A171418 -id:A171418 - cp's OEIS frontend

A040000 a(0)=1; a(n)=2 for n >= 1.

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

N. J. A. Sloane, Dec 11 1999

Continued fraction expansion of sqrt(2) is 1 + 1/(2 + 1/(2 + 1/(2 + ...))).
Inverse binomial transform of Mersenne numbers A000225(n+1) = 2^(n+1) - 1. - Paul Barry, Feb 28 2003
A Chebyshev transform of 2^n: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)). - Paul Barry, Oct 31 2004
An inverse Catalan transform of A068875 under the mapping g(x)->g(x(1-x)). A068875 can be retrieved using the mapping g(x)->g(xc(x)), where c(x) is the g.f. of A000108. A040000 and A068875 may be described as a Catalan pair. - Paul Barry, Nov 14 2004
Sequence of electron arrangement in the 1s 2s and 3s atomic subshells. Cf. A001105, A016825. - Jeremy Gardiner, Dec 19 2004
Binomial transform of A165326. - Philippe Deléham, Sep 16 2009
Let m=2. We observe that a(n) = Sum_{k=0..floor(n/2)} binomial(m,n-2*k). Then there is a link with A113311 and A115291: it is the same formula with respectively m=3 and m=4. We can generalize this result with the sequence whose g.f. is given by (1+z)^(m-1)/(1-z). - Richard Choulet, Dec 08 2009
With offset 1: number of permutations where |p(i) - p(i+1)| <= 1 for n=1,2,...,n-1. This is the identical permutation and (for n>1) its reversal.
Equals INVERT transform of bar(1, 1, -1, -1, ...).
Eventual period is (2). - Zak Seidov, Mar 05 2011
Also decimal expansion of 11/90. - Vincenzo Librandi, Sep 24 2011
a(n) = 3 - A054977(n); right edge of the triangle in A182579. - Reinhard Zumkeller, May 07 2012
With offset 1: minimum cardinality of the range of a periodic sequence with (least) period n. Of course the range's maximum cardinality for a purely periodic sequence with (least) period n is n. - Rick L. Shepherd, Dec 08 2014
With offset 1: n*a(1) + (n-1)*a(2) + ... + 2*a(n-1) + a(n) = n^2. - Warren Breslow, Dec 12 2014
With offset 1: decimal expansion of gamma(4) = 11/9 where gamma(n) = Cp(n)/Cv(n) is the n-th Poisson's constant. For the definition of Cp and Cv see A272002. - Natan Arie Consigli, Sep 11 2016
a(n) equals the number of binary sequences of length n where no two consecutive terms differ. Also equals the number of binary sequences of length n where no two consecutive terms are the same. - David Nacin, May 31 2017
a(n) is the period of the continued fractions for sqrt((n+2)/(n+1)) and sqrt((n+1)/(n+2)). - A.H.M. Smeets, Dec 05 2017
Also, number of self-avoiding walks and coordination sequence for the one-dimensional lattice Z. - Sean A. Irvine, Jul 27 2020

			sqrt(2) = 1.41421356237309504... = 1 + 1/(2 + 1/(2 + 1/(2 + 1/(2 + ...)))). - _Harry J. Smith_, Apr 21 2009
G.f. = 1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + ...
11/90 = 0.1222222222222222222... - _Natan Arie Consigli_, Sep 11 2016

A. Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.4 Powers and Roots, p. 144.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 276-278.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Bruce Fang, Pamela E. Harris, Brian M. Kamau, and David Wang, Vacillating parking functions, arXiv:2402.02538 [math.CO], 2024.
Kshitij Education, Molar specific heat.
MathPath, Square-roots via Continued Fractions.
Narad Rampersad and Max Wiebe, Sums of products of binomial coefficients mod 2 and 2-regular sequences, arXiv:2309.04012 [math.NT], 2023.
Eric Weisstein's World of Mathematics, Square root.
Eric Weisstein's World of Mathematics, Pythagoras's Constant.
Wikipedia, Poisson's constant.
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for eventually constant sequences.
Index to divisibility sequences.
Index entries for linear recurrences with constant coefficients, signature (1).

