A095121 -id:A095121 - cp's OEIS frontend

A000918 a(n) = 2^n - 2.

-1, 0, 2, 6, 14, 30, 62, 126, 254, 510, 1022, 2046, 4094, 8190, 16382, 32766, 65534, 131070, 262142, 524286, 1048574, 2097150, 4194302, 8388606, 16777214, 33554430, 67108862, 134217726, 268435454, 536870910, 1073741822, 2147483646, 4294967294, 8589934590, 17179869182, 34359738366, 68719476734, 137438953470

Offset: 0

N. J. A. Sloane

For n > 1, a(n) is the expected number of tosses of a fair coin to get n-1 consecutive heads. - Pratik Poddar, Feb 04 2011
For n > 2, Sum_{k=1..a(n)} (-1)^binomial(n, k) = A064405(a(n)) + 1 = 0. - Benoit Cloitre, Oct 18 2002
For n > 0, the number of nonempty proper subsets of an n-element set. - Ross La Haye, Feb 07 2004
Numbers j such that abs( Sum_{k=0..j} (-1)^binomial(j, k)*binomial(j + k, j - k) ) = 1. - Benoit Cloitre, Jul 03 2004
For n > 2 this formula also counts edge-rooted forests in a cycle of length n. - Woong Kook (andrewk(AT)math.uri.edu), Sep 08 2004
For n >= 1, conjectured to be the number of integers from 0 to (10^n)-1 that lack 0, 1, 2, 3, 4, 5, 6 and 7 as a digit. - Alexandre Wajnberg, Apr 25 2005
Beginning with a(2) = 2, these are the partial sums of the subsequence of A000079 = 2^n beginning with A000079(1) = 2. Hence for n >= 2 a(n) is the smallest possible sum of exactly one prime, one semiprime, one triprime, ... and one product of exactly n-1 primes. A060389 (partial sums of the primorials, A002110, beginning with A002110(1)=2) is the analog when all the almost primes must also be squarefree. - Rick L. Shepherd, May 20 2005
From the second term on (n >= 1), the binary representation of these numbers is a 0 preceded by (n - 1) 1's. This pattern (0)111...1110 is the "opposite" of the binary 2^n+1: 1000...0001 (cf. A000051). - Alexandre Wajnberg, May 31 2005
The numbers 2^n - 2 (n >= 2) give the positions of 0's in A110146. Also numbers k such that k^(k + 1) = 0 mod (k + 2). - Zak Seidov, Feb 20 2006
Partial sums of A155559. - Zerinvary Lajos, Oct 03 2007
Number of surjections from an n-element set onto a two-element set, with n >= 2. - Mohamed Bouhamida, Dec 15 2007
It appears that these are the numbers n such that 3*A135013(n) = n*(n + 1), thus answering Problem 2 on the Mathematical Olympiad Foundation of Japan, Final Round Problems, Feb 11 1993 (see link Japanese Mathematical Olympiad).
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if x is a proper subset of y or y is a proper subset of x and x and y are disjoint. Then a(n+1) = |R|. - Ross La Haye, Mar 19 2009
The permutohedron Pi_n has 2^n - 2 facets [Pashkovich]. - Jonathan Vos Post, Dec 17 2009
First differences of A005803. - Reinhard Zumkeller, Oct 12 2011
For n >= 1, a(n + 1) is the smallest even number with bit sum n. Cf. A069532. - Jason Kimberley, Nov 01 2011
a(n) is the number of branches of a complete binary tree of n levels. - Denis Lorrain, Dec 16 2011
For n>=1, a(n) is the number of length-n words on alphabet {1,2,3} such that the gap(w)=1. For a word w the gap g(w) is the number of parts missing between the minimal and maximal elements of w. Generally for words on alphabet {1,2,...,m} with g(w)=g>0 the e.g.f. is Sum_{k=g+2..m} (m - k + 1)*binomial((k - 2),g)*(exp(x) - 1)^(k - g). a(3)=6 because we have: 113, 131, 133, 311, 313, 331. Cf. A240506. See the Heubach/Mansour reference. - Geoffrey Critzer, Apr 13 2014
For n > 0, a(n) is the minimal number of internal nodes of a red-black tree of height 2*n-2. See the Oct 02 2015 comment under A027383. - Herbert Eberle, Oct 02 2015
Conjecture: For n>0, a(n) is the least m such that A007814(A000108(m)) = n-1. - L. Edson Jeffery, Nov 27 2015
Actually this follows from the procedure for determining the multiplicity of prime p in C(n) given in A000108 by Franklin T. Adams-Watters: For p=2, the multiplicity is the number of 1 digits minus 1 in the binary representation of n+1. Obviously, the smallest k achieving "number of 1 digits" = k is 2^k-1. Therefore C(2^k-2) is divisible by 2^(k-1) for k > 0 and there is no smaller m for which 2^(k-1) divides C(m) proving the conjecture. - Peter Schorn, Feb 16 2020
For n >= 0, a(n) is the largest number you can write in bijective base-2 (a.k.a. the dyadic system, A007931) with n digits. - Harald Korneliussen, May 18 2019
The terms of this sequence are also the sum of the terms in each row of Pascal's triangle other than the ones. - Harvey P. Dale, Apr 19 2020
For n > 1, binomial(a(n),k) is odd if and only if k is even. - Charlie Marion, Dec 22 2020
For n >= 2, a(n+1) is the number of n X n arrays of 0's and 1's with every 2 X 2 square having density exactly 2. - David desJardins, Oct 27 2022
For n >= 1, a(n+1) is the number of roots of unity in the unique degree-n unramified extension of the 2-adic field Q_2. Note that for each p, the unique degree-n unramified extension of Q_p is the splitting field of x^(p^n) - x, hence containing p^n - 1 roots of unity for p > 2 and 2*(2^n - 1) for p = 2. - Jianing Song, Nov 08 2022

