A082639 -id:A082639 - cp's OEIS frontend

A113224 a(2n) = A002315(n), a(2n+1) = A082639(n+1).

1, 2, 7, 16, 41, 98, 239, 576, 1393, 3362, 8119, 19600, 47321, 114242, 275807, 665856, 1607521, 3880898, 9369319, 22619536, 54608393, 131836322, 318281039, 768398400, 1855077841, 4478554082, 10812186007, 26102926096, 63018038201

Offset: 0

Creighton Dement, Oct 18 2005

From Paul D. Hanna, Oct 22 2005: (Start)
The logarithmic derivative of this sequence is twice the g.f. of A113282, where a(2*n) = A113282(2*n), a(4*n+1) = A113282(4*n+1) - 3, a(4*n+3) = A113282(4*n+3) - 1.
Equals the self-convolution of integer sequence A113281. (End)
With an offset of 1, this sequence is the case P1 = 2, P2 = 0, Q = -1 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 19 2015
Floretion Algebra Multiplication Program, FAMP Code: -2ibaseiseq[B*C], B = - .5'i + .5'j - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj'; C = + .5'i + .5i' + .5'ii' + .5e

Creighton Dement, Floretion Online Multiplier. [broken link]
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).

Cf. A113225, A002315, A082639, A100828, A113281, A113282, A113283, A113284, A129744.

[Floor((1+Sqrt(2))^(n+1)/2): n in [0..30]]; // Vincenzo Librandi, Mar 20 2015

a[n_] := n*Sum[ Sum[ Binomial[i, n-k-i]*Binomial[k+i-1, k-1], {i, Ceiling[(n-k)/2], n-k}]*(1-(-1)^k)/(2*k), {k, 1, n}]; Table[a[n], {n, 1, 29}] (* Jean-François Alcover, Feb 26 2013, after Vladimir Kruchinin *)
CoefficientList[Series[(1 + x^2) / ((x^2 - 1) (x^2 + 2 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Mar 20 2015 *)
LinearRecurrence[{2,2,-2,-1},{1,2,7,16},30] (* Harvey P. Dale, Oct 10 2017 *)

a(n):=n*sum(sum(binomial(i,n-k-i)*binomial(k+i-1,k-1),i,ceiling((n-k)/2),n-k)*(1-(-1)^k)/(2*k),k,1,n); /* Vladimir Kruchinin, Apr 11 2011 */

{a(n)=local(x=X+X*O(X^n));polcoeff((1+x^2)/(1-x^2)/(1-2*x-x^2),n,X)} \\ Paul D. Hanna

G.f.: (1+x^2)/((x-1)*(x+1)*(x^2+2*x-1)).
a(n+2) - a(n+1) - a(n) = A100828(n+1).
a(n) = -(u^(n+1)-1)*(v^(n+1)-1)/2 with u = 1+sqrt(2), v = 1-sqrt(2). - Vladeta Jovovic, May 30 2007
a(n) = n * Sum_{k=1..n} Sum_{i=ceiling((n-k)/2)..n-k} binomial(i,n-k-i)*binomial(k+i-1,k-1)*(1-(-1)^k)/(2*k). - Vladimir Kruchinin, Apr 11 2011
a(n) = A001333(n+1) - A000035(n). - R. J. Mathar, Apr 12 2011
a(n) = floor((1+sqrt(2))^(n+1)/2). - Bruno Berselli, Feb 06 2013
From Peter Bala, Mar 19 2015: (Start)
a(n) = (1/2) * A129744(n+1).
exp( Sum_{n >= 1} 2*a(n-1)*x^n/n ) = 1 + 2*Sum_{n >= 1