Convolution square is A008574.
See A003945 etc. for (1+x)/(1-k*x).
From Jaume Oliver Lafont, Mar 26 2009: (Start)
Sum_{0<=k<=n} a(k) = A005408(n).
Prod_{0<=k<=n} a(k) = A000079(n). (End)
Cf. A113311, A115291, A171418, A171440, A171441, A171442, A171443.
Cf. A000674 (boustrophedon transform).
Cf. A001333/A000129 (continued fraction convergents).
Cf. A000122, A002193 (sqrt(2) decimal expansion), A006487 (Egyptian fraction).
Cf. Other continued fractions for sqrt(a^2+1) = (a, 2a, 2a, 2a....): A040002 (contfrac(sqrt(5)) = (2,4,4,...)), A040006, A040012, A040020, A040030, A040042, A040056, A040072, A040090, A040110 (contfrac(sqrt(122)) = (11,22,22,...)), A040132, A040156, A040182, A040210, A040240, A040272, A040306, A040342, A040380, A040420 (contfrac(sqrt(442)) = (21,42,42,...)), A040462, A040506, A040552, A040600, A040650, A040702, A040756, A040812, A040870, A040930 (contfrac(sqrt(962)) = (31,62,62,...)).

a040000 0 = 1; a040000 n = 2
a040000_list = 1 : repeat 2  -- Reinhard Zumkeller, May 07 2012

Digits := 100: convert(evalf(sqrt(2)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[2],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
a[ n_] := 2 - Boole[n == 0]; (* Michael Somos, Dec 28 2014 *)
PadRight[{1},120,2] (* or *) RealDigits[11/90, 10, 120][[1]] (* Harvey P. Dale, Jul 12 2025 *)

{a(n) = 2-!n}; /* Michael Somos, Apr 16 2007 */

a(n)=1+sign(n)  \\ Jaume Oliver Lafont, Mar 26 2009

allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(2)); for (n=0, 20000, write("b040000.txt", n, " ", x[n+1]));  \\ Harry J. Smith, Apr 21 2009

G.f.: (1+x)/(1-x). - Paul Barry, Feb 28 2003
a(n) = 2 - 0^n; a(n) = Sum_{k=0..n} binomial(1, k). - Paul Barry, Oct 16 2004
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*2^(n-2*k)/(n-k). - Paul Barry, Oct 31 2004
A040000(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*A068875(n-k). - Paul Barry, Nov 14 2004
From Michael Somos, Apr 16 2007: (Start)
Euler transform of length 2 sequence [2, -1].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u-v)*(u+v) - 2*v*(u-w).
E.g.f.: 2*exp(x) - 1.
a(n) = a(-n) for all n in Z (one possible extension to n<0). (End)
G.f.: (1-x^2)/(1-x)^2. - Jaume Oliver Lafont, Mar 26 2009
G.f.: exp(2*atanh(x)). - Jaume Oliver Lafont, Oct 20 2009
a(n) = Sum_{k=0..n} A108561(n,k)*(-1)^k. - Philippe Deléham, Nov 17 2013
a(n) = 1 + sign(n). - Wesley Ivan Hurt, Apr 16 2014
10 * 11/90 = 11/9 = (11/2 R)/(9/2 R) = Cp(4)/Cv(4) = A272005/A272004, with R = A081822 (or A070064). - Natan Arie Consigli, Sep 11 2016
a(n) = A001227(A000040(n+1)). - Omar E. Pol, Feb 28 2018

A113311 Expansion of (1+x)^2/(1-x).

Original entry on oeis.org

1, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Paul Barry, Oct 25 2005

Row sums of A113310.
Let m=3. We observe that a(n)=Sum_{k=0..floor(n/2)} C(m,n-2*k). Then there is a link with A040000 and A115291: it is the same formula with respectively m=2 and m=4. We can generalize this result with the sequence whose g.f. is given by (1+z)^(m-1)/(1-z). - Richard Choulet, Dec 08 2009
Also continued fraction expansion of (3+sqrt(5))/4. - Bruno Berselli, Sep 23 2011
Also decimal expansion of 121/900. - Vincenzo Librandi, Sep 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A040000, A115291, A171418, A171440-A171443.