			a(4) = 14 because the 14 = 6 + 4 + 4 rationals (in lowest terms) for n = 3 are (see the Jun 14 2017 formula above): 1/2, 1, 3/2, 2, 5/2, 3; 1/4, 3/4, 5/4, 7/4; 1/8, 3/8, 5/8, 7/8. - _Wolfdieter Lang_, Jun 14 2017

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
Ralph P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction, Fifth Edition, Addison-Wesley, 2004, p. 134. - Mohammad K. Azarian, Oct 27 2011
S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Chapman and Hall, 2009 page 86, Exercise 3.16.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
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).
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Cyril Banderier, and Benjamin Hackl, Flip-sort and combinatorial aspects of pop-stack sorting, arXiv:2003.04912 [math.CO], 2020.
O. Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91--100.
S. Bilotta, E. Grazzini, and E. Pergola, Enumeration of Two Particular Sets of Minimal Permutations, J. Int. Seq. 18 (2015) 15.10.2
R. B. Campbell, The effect of inbreeding constraints and offspring distribution on time to the most recent common ancestor, Journal of Theoretical Biology, Volume 382, 7 October 2015, Pages 74-80.
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012
M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one. - _N. J. A. Sloane_, Sep 27 2010
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 77
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Milan Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Japanese Mathematical Olympiad 1993, Final Round - Problem 2, Feb 11 1993.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
T. Manneville, V. Pilaud, Compatibility fans for graphical nested complexes, arXiv preprint arXiv:1501.07152 [math.CO], 2015.
Kanstantsin Pashkovich, Symmetry in Extended Formulations of the Permutahedron [sic], arXiv:0912.3446 [math.CO], 2009-2013. [_Jonathan Vos Post_, Dec 17 2009]
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
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
Pratik Poddar, Consecutive Heads Puzzle, Oct 2009.
H. P. Robinson, Letter to N. J. A. Sloane, Sep 1975
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
Eric Weisstein's World of Mathematics, Sphere Line Picking
Index entries for linear recurrences with constant coefficients, signature (3,-2).
Index to sequences related to Olympiads.

