W. Adamson, Jul 08 2019
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k-1,k). - Seiichi Manyama, Jun 12 2024

Extended by R. J. Mathar, Apr 02 2009

A289843 p-INVERT of (1,0,2,0,3,0,4,0,5,...) (A027656), where p(S) = 1 - S - S^2.

Original entry on oeis.org

1, 2, 5, 13, 29, 73, 168, 410, 962, 2317, 5483, 13131, 31193, 74509, 177311, 423025, 1007505, 2402354, 5723761, 13644587, 32514730, 77501115, 184698088, 440216833, 1049148789, 2500520812, 5959478837, 14203542282, 33851496564, 80679640434, 192285583548

Offset: 0

Clark Kimberling, Aug 12 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

See A289780 for a guide to related sequences.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 5, -2, -6, 1, 4, 0, -1)

Cf. A027656, A289780.

z = 60; s = x/(1 - x^2)^2; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A027656 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289843 *)

G.f.: (1 + x - 2 x^2 + x^4)/(1 - x - 5 x^2 + 2 x^3 + 6 x^4 - x^5 - 4 x^6 + x^8).

a(n) = a(n-1) + 5*a(n-2) - 2*a(n-3) - 6*a(n-4) + a(n-5) + 4*a(n-6) - a(n-8).

A081037 Inverse binary transform of A027656.

Original entry on oeis.org

1, -1, 3, -7, 16, -36, 80, -176, 384, -832, 1792, -3840, 8192, -17408, 36864, -77824, 163840, -344064, 720896, -1507328, 3145728, -6553600, 13631488, -28311552, 58720256, -121634816, 251658240, -520093696, 1073741824, -2214592512, 4563402752, -9395240960, 19327352832

Offset: 0

Paul Barry, Mar 03 2003

Vincenzo Librandi and Evert Provoost, Table of n, a(n) for n = 0..1000 [Terms 0 through 300 were computed by Vincenzo Librandi; terms 301 through 1000 by Evert Provoost, Jan 25 2016]
F. Ellermann, Illustration of binomial transforms
Index entries for linear recurrences with constant coefficients, signature (-4,-4).

Cf. A027656, A045623, A045891.

I:=[1, -1, 3, -7]; [n le 4 select I[n] else -4*Self(n-1)-4*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 06 2013

CoefficientList[Series[(1 + x)^3 / (1 + 2 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{-4,-4},{1,-1,3,-7},40] (* Harvey P. Dale, Apr 14 2019 *)

Vec((1+x)^3/(1+2*x)^2 + O(x^40)) \\ Michel Marcus, Jan 25 2016

G.f.: (1+x)^3/(1+2*x)^2.
a(n) = (-1)^n*A045623(n+1)/4, n>1. - R. J. Mathar, Sep 27 2012
Recurrence: a(n) = -4a(n-1) - 4a(n-2), a(0)=1, a(1)=-1, a(2)=3, a(3)=-7. - Ralf Stephan, Jul 14 2013
E.g.f.: exp(-x)*(2*cosh(x) + x*sinh(x))/2. - Stefano Spezia, Apr 24 2023

A136520 a(n) = Sum_{k=1..n} A001263(n,k) * A027656(k).

Original entry on oeis.org

1, 1, 3, 13, 44, 146, 530, 1975, 7314, 27262, 102802, 390138, 1486064, 5682756, 21812436, 83976075, 324115550, 1253795510, 4859960402, 18871869302, 73398851448, 285882923196, 1114943553308, 4353426835238, 17016813133124, 66581653586476, 260750813149140, 1022023318047220

Offset: 1

Gary W. Adamson, Jan 02 2008

Narayana transform of A027656.