CoefficientList[Series[(1+x)^2/(1-x),{x,0,110}],x] (* Harvey P. Dale, Aug 19 2011 *)

a(n)=if(n>1,4,2*n+1) \\ Charles R Greathouse IV, Jun 12 2015

a(n) = Sum_{k=0..n} Sum_{i=0..n-k} (-1)^i*C(i+k-2, i).

E.g.f.: 4*exp(x) - x - 3. - Elmo R. Oliveira, Aug 08 2024

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A002663 a(n) = 2^n - C(n,0) - C(n,1) - C(n,2) - C(n,3).

Original entry on oeis.org

0, 0, 0, 0, 1, 6, 22, 64, 163, 382, 848, 1816, 3797, 7814, 15914, 32192, 64839, 130238, 261156, 523128, 1047225, 2095590, 4192510, 8386560, 16774891, 33551806, 67105912, 134214424, 268431773, 536866822, 1073737298

Offset: 0

N. J. A. Sloane

Starting with "1" = eigensequence of a triangle with bin(n,4), A000332 as the left border: (1, 5, 15, 35, 70, ...) and the rest 1's. - Gary W. Adamson, Jul 24 2010
The Kn25 sums, see A180662, of triangle A065941 equal the terms (doubled) of this sequence minus the four leading zeros. - Johannes W. Meijer, Aug 14 2011
(1 + 6x + 22x^2 + 64x^3 + ...) = (1 + 3x + 6x^2 + 10x^3 + ...) * (1 + 3x + 7x^2 + 15x^3 + ...). - Gary W. Adamson, Mar 14 2012
The sequence starting (1, 6, 22, ...) is the binomial transform of A171418 and starting (0, 0, 0, 1, 6, 22, ...) is the binomial transform of (0, 0, 0, 1, 2, 2, 2, 2, 2, ...). - Gary W. Adamson, Jul 27 2015
Number of binary sequences with at least four 0's. - Enrique Navarrete, Jul 23 2025

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
J. Eckhoff, Der Satz von Radon in konvexen Productstrukturen II, Monat. f. Math., 73 (1969), 7-30.
R. K. Guy, Letter to N. J. A. Sloane
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-9,2).

a(n)= A055248(n, 4). Partial sums of A002662.

Cf. A000079, A000225, A000295, A002664, A035038-A035042, A007318.

a002663 n = a002663_list !! n
a002663_list = map (sum . drop 4) a007318_tabl
-- Reinhard Zumkeller, Jun 20 2015

[2^n - Binomial(n,0)- Binomial(n,1) - Binomial(n,2) - Binomial(n,3): n in [0..35]]; // Vincenzo Librandi, May 20 2011

A002663 := proc(n): 2^n - add(binomial(n,k),k=0..3) end: seq(A002663(n), n=0..30); # Johannes W. Meijer, Aug 14 2011

a=1;lst={};s1=s2=s3=s4=0;Do[s1+=a;s2+=s1;s3+=s2;s4+=s3;AppendTo[lst,s4];a=a*2,{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 10 2009 *)
Table[Sum[ Binomial[n + 4, k + 4], {k, 0, n}], {n, -4, 26}] (* Zerinvary Lajos, Jul 08 2009 *)

a(n)=(6*2^n-n^3-5*n-6)/6 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 2^n - A000125(n).
G.f.: x^4/((1-2*x)*(1-x)^4). - Simon Plouffe in his 1992 dissertation
a(n) = Sum_{k=0..n} binomial(n,k+4) = Sum_{k=4..n} binomial(n,k). - Paul Barry, Aug 23 2004
a(n) = 2*a(n-1) + binomial(n-1,3). - Paul Barry, Aug 23 2004
a(n) = (6*2^n - n^3 - 5*n - 6)/6. - Mats Granvik, Gary W. Adamson, Feb 17 2010
From Enrique Navarrete, Jul 23 2025: (Start)
a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 9*a(n-4) + 2*a(n-5).
E.g.f.: exp(x)*(exp(x) - 1 - x - x^2/2 - x^3/6). (End)

A115291 Expansion of (1+x)^3/(1-x).