Row sums of triangle A026998.
Cf. A000919, A001117, A001118, A095121, A110146.
Cf. A033484, A000225, A000325, A095151, A125128.
Cf. A058809 (3^n-3, n>0).

a000918 = (subtract 2) . (2 ^)
a000918_list = iterate ((subtract 2) . (* 2) . (+ 2)) (- 1)
-- Reinhard Zumkeller, Apr 23 2013

[2^n - 2: n in [0..40]]; // Vincenzo Librandi, Jun 23 2011

Maple
```
seq(2^n-2,n=0..20) ;
```

Table[2^n - 2, {n, 0, 29}] (* Alonso del Arte, Dec 01 2012 *)

a(n)=2^n-2 \\ Charles R Greathouse IV, Jun 16 2011

def A000918(n): return (1<Chai Wah Wu, Jun 10 2025

a(n) = 2*A000225(n-1).
G.f.: 1/(1 - 2*x) - 2/(1 - x), e.g.f.: (e^x - 1)^2 - 1. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001
For n >= 1, a(n) = A008970(n + 1, 2). - Philippe Deléham, Feb 21 2004
G.f.: (3*x - 1)/((2*x - 1)*(x - 1)). - Simon Plouffe in his 1992 dissertation for the sequence without the leading -1
a(n) = 2*a(n - 1) + 2. - Alexandre Wajnberg, Apr 25 2005
a(n) = A000079(n) - 2. - Omar E. Pol, Dec 16 2008
a(n) = A058896(n)/A052548(n). - Reinhard Zumkeller, Feb 14 2009
a(n) = A164874(n - 1, n - 1) for n > 1. - Reinhard Zumkeller, Aug 29 2009
a(n) = A173787(n,1); a(n) = A028399(2*n)/A052548(n), n > 0. - Reinhard Zumkeller, Feb 28 2010
a(n + 1) = A027383(2*n - 1). - Jason Kimberley, Nov 02 2011
G.f.: U(0) - 1, where U(k) = 1 + x/(2^k + 2^k/(x - 1 - x^2*2^(k + 1)/(x*2^(k + 1) - (k + 1)/U(k + 1) ))); (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Dec 01 2012
a(n+1) is the sum of row n in triangle A051601. - Reinhard Zumkeller, Aug 05 2013
a(n+1) = A127330(n,0). - Reinhard Zumkeller, Nov 16 2013
a(n) = Sum_{k=1..n-1} binomial(n, k) for n > 0. - Dan McCandless, Nov 14 2015
From Miquel Cerda, Aug 16 2016: (Start)
a(n) = A000225(n) - 1.
a(n) = A125128(n-1) - A000325(n).
a(n) = A095151(n) - A125128(n) - 1. (End)
a(n+1) = 2*(n + Sum_{j=1..n-1} (n-j)*2^(j-1)), n >= 1. This is the number of the rationals k/2, k = 1..2*n for n >= 1 and (2*k+1)/2^j for j = 2..n, n >= 2, and 2*k+1 < n-(j-1). See the example for n = 3 below. Motivated by the proposal A287012 by Mark Rickert. - Wolfdieter Lang, Jun 14 2017

Maple programs fixed by Vaclav Kotesovec, Dec 13 2014

A151821 Powers of 2, omitting 2 itself.