			a(4) = 13 = (1, 6, 6, 1) dot (1, 0, 2, 0) = (1 + 0 + 12 + 0).
Triangle A001263(n,k) * A027656(k+1) and the rows sums:
  1;                                              :     1;
  1,  0;                                          :     1;
  1,  0,    2;                                    :     3;
  1,  0,   12,  0;                                :    13;
  1,  0,   40,  0,     3;                         :    44;
  1,  0,  100,  0,    45,  0;                     :   146;
  1,  0,  210,  0,   315,  0,     4;              :   530;
  1,  0,  392,  0,  1470,  0,   112,  0;          :  1975;
  1,  0,  672,  0,  5292,  0,  1344,  0,    5;    :  7314;
  1,  0, 1080,  0, 15876,  0, 10080,  0,  225,  0 : 27262;

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001263, A027656.

A136520:= func< n | (&+[((j+1)/(2*j+1))*Binomial(n,2*j)*Binomial(n-1,2*j): j in [0..Floor((n-1)/2)]]) >;
[A136520(n): n in [1..40]]; // G. C. Greubel, Jul 27 2023

A136520[n_]:= Sum[Binomial[n-1, 2*k]*Binomial[n, 2*k]*((k+1)/(2*k+1)), {k,0,Floor[(n-1)/2]}];
Table[A136520[n], {n, 40}] (* G. C. Greubel, Jul 27 2023 *)

def A136520(n): return sum(((j+1)/(2*j+1))*binomial(n,2*j)*binomial(n-1, 2*j) for j in range((n+1)//2))
[A136520(n) for n in range(1,41)] # G. C. Greubel, Jul 27 2023

a(n) = Sum_{k=1..n} A001263(n,k) * A027656(k).

a(n) = Sum_{j=0..floor((n-1)/2)} ((j+1)/(2*j+1))*binomial(n, 2*j) * binomial(n-1, 2*j). - G. C. Greubel, Jul 27 2023

Terms a(11) onward added by G. C. Greubel, Jul 27 2023

A213042 Convolution of (1,0,2,0,3,0,...) and (1,0,0,2,0,0,3,0,0,...); i.e., (A027656(n)) and (A175676(n+2)).

Original entry on oeis.org

1, 0, 2, 2, 3, 4, 7, 6, 11, 12, 15, 18, 24, 24, 33, 36, 42, 48, 58, 60, 74, 80, 90, 100, 115, 120, 140, 150, 165, 180, 201, 210, 237, 252, 273, 294, 322, 336, 371, 392, 420, 448, 484, 504, 548, 576, 612, 648, 693, 720, 774, 810, 855, 900, 955, 990, 1055

Offset: 0

Clark Kimberling, Jun 10 2012

			a(6) = (1,0,2,0,3,0,4)**(1,0,0,2,0,0,3) = 1*3 + 0*0 + 2*0 + 0*2 + 3*0 + 0*0 + 4*1 = 7.

Index entries for linear recurrences with constant coefficients, signature (0,2,2,-1,-4,-1,2,2,0,-1).

s = Normal[Series[1/((1 - x^2)^2 (1 - x^3)^2),
{x, 0, 80}]]
c = CoefficientList[s, t]  (* A213042 *)

a(n) = 2*a(n-2)+2*a(n-3)-a(n-4)-4*a(n-5)-a(n-6)+2*a(n-7)+2*a(n-8)-a(n-10).

G.f.: 1/(((1 - x^2)^2)*(1 - x^3)^2).

A258369 Stirling-Bernoulli transform of A027656.

Original entry on oeis.org

1, 1, 5, 25, 173, 1441, 14165, 160105, 2044733, 29105521, 456781925, 7834208185, 145760370893, 2923764916801, 62891469229685, 1444055265984265, 35250519098274653, 911569049328779281, 24893164161460525445, 715822742720760256345, 21620050147748210572013

Offset: 0

Philippe Deléham, May 28 2015

Also called Akiyama-Tanigawa transform of A027656.

			a(0) = 1*1 = 1.
a(1) = 1*1 = 1.
a(2) = 1*1 + 2*2 = 5.
a(3) = 1*1 + 12*2 = 25.
a(4) = 1*1 + 50*2 + 24*3 = 173.