Original entry on oeis.org

1, 4, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

Paul Barry, Jan 19 2006

Partial sums are A086570. Partial sums of squares are A115295. Correlation triangle is A115292.
Let m=4. We observe that a(n) = Sum_{k=0..floor(n/2)} C(m,n-2*k). Then there is a link with A113311 and A040000: it is the same formula with respectively m=3 and m=2. We can generalize this result with the sequence whose G.f is given by (1+z)^(m-1)/(1-z). - Richard Choulet, Dec 08 2009
Also continued fraction expansion of (132-sqrt(17))/103. - Bruno Berselli, Sep 23 2011
Also decimal expansion of 1331/9000. - Vincenzo Librandi, Sep 23 2011

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

Cf. A040000, A113311, A171418, A171440, A171441, A171442, A171443.

CoefficientList[Series[(1+x)^3/(1-x),{x,0,100}],x] (* or *) PadRight[ {1,4,7},120,{8}] (* Harvey P. Dale, May 23 2016 *)

a(n) = 8 - C(2, n) - 2*C(1, n) - 4*C(0, n).
a(n) = Sum_{k=0..n} C(3, k).
a(n) = A004070(n, 3).
From Elmo R. Oliveira, Aug 09 2024: (Start)
E.g.f.: 8*exp(x) - 7 - 4*x - x^2/2.
a(n) = 8, n > 2. (End)

A171440 Expansion of (1+x)^5/(1-x).

Original entry on oeis.org

1, 6, 16, 26, 31, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32

Offset: 0

Richard Choulet, Dec 09 2009

a(n)=2^5=32 for n>=5. We observe that this sequence is the transform of A171418 by T such that: T(u_0,u_1,u_2,u_3,u_4,u_5,...)=(u_0,u_0+u_1,u_1+u_2,u_2+u_3,u_3+u_4,...).

Also continued fraction expansion of (229657824-sqrt(257))/197139199. - Bruno Berselli, Sep 23 2011

			a(4) = C(6,4-0)+C(6,4-2)+C(6,4-4) = 15+15+1 = 31.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Richard Choulet, Une nouvelle formule de combinatoire?, Mathématique et Pédagogie, 157 (2006), p. 53-60. In French.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A040000, A113311, A115291, A171418, A171441, A171442, A171443.

PadRight[{1,6,16,26,31},100,32] (* Harvey P. Dale, Oct 01 2013 *)

With m=6, a(n) = Sum_{k=0..floor(n/2)} binomial(m,n-2*k).

Definition rewritten by Bruno Berselli, Sep 23 2011

A171443 Expansion of g.f. (1+x)^8/(1-x).

Original entry on oeis.org

1, 9, 37, 93, 163, 219, 247, 255, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256, 256

Offset: 0

Richard Choulet, Dec 09 2009

a(n)=2^8=256 for n>=8. We observe that this sequence is the transform of A171442 by T such that: T(u_0,u_1,u_2,u_3,u_4,u_5,...)=(u_0,u_0+u_1,u_1+u_2,u_2+u_3,u_3+u_4,...).

			a(7) = C(9,7-0)+C(9,7-2)+C(9,7-4)+C(9,7-6) = 36+126+84+9 = 255.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard Choulet, Une nouvelle formule de combinatoire?, Mathématique et Pédagogie, 157 (2006), p. 53-60. In French.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A040000, A113311, A115291, A171418, A171440, A171441, A171442.

m:=9:for n from 0 to 40 do a(n):=sum('binomial(m,n-2*k)',k=0..floor(n/2)): od : seq(a(n),n=0..40);

CoefficientList[Series[(1+x)^8/(1-x),{x,0,80}],x] (* Harvey P. Dale, Jul 22 2014 *)

With m=9, a(n) = Sum_{k=0..floor(n/2)} binomial(m,n-2*k).

Definition rewritten by Bruno Berselli, Sep 23 2011

A171441 Expansion of g.f. (1+x)^6/(1-x).

Original entry on oeis.org

1, 7, 22, 42, 57, 63, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64

Offset: 0

Richard Choulet, Dec 09 2009

a(n)=2^6=64 for n>=6. We observe that this sequence is the transform of A171440 by T such that: T(u_0,u_1,u_2,u_3,u_4,u_5,...)=(u_0,u_0+u_1,u_1+u_2,u_2+u_3,u_3+u_4,...).