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 1

N. J. A. Sloane, Jul 08 2009

Different from A046055.
An elephant sequence, see A175655. For the central square just one A[5] vector, with decimal value 170, leads to this sequence. For the corner squares this vector leads to the companion sequence A095121. - Johannes W. Meijer, Aug 15 2010
This is a subsequence of A055744, numbers n such that n and phi(n) have same prime factors. - Michel Marcus, Mar 20 2015
INVERTi transform of A007483: (1, 5, 17, 61, 217, 773, ...). - Gary W. Adamson, Aug 06 2016
Nonprimes that are also powers of 2. Intersection of A000079 and A018252. - Omar E. Pol, Jan 27 2017
Also the chromatic number of the n-Keller graph. - Eric W. Weisstein, Nov 17 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Chromatic Number
Eric Weisstein's World of Mathematics, Keller Graph
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000079, A007483, A018252, A020707, A046055, A055744, A063759, A095121, A175655.

Partial sums are given by 2*A000225(n)-1, which is not the same as A000918.

a151821 n = a151821_list !! (n-1)
a151821_list = x : xs where (x : _ : xs) = a000079_list
-- Reinhard Zumkeller, Dec 16 2013

[1] cat [2^n: n in [2..35]]; // Vincenzo Librandi, Jul 21 2013

CoefficientList[Series[(1 + 2 x)/(1 - 2 x), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)
DeleteCases[2^Range[0, 33], p_ /; PrimeQ @ p] (* Michael De Vlieger, Aug 06 2016 *)
Join[{1}, 2^Range[2, 20]] (* Eric W. Weisstein, Nov 17 2017 *)

a(n)=if(n>1,2^n,1) \\ Charles R Greathouse IV, Dec 08 2015

Vec(x*(1+2*x)/(1-2*x) + O(x^100)) \\ Altug Alkan, Dec 09 2015

def A151821(n): return 1<1 else 1 # Chai Wah Wu, Jun 10 2025

G.f.: x*(1+2*x)/(1-2*x). - Philippe Deléham, Sep 17 2009
a(1) = 1 and a(n) = 3 + Sum_{k=1..n-1} a(k) for n>=2. - Joerg Arndt, Aug 15 2012
E.g.f.: exp(2*x) - x - 1. - Stefano Spezia, Jan 31 2023

A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3x - x^2 + 6x^3).

Original entry on oeis.org

1, 2, 6, 14, 36, 86, 210, 500, 1194, 2822, 6660, 15638, 36642, 85604, 199626, 464630, 1079892, 2506550, 5811762, 13462484, 31159914, 72071654, 166599972, 384912086, 888906306, 2052031172, 4735527306, 10925175254, 25198866036, 58108609526, 133973643090

Offset: 0

Johannes W. Meijer, Aug 06 2010; edited Jun 21 2013

a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 or 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the center square the bishop flies into a rage and turns into a raging elephant.
In chaturanga, the old Indian version of chess, one of the pieces was called gaja, elephant in Sanskrit. The Arabs called the game shatranj and the elephant became el fil in Arabic. In Spain chess became chess as we know it today but surprisingly in Spanish a bishop isn't a Christian bishop but a Moorish elephant and it still goes by its original name of el alfil.
On a 3 X 3 chessboard there are 2^9 = 512 ways for an elephant to fly into a rage on the central square (off the center the piece behaves like a normal bishop). The elephant is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the corner squares the 512 elephants lead to 46 different elephant sequences, see the overview of elephant sequences and the crossreferences.
The sequence above corresponds to 16 A[5] vectors with decimal values 71, 77, 101, 197, 263, 269, 293, 323, 326, 329, 332, 353, 356, 389, 449 and 452. These vectors lead for the side squares to A000079 and for the central square to A175655.

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.
David Hooper and Kenneth Whyld, The Oxford Companion to Chess, pp. 74, 366, 1992.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Viswanathan Anand, The Indian Defense, Time, Jun 19 2008.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Johannes W. Meijer, The elephant sequences.
Wikipedia, War Elephant.
Index entries for linear recurrences with constant coefficients, signature (3,1,-6).

Cf. Elephant sequences corner squares [decimal value A[5]]: A040000 [0], A000027 [16], A000045 [1], A094373 [2], A000079 [3], A083329 [42], A027934 [11], A172481 [7], A006138 [69], A000325 [26], A045623 [19], A000129 [21], A095121 [170], A074878 [43], A059570 [15], A175654 [71, this sequence], A026597 [325], A097813 [58], A057711 [27], 2*A094723 [23; n>=-1], A002605 [85], A175660 [171], A123203 [186], A066373 [59], A015518 [341], A134401 [187], A093833 [343].

Cf. A000244, A006130, A006138, A075188, A175655, A180140.

[n le 3 select Factorial(n) else 3*Self(n-1) +Self(n-2) -6*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 08 2021

nmax:=28; m:=1; A[1]:=[0,0,0,0,1,0,0,0,1]: A[2]:=[0,0,0,1,0,1,0,0,0]: A[3]:=[0,0,0,0,1,0,1,0,0]: A[4]:=[0,1,0,0,0,0,0,1,0]: A[5]:=[0,0,1,0,0,0,1,1,1]: A[6]:=[0,1,0,0,0,0,0,1,0]: A[7]:=[0,0,1,0,1,0,0,0,0]: A[8]:=[0,0,0,1,0,1,0,0,0]: A[9]:=[1,0,0,0,1,0,0,0,0]: A:=Matrix([A[1], A[2], A[3], A[4], A[5], A[6], A[7], A[8], A[9]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

LinearRecurrence[{3,1,-6}, {1,2,6}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2012 *)

a(n)=([0,1,0; 0,0,1; -6,1,3]^n*[1;2;6])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[( (1-x-x^2)/((1-2*x)*(1-x-3*x^2)) ).series(x,n+1).list()[n] for n in (0..40)] # G. C. Greubel, Dec 08 2021

G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3).
a(n) = 3*a(n-1) + a(n-2) - 6*a(n-3) with a(0)=1, a(1)=2 and a(2)=6.
a(n) = ((6+10*A)*A^(-n-1) + (6+10*B)*B^(-n-1))/13 - 2^n with A = (-1+sqrt(13))/6 and B = (-1-sqrt(13))/6.
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n)*2*A000244(n)/(A075118(n) - A006130(n-1)*sqrt(13)).
a(n) = b(n) - b(n-1) - b(n-2), where b(n) = Sum_{k=1..n} Sum_{j=0..k} binomial(j,n-3*k+2*j)*(-6)^(k-j)*binomial(k,j)*3^(3*k-n-j), n>0, b(0)=1, with a(0) = b(0), a(1) = b(1) - b(0). - Vladimir Kruchinin, Aug 20 2010
a(n) = 2*A006138(n) - 2^n = 2*(A006130(n) + A006130(n-1)) - 2^n. - G. C. Greubel, Dec 08 2021
E.g.f.: 2*exp(x/2)*(13*cosh(sqrt(13)*x/2) + 3*sqrt(13)*sinh(sqrt(13)*x/2))/13 - cosh(2*x) - sinh(2*x). - Stefano Spezia, Feb 12 2023

A026998 Triangular array T read by rows: T(n, k) = t(n, 2k), t given by A027960, 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 4, 8, 1, 1, 4, 11, 13, 1, 1, 4, 11, 26, 19, 1, 1, 4, 11, 29, 54, 26, 1, 1, 4, 11, 29, 73, 101, 34, 1, 1, 4, 11, 29, 76, 171, 174, 43, 1, 1, 4, 11, 29, 76, 196, 370, 281, 53, 1, 1, 4, 11, 29, 76, 199, 487, 743, 431, 64, 1

Offset: 0

Clark Kimberling

Right-edge columns are polynomials approximating Lucas(2n+1).

			  .................................... 1;
  ................................. 1, 1;
  ............................. 1,  4, 1;
  ........................ 1,   4,  8, 1;
  ................... 1,   4,  11, 13, 1;
  .............. 1,   4,  11,  26, 19, 1;
  .......... 1,  4,  11,  29,  54, 26, 1;
  ...... 1,  4, 11,  29,  73, 101, 34, 1;
  .. 1,  4, 11, 29,  76, 171, 174, 43, 1;
  1, 4, 11, 29, 76, 196, 370, 281, 53, 1;

G. C. Greubel, Table of n, a(n) for n = 0..1325

This is a bisection of the "Lucas array" A027960, see A027011 for the other bisection.
Row sums give A095121.
Signed row sums give A090132.
Diagonal sums give A027010.
Right-edge columns include A034856, A027966, A027968, A027970, A027972.
Cf. A000032.

function t(n, k) // t = A027960
      if k le n then return Lucas(k+1);
      elif k gt 2*n then return 0;
      else return t(n-1, k-2) + t(n-1, k-1);
      end if;
end function;
A026998:= func< n,k | t(n, 2*k) >;
[A026998(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2025

f[n_, k_]:= f[n, k]= Sum[Binomial[2*n-k+j,j]*LucasL[2*(k-n-j)], {j,0,k-n-1}];
A027960[n_, k_]:= LucasL[k+1] - f[n,k]*Boole[k>n];
A026998[n_, k_]:= A027960[n,2*k];
Table[A026998[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 09 2025 *)

@CachedFunction
def t(n, k): # t = A027960
    if (k>2*n): return 0
    elif (kA026998(n,k): return t(n, 2*k)
print(flatten([[A026998(n, k) for k in (0..n)] for n in (0..12)])) # G. C. Greubel, Jul 09 2025

T(n, k) = Lucas(2*n+1) = A002878(n) for 2*k <= n, otherwise the (2*n-2*k)-th coefficient of the power series for (1+2*x)/( (1-x-x^2)*(1-x)^(2*k-n) ).

Edited by Ralf Stephan, May 05 2005

A123110 Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,0,0,0,0,0,0,0,0,...] DELTA [1,0,-1,1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Philippe Deléham, Sep 28 2006

Diagonal sums give A123108. - Philippe Deléham, Oct 08 2009

			Triangle begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1, 1;
  0, 1, 1, 1, 1;
  0, 1, 1, 1, 1, 1;
  0, 1, 1, 1, 1, 1, 1;
  0, 1, 1, 1, 1, 1, 1, 1;
  0, 1, 1, 1, 1, 1, 1, 1, 1;
  0, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Antti Karttunen, Table of n, a(n) for n = 0..22154; the first 210 rows of the triangle
Index entries for characteristic functions.

Essentially the same sequence as A114607.
Also essentially the same as A023532. - R. J. Mathar, Jun 18 2008
After the initial a(0)=1, the characteristic function of A014132.
Cf. A010054.

A123110(n) = (!n || !ispolygonal(n,3)); \\ Antti Karttunen, Jan 19 2025

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A028310(n), A095121(n), A123109(n) for x=0,1,2,3 respectively.
G.f.: (1-x+y*x^2)/(1-(1+y)*x+y*x^2). - Philippe Deléham, Nov 01 2011
From Tom Copeland, Nov 10 2012: (Start)
O.g.f. for row polynomials: 1 + (t/(1-t))*(1/(1-x)-1/(1-x*t)) = 1 + t*x + (t+t^2)*x^2 + ....
E.g.f. for row polynomials: 1 + (t/(1-t))*(e^x-e^(t*x)) = 1 + t*x + (t+t^2)*x^2/2 + .... (End)
a(0) = 1; for n > 0, a(n) = 1 - A010054(n). [As a flat sequence]  - Antti Karttunen, Jan 19 2025

A131128 Binomial transform of [1, 1, 5, 1, 5, 1, 5, ...].

Original entry on oeis.org

1, 2, 8, 20, 44, 92, 188, 380, 764, 1532, 3068, 6140, 12284, 24572, 49148, 98300, 196604, 393212, 786428, 1572860, 3145724, 6291452, 12582908, 25165820, 50331644, 100663292, 201326588, 402653180, 805306364, 1610612732, 3221225468

Offset: 0

Gary W. Adamson, Jun 16 2007

Row sums of triangle A131129. - Emeric Deutsch, Jun 19 2007

For n >= 4, a(n) is the number of vertices in the dendrimer nanostar NS1[n-3] defined pictorially in the Ashrafi et al. reference (Ns1[3] is shown in Fig. 1) or in the Ahmadi et al. reference (Fig. 1). - Emeric Deutsch, May 17 2018

			a(3) = 20 = (1, 3, 3, 1) dot (1, 1, 5, 1) = (1 + 3 + 15 + 1).

B. Monjardet, Acyclic domains of linear orders: a survey, in "The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn", edited by Steven Brams, William V. Gehrlein and Fred S. Roberts, Springer, 2009, pp. 139-160. [From N. J. A. Sloane, Feb 07 2009]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. B. Ahmadi and M. Sadeghimehr, Atom bond connectivity index of an infinite class NS1[n] of dendrimer nanostars, Optoelectronics and Advanced Materials, 4(7):1040-1042 July 2010.
Ali Reza Ashrafi and Parisa Nikzad, Kekulé index and bounds of energy for nanostar dendrimers, Digest J. of Nanomaterials and Biostructures, 4, No. 2, 2009, 383-388.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A131129, A095121, A132047.

Concatenation([1],List([1..30], n->3*2^n-4)); # Muniru A Asiru, May 17 2018

1, seq(3*2^n-4, n = 1 .. 30); # Emeric Deutsch, Jun 19 2007

CoefficientList[Series[(1-x+4x^2)/((1-x)(1-2x)),{x,0,40}],x] (* Vincenzo Librandi, Apr 11 2012 *)

a(n) = 3*2^n - 4 for n >= 1; a(0)=1. Formula follows by replacing [1,1,5,1,5,1,...] with [1,3-2,3+2,3-2,3+2,3-2,...]. - Emeric Deutsch, Jun 19 2007
G.f.: (1 - x + 4x^2)/((1-x)(1-2x)). - Emeric Deutsch, Jul 09 2007
Row sums of triangle A132047. - Gary W. Adamson, Aug 08 2007
a(n) = 2*a(n-1) + 4 for n >= 2, a(0)=1, a(1)=2. - Philippe Deléham, Sep 23 2009
a(n) = 2*A033484(n-1) for n>0. - R. J. Mathar, Feb 27 2019

For n >= 3, number of vertices of 4,4'-bipyridinium dendrimers (see the Arjomanfar and Gholami reference, p. 71). - Emeric Deutsch, Apr 12 2015

Number of ways to color a (2n-1) X (2n-1) chess board in a "balanced" way. A coloring is called balanced if, within every square subgrid made up of k^2 cells for 1 <= k <= 2*n-1, the number of black cells differs from the number of white cells by at most one. It is problem 3 from the British Maths Olympiad 2020. - Ruediger Jehn, Jan 27 2021

T(2*n,n) is A100320 (with Hankel transform A144704). - Paul Barry, Sep 19 2008
Double the internal elements of Pascal's triangle. - Paul Barry, Jan 07 2009
Coefficients of 2*(x + 1)^n - (x^n + 1) as a triangle (except for the very first term). - Thomas Baruchel, Jun 02 2018

cp's OEIS Frontend

A000918 a(n) = 2^n - 2.

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A151821 Powers of 2, omitting 2 itself.

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A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3).

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A026998 Triangular array T read by rows: T(n, k) = t(n, 2k), t given by A027960, 0 <= k <= n, n >= 0.

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A123110 Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,0,0,0,0,0,0,0,0,...] DELTA [1,0,-1,1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A131128 Binomial transform of [1, 1, 5, 1, 5, 1, 5, ...].

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A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3x - x^2 + 6x^3).

A154117 Expansion of (1 - x + 3x^2)/((1-x)(1-2*x)).