Cf. A027656, A163626, A249163.

a(n) = Sum_{k = 0..n} A163626(n,k)*A027656(k).
a(n) = Sum_{k>=0} A249163(n,k) * (k+1).
E.g.f.: 1/(exp(x)*(2 - exp(x))^2).
a(n) ~ n! * n / (8 * (log(2))^(n+2)). - Vaclav Kotesovec, Jul 01 2018

A006918 a(n) = binomial(n+3, 3)/4 for odd n, n(n+2)(n+4)/24 for even n.

Original entry on oeis.org

0, 1, 2, 5, 8, 14, 20, 30, 40, 55, 70, 91, 112, 140, 168, 204, 240, 285, 330, 385, 440, 506, 572, 650, 728, 819, 910, 1015, 1120, 1240, 1360, 1496, 1632, 1785, 1938, 2109, 2280, 2470, 2660, 2870, 3080, 3311, 3542, 3795, 4048, 4324, 4600, 4900, 5200, 5525, 5850, 6201, 6552, 6930

Offset: 0

N. J. A. Sloane

Maximal number of inconsistent triples in a tournament on n+2 nodes [Kac]. - corrected by Leen Droogendijk, Nov 10 2014
a(n-4) is the number of aperiodic necklaces (Lyndon words) with 4 black beads and n-4 white beads.
a(n-3) is the maximum number of squares that can be formed from n lines, for n>=3. - Erich Friedman; corrected by Leen Droogendijk, Nov 10 2014
Number of trees with diameter 4 where at most 2 vertices 1 away from the graph center have degree > 2. - Jon Perry, Jul 11 2003
a(n+1) is the number of partitions of n into parts of two kinds, with at most two parts of each kind. Also a(n-3) is the number of partitions of n with Durfee square of size 2. - Franklin T. Adams-Watters, Jan 27 2006
Factoring the g.f. as x/(1-x)^2 times 1/(1-x^2)^2 we find that the sequence equals (1, 2, 3, 4, ...) convolved with (1, 0, 2, 0, 3, 0, 4, ...), A000027 convolved with its aerated variant. -  Gary W. Adamson, May 01 2009
Starting with "1" = triangle A171238 * [1,2,3,...]. - Gary W. Adamson, Dec 05 2009
The Kn21, Kn22, Kn23, Fi2 and Ze2 triangle sums, see A180662 for their definitions, of the Connell-Pol triangle A159797 are linear sums of shifted versions of this sequence, e.g., Kn22(n) = a(n+1) + a(n) + 2*a(n-1) + a(n-2) and Fi2(n) = a(n) + 4*a(n-1) + a(n-2). - Johannes W. Meijer, May 20 2011
For n>3, a(n-4) is the number of (w,x,y,z) having all terms in {1,...,n} and w+x+y+z=|x-y|+|y-z|. - Clark Kimberling, May 23 2012
a(n) is the number of (w,x,y) having all terms in {0,...,n} and w+x+y < |w-x|+|x-y|. - Clark Kimberling, Jun 13 2012
For n>0 number of inequivalent (n-1) X 2 binary matrices, where equivalence means permutations of rows or columns or the symbol set. - Alois P. Heinz, Aug 17 2014
Number of partitions p of n+5 such that p[3] = 2. Examples: a(1)=1 because we have (2,2,2); a(2)=2 because we have (2,2,2,1) and (3,2,2); a(3)=5 because we have (2,2,2,1,1), (2,2,2,2), (3,2,2,1), (3,3,2), and (4,2,2). See the R. P. Stanley reference. - Emeric Deutsch, Oct 28 2014
Sum over each antidiagonal of A243866. - Christopher Hunt Gribble, Apr 02 2015
Number of nonisomorphic outer planar graphs of order n>=3, size n+2, and maximum degree 3. - Christian Barrientos and Sarah Minion, Feb 27 2018
a(n) is the number of 2413-avoiding odd Grassmannian permutations of size n+1. - Juan B. Gil, Mar 09 2023

			G.f. = x + 2*x^2 + 5*x^3 + 8*x^4 + 14*x^5 + 20*x^6 + 30*x^7 + 40*x^8 + 55*x^9 + ...
From _Gus Wiseman_, Apr 06 2019: (Start)
The a(4 - 3) = 1 through a(8 - 3) = 14 integer partitions with Durfee square of length 2 are the following (see Franklin T. Adams-Watters's second comment). The Heinz numbers of these partitions are given by A325164.
  (22)  (32)   (33)    (43)     (44)
        (221)  (42)    (52)     (53)
               (222)   (322)    (62)
               (321)   (331)    (332)
               (2211)  (421)    (422)
                       (2221)   (431)
                       (3211)   (521)
                       (22111)  (2222)
                                (3221)
                                (3311)
                                (4211)
                                (22211)
                                (32111)
                                (221111)
The a(0 + 1) = 1 through a(4 + 1) = 14 integer partitions of n into parts of two kinds with at most two parts of each kind are the following (see Franklin T. Adams-Watters's first comment).
  ()()  ()(1)  ()(2)   ()(3)    ()(4)
        (1)()  (2)()   (3)()    (4)()
               ()(11)  (1)(2)   (1)(3)
               (1)(1)  ()(21)   ()(22)
               (11)()  (2)(1)   (2)(2)
                       (21)()   (22)()
                       (1)(11)  ()(31)
                       (11)(1)  (3)(1)
                                (31)()
                                (11)(2)
                                (1)(21)
                                (2)(11)
                                (21)(1)
                                (11)(11)
The a(6 - 5) = 1 through a(10 - 5) = 14 integer partitions whose third part is 2 are the following (see Emeric Deutsch's comment). The Heinz numbers of these partitions are given by A307373.
  (222)  (322)   (332)    (432)     (442)
         (2221)  (422)    (522)     (532)
                 (2222)   (3222)    (622)
                 (3221)   (3321)    (3322)
                 (22211)  (4221)    (4222)
                          (22221)   (4321)
                          (32211)   (5221)
                          (222111)  (22222)
                                    (32221)
                                    (33211)
                                    (42211)
                                    (222211)
                                    (322111)
                                    (2221111)
(End)

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 147.
M. Kac, An example of "counting without counting", Philips Res. Reports, 30 (1975), 20*-22* [Special issue in honour of C. J. Bouwkamp].
E. V. McLaughlin, Numbers of factorizations in non-unique factorial domains, Senior Thesis, Allegeny College, Meadville, PA, 2004.
K. B. Reid and L. W. Beineke "Tournaments", pp. 169-204 in L. W. Beineke and R. J. Wilson, editors, Selected Topics in Graph Theory, Academic Press, NY, 1978, p. 186, Theorem 6.11.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 2nd ed., 2012, Exercise 4.16, pp. 530, 552.
W. A. Whitworth, DCC Exercises in Choice and Chance, Stechert, NY, 1945, p. 33.

T. D. Noe, Table of n, a(n) for n = 0..1000
Jean-Luc Baril, Alexander Burstein, and Sergey Kirgizov, Pattern statistics in faro words and permutations, arXiv:2010.06270 [math.CO], 2020.
David J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
Yuriy Choliy and Andrew V. Sills, A formula for the partition function that “counts”, Preprint 2015.
L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
S. J. Cyvin et al., Polygonal systems including the corannulene and coronene homologs: novel applications of Pólya's theorem, Z. Naturforsch., 52a (1997), 867-873.
Steven Edwards and William Griffiths, Generalizations of Delannoy and cross polytope numbers, Fib. Q., Vol. 55, No. 4 (2017), pp. 356-366.
B. G. Eke, Monotonic triads, Discrete Math., Vol. 9, No. 4 (1974), pp. 359-363. MR0354390 (50 #6869)
Irene Erazo, John López, and Carlos Trujillo, A combinatorial problem that arose in integer B_3 Sets, Revista Colombiana de Matemáticas, Vol. 53, No. 2 (2019), pp. 195-203.
Juan B. Gil and Jessica A. Tomasko, Pattern-avoiding even and odd Grassmannian permutations, arXiv:2207.12617 [math.CO], 2022.
John Golden and Marcus Spradlin, Collinear and Soft Limits of Multi-Loop Integrands in N= 4 Yang-Mills, arXiv preprint arXiv:1203.1915 [hep-th], 2012. - From _N. J. A. Sloane_, Sep 14 2012
Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T. Ricolfi, Enumeration of partitions via socle reduction, arXiv:2501.10267 [math.CO], 2025. See p. 40.
Brian Hopkins and Aram Tangboonduangjit, Water Cells in Compositions of 1s and 2s, arXiv:2412.11528 [math.CO], 2024. See p. 3.
Mohammed L. Nadji, Moussa Ahmia, Daniel F. Checa, and José L. Ramírez, Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants, J. Int. Seq. (2025) Vol. 28, Issue 3, Article 25.3.6. See p. 11.
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=2]
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).
Index entries for sequences related to Lyndon words.

Cf. A000031, A001037, A028723, A051168. a(n) = T(n,4), array T as in A051168.
Cf. A000094.
Cf. A171238. - Gary W. Adamson, Dec 05 2009
Row sums of A173997. - Gary W. Adamson, Mar 05 2010
Column k=2 of A242093. Column k=2 of A115720 and A115994.
Cf. A117485, A257990, A307373, A325164, A325168.

a006918 n = a006918_list !! n
a006918_list = scanl (+) 0 a008805_list
-- Reinhard Zumkeller, Feb 01 2013

[Floor(Binomial(n+4, 4)/(n+4))-Floor((n+2)/8)*(1+(-1)^n)/2: n in [0..60]]; // Vincenzo Librandi, Nov 10 2014

with(combstruct):ZL:=[st,{st=Prod(left,right),left=Set(U,card=r),right=Set(U,card=r),U=Sequence(Z,card>=3)}, unlabeled]: subs(r=1,stack): seq(count(subs(r=2,ZL),size=m),m=11..58) ; # Zerinvary Lajos, Mar 09 2007
A006918 := proc(n)
    if type(n,'even') then
        n*(n+2)*(n+4)/24 ;
    else
        binomial(n+3,3)/4 ;
    fi ;
end proc: # R. J. Mathar, May 17 2016

f[n_]:=If[EvenQ[n],(n(n+2)(n+4))/24,Binomial[n+3,3]/4]; Join[{0},Array[f,60]]  (* Harvey P. Dale, Apr 20 2011 *)
durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
Table[Length[Select[IntegerPartitions[n],durf[#]==2&]],{n,0,30}] (* Gus Wiseman, Apr 06 2019 *)

{ parttrees(n)=local(pt,k,nk); if (n%2==0, pt=(n/2+1)^2, pt=ceil(n/2)*(ceil(n/2)+1)); pt+=floor(n/2); for (x=1,floor(n/2),pt+=floor(x/2)+floor((n-x)/2)); if (n%2==0 && n>2, pt-=floor(n/4)); k=1; while (3*k<=n, for (x=k,floor((n-k)/2), pt+=floor(k/2); if (x!=k, pt+=floor(x/2)); if ((n-x-k)!=k && (n-x-k)!=x, pt+=floor((n-x-k)/2))); k++); pt }

{a(n) = n += 2; (n^3 - n * (2-n%2)^2) / 24}; /* Michael Somos, Aug 15 2009 */

G.f.: x/((1-x)^2*(1-x^2)^2) = x/((1+x)^2*(1-x)^4).
0, 0, 0, 1, 2, 5, 8, 14, ... has a(n) = (Sum_{k=0..n} floor(k(n-k)/2))/2. - Paul Barry, Sep 14 2003
0, 0, 0, 0, 0, 1, 2, 5, 8, 14, 20, 30, 40, 55, ... has a(n) = binomial(floor(1/2 n), 3) + binomial(floor(1/2 n + 1/2), 3) [Eke]. - N. J. A. Sloane, May 12 2012
a(0)=0, a(1)=1, a(n) = (2/(n-1))*a(n-1) + ((n+3)/(n-1))*a(n-2). - Benoit Cloitre, Jun 28 2004
a(n) = floor(binomial(n+4, 4)/(n+4)) - floor((n+2)/8)(1+(-1)^n)/2. - Paul Barry, Jan 01 2005
a(n+1) = a(n) + binomial(floor(n/2)+2,2), i.e., first differences are A008805. Convolution of A008619 with itself, then shifted right (or A004526 with itself, shifted left by 3). - Franklin T. Adams-Watters, Jan 27 2006
a(n+1) = (A027656(n) + A003451(n+5))/2 with a(1)=0. - Yosu Yurramendi, Sep 12 2008
Linear recurrence: a(n) = 2a(n-1) + a(n-2) - 4a(n-3) + a(n-4) + 2a(n-5) - a(n-6). - Jaume Oliver Lafont, Dec 05 2008
Euler transform of length 2 sequence [2, 2]. - Michael Somos, Aug 15 2009
a(n) = -a(-4-n) for all n in Z.
a(n+1) + a(n) = A002623(n). - Johannes W. Meijer, May 20 2011
a(n) = (n+2)*(2*n*(n+4)-3*(-1)^n+3)/48. - Bruno Berselli, May 21 2011
a(2n) = A007290(n+2). - Jon Perry, Nov 10 2014
G.f.: (1/(1-x)^4-1/(1-x^2)^2)/4. - Herbert Kociemba, Oct 23 2016
E.g.f.: (x*(18 + 9*x + x^2)*cosh(x) + (6 + 15*x + 9*x^2 + x^3)*sinh(x))/24. - Stefano Spezia, Dec 07 2021
From Amiram Eldar, Mar 20 2022: (Start)
Sum_{n>=1} 1/a(n) = 75/4 - 24*log(2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 69/4 - 24*log(2). (End)

A289780 p-INVERT of the positive integers (A000027), where p(S) = 1 - S - S^2.

Original entry on oeis.org

1, 4, 14, 47, 156, 517, 1714, 5684, 18851, 62520, 207349, 687676, 2280686, 7563923, 25085844, 83197513, 275925586, 915110636, 3034975799, 10065534960, 33382471801, 110713382644, 367182309614, 1217764693607, 4038731742156, 13394504020957, 44423039068114

Offset: 0

Clark Kimberling, Aug 10 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x).
Taking p(S) = 1 - S gives the INVERT transform of s, so that p-INVERT is a generalization of the INVERT transform (e.g., A033453).
Guide to p-INVERT sequences using p(S) = 1 - S - S^2:
t(A000012) = t(1,1,1,1,1,1,1,...)    = A001906
t(A000290) = t(1,4,9,16,25,36,...)   = A289779
t(A000027) = t(1,2,3,4,5,6,7,8,...)  = A289780
t(A000045) = t(1,2,3,5,8,13,21,...)  = A289781
t(A000032) = t(2,1,3,4,7,11,14,...)  = A289782
t(A000244) = t(1,3,9,27,81,243,...)  = A289783
t(A000302) = t(1,4,16,64,256,...)    = A289784
t(A000351) = t(1,5,25,125,625,...)   = A289785
t(A005408) = t(1,3,5,7,9,11,13,...)  = A289786
t(A005843) = t(2,4,6,8,10,12,14,...) = A289787
t(A016777) = t(1,4,7,10,13,16,...)   = A289789
t(A016789) = t(2,5,8,11,14,17,...)   = A289790
t(A008585) = t(3,6,9,12,15,18,...)   = A289795
t(A000217) = t(1,3,6,10,15,21,...)   = A289797
t(A000225) = t(1,3,7,15,31,63,...)   = A289798
t(A000578) = t(1,8,27,64,625,...)    = A289799
t(A000984) = t(1,2,6,20,70,252,...)  = A289800
t(A000292) = t(1,4,10,20,35,56,...)  = A289801
t(A002620) = t(1,2,4,6,9,12,16,...)  = A289802
t(A001906) = t(1,3,8,21,55,144,...)  = A289803
t(A001519) = t(1,1,2,5,13,34,...)    = A289804
t(A103889) = t(2,1,4,3,6,5,8,7,,...) = A289805
t(A008619) = t(1,1,2,2,3,3,4,4,...)  = A289806
t(A080513) = t(1,2,2,3,3,4,4,5,...)  = A289807
t(A133622) = t(1,2,1,3,1,4,1,5,...)  = A289809
t(A000108) = t(1,1,2,5,14,42,...)    = A081696
t(A081696) = t(1,1,3,9,29,97,...)    = A289810
t(A027656) = t(1,0,2,0,3,0,4,0,5...) = A289843
t(A175676) = t(1,0,0,2,0,0,3,0,...)  = A289844
t(A079977) = t(1,0,1,0,2,0,3,...)    = A289845
t(A059841) = t(1,0,1,0,1,0,1,...)    = A289846
t(A000040) = t(2,3,5,7,11,13,...)    = A289847
t(A008578) = t(1,2,3,5,7,11,13,...)  = A289828
t(A000142) = t(1!, 2!, 3!, 4!, ...)  = A289924
t(A000201) = t(1,3,4,6,8,9,11,...)   = A289925
t(A001950) = t(2,5,7,10,13,15,...)   = A289926
t(A014217) = t(1,2,4,6,11,17,29,...) = A289927
t(A000045*) = t(0,1,1,2,3,5,...)     = A289975 (* indicates prepended 0's)
t(A000045*) = t(0,0,1,1,2,3,5,...)   = A289976
t(A000045*) = t(0,0,0,1,1,2,3,5,...) = A289977
t(A290990*) = t(0,1,2,3,4,5,...)     = A290990
t(A290990*) = t(0,0,1,2,3,4,5,...)   = A290991
t(A290990*) = t(0,0,01,2,3,4,5,...)  = A290992

			Example 1:  s = (1,2,3,4,5,6,...) = A000027 and p(S) = 1 - S.
S(x) = x + 2x^2 + 3x^3 + 4x^4 + ...
p(S(x)) = 1 - (x + 2x^2 + 3x^3 + 4x^4 + ... )
- p(0) + 1/p(S(x)) = -1 + 1 + x + 3x^2 + 8x^3 + 21x^4 + ...
T(x) = 1 + 3x + 8x^2 + 21x^3 + ...
t(s) = (1,3,8,21,...) = A001906.
***
Example 2:  s = (1,2,3,4,5,6,...) = A000027 and p(S) = 1 - S - S^2.
S(x) =  x + 2x^2 + 3x^3 + 4x^4 + ...
p(S(x)) = 1 - ( x + 2x^2 + 3x^3 + 4x^4 + ...) - ( x + 2x^2 + 3x^3 + 4x^4 + ...)^2
- p(0) + 1/p(S(x)) = -1 + 1 + x + 4x^2 + 14x^3 + 47x^4 + ...
T(x) = 1 + 4x + 14x^2 + 47x^3 + ...
t(s) = (1,4,14,47,...) = A289780.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -7, 5, -1)

Cf. A000027.

P:=[1,4,14,47];; for n in [5..10^2] do P[n]:=5*P[n-1]-7*P[n-2]+5*P[n-3]-P[n-4]; od; P; # Muniru A Asiru, Sep 03 2017

z = 60; s = x/(1 - x)^2; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289780 *)

x='x+O('x^99); Vec((1-x+x^2)/(1-5*x+7*x^2-5*x^3+x^4)) \\ Altug Alkan, Aug 13 2017

G.f.: (1 - x + x^2)/(1 - 5 x + 7 x^2 - 5 x^3 + x^4).

a(n) = 5*a(n-1) - 7*a(n-2) + 5*a(n-3) - a(n-4).

A299257 Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.12.12 2D tiling (cf. A250122).

Original entry on oeis.org

1, 5, 12, 22, 36, 56, 82, 111, 144, 183, 226, 272, 324, 382, 442, 505, 576, 653, 730, 810, 900, 996, 1090, 1187, 1296, 1411, 1522, 1636, 1764, 1898, 2026, 2157, 2304, 2457, 2602, 2750, 2916, 3088, 3250, 3415, 3600, 3791, 3970, 4152, 4356, 4566, 4762, 4961, 5184, 5413, 5626, 5842, 6084, 6332

Offset: 0

N. J. A. Sloane, Feb 07 2018

B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56. See tiling #19.

Colin Barker, Table of n, a(n) for n = 0..1000
Reticular Chemistry Structure Resource (RCSR), The ttw tiling (or net)
Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).

Cf. A250122.
Partial sums: A299263.
The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e:  A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.

LinearRecurrence[{3, -5, 7, -7, 5, -3, 1}, {1, 5, 12, 22, 36, 56, 82, 111, 144, 183}, 60] (* Paolo Xausa, Jun 20 2024 *)

Vec((1 + x)*(1 + x + x^2 + 3*x^3 - x^4 + 5*x^5 - 3*x^6 + 4*x^7 - 2*x^8) / ((1 - x)^3*(1 + x^2)^2) + O(x^60)) \\ Colin Barker, Feb 09 2018

G.f.: (2*x^8 - 4*x^7 + 3*x^6 - 5*x^5 + x^4 - 3*x^3 - x^2 - x - 1)*(x + 1) / ((x - 1)^3*(x^2 + 1)^2).
From Colin Barker, Feb 09 2018: (Start)
a(n) = (4 - (2+8*i)*(-i)^n - (2-8*i)*i^n + i*((-i)^n-i^n)*n + 18*n^2) / 8 for n>2, where i=sqrt(-1).
a(n) = 3*a(n-1) - 5*a(n-2) + 7*a(n-3) - 7*a(n-4) + 5*a(n-5) - 3*a(n-6) + a(n-7) for n>9. (End)
a(n) = 1/2 + 9*n^2/4 + (-1)^floor(n/2)*(A027656(n-1)/2 - A010699(n)/4). - R. J. Mathar, Feb 12 2021

cp's OEIS Frontend

A035528 Euler transform of A027656(n-1).

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A158943 INVERT transform of A027656: (1, 0, 2, 0, 3, 0, 4, 0, 5, ...).

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A289843 p-INVERT of (1,0,2,0,3,0,4,0,5,...) (A027656), where p(S) = 1 - S - S^2.

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A081037 Inverse binary transform of A027656.

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A136520 a(n) = Sum_{k=1..n} A001263(n,k) * A027656(k).

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A213042 Convolution of (1,0,2,0,3,0,...) and (1,0,0,2,0,0,3,0,0,...); i.e., (A027656(n)) and (A175676(n+2)).

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A258369 Stirling-Bernoulli transform of A027656.

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A006918 a(n) = binomial(n+3, 3)/4 for odd n, n(n+2)(n+4)/24 for even n.