Also continued fraction expansion of 1+(1233212607598+5*sqrt(41))/8688482797079. - Bruno Berselli, Sep 23 2011

			a(4) = C(7,4-0) + C(7,4-2) + C(7,4-4) = 35+21+1 = 57.

Richard Choulet, Une nouvelle formule de combinatoire?, Mathématique et Pédagogie, 157 (2006), p. 53-60. In French.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A040000, A113311, A115291, A171418, A171440, A171442, A171443.

m:=7:for n from 0 to 40 do a(n):=sum('binomial(m,n-2*k)',k=0..floor(n/2)): od : seq(a(n),n=0..40);

With m=7, a(n) = Sum_{k=0..floor(n/2)} binomial(m,n-2*k).

Definition rewritten by Bruno Berselli, Sep 23 2011

A171442 Expansion of (1+x)^7/(1-x).

Original entry on oeis.org

1, 8, 29, 64, 99, 120, 127, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128, 128

Offset: 0

Richard Choulet, Dec 09 2009

a(n)=2^7=128 for n>=7. We observe that this sequence is the transform of A171441 by T such that: T(u_0,u_1,u_2,u_3,u_4,u_5,...)=(u_0,u_0+u_1,u_1+u_2,u_2+u_3,u_3+u_4,...).

			a(5) = C(8,5-0)+C(8,5-2)+C(8,5-4) = 56+56+8 = 120.

Richard Choulet, Une nouvelle formule de combinatoire?, Mathématique et Pédagogie, 157 (2006), p. 53-60. In French.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A040000, A113311, A115291, A171418, A171440, A171441, A171443.

m:=8:for n from 0 to 40 do a(n):=sum('binomial(m,n-2*k)',k=0..floor(n/2)): od : seq(a(n),n=0..40);

CoefficientList[Series[(1+x)^7/(1-x),{x,0,60}],x] (* Harvey P. Dale, Apr 30 2012 *)

With m=8, a(n) = Sum_{k=0..floor(n/2)} binomial(m,n-2*k).

Definition rewritten by Bruno Berselli, Sep 23 2011

A171445 Expansion of g.f. (1+z)^(24)/(1-z).

Original entry on oeis.org

1, 25, 301, 2325, 12951, 55455, 190051, 536155, 1271626, 2579130, 4540386, 7036530, 9740686, 12236830, 14198086, 15505590, 16241061, 16587165, 16721761, 16764265, 16774891, 16776915, 16777191, 16777215, 16777216, 16777216

Offset: 0

Richard Choulet, Dec 09 2009

a(n)=2^(24)=16777216 for n>=24. We observe that this sequence is the transform of A171443 by the iterated T^(16) of T such that: T(u_0,u_1,u_2,u_3,u_4,u_5,...)=(u_0,u_0+u_1,u_1+u_2,u_2+u_3,u_3+u_4,...).

			a(3) = C(25,3)+C(25,3-2) = 2325.

Richard Choulet, Une nouvelle formule de combinatoire?, Mathématique et Pédagogie, 157 (2006), p. 53-60. In French.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A040000, A113311, A115291, A171418, A171440, A171441, A171442, A171443.

m:=25:for n from 0 to 40 do a(n):=sum('binomial(m,n-2*k)',k=0..floor(n/2)): od : seq(a(n),n=0..40);

CoefficientList[Series[(1+x)^24/(1-x),{x,0,30}],x] (* Harvey P. Dale, Jun 11 2019 *)

With m=25, a(n) = Sum_{k=0..floor(n/2)} binomial(m,n-2*k).

cp's OEIS Frontend

A040000 a(0)=1; a(n)=2 for n >= 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

Formula

A113311 Expansion of (1+x)^2/(1-x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A002663 a(n) = 2^n - C(n,0) - C(n,1) - C(n,2) - C(n,3).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

A115291 Expansion of (1+x)^3/(1-x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A171440 Expansion of (1+x)^5/(1-x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A171443 Expansion of g.f. (1+x)^8/(1-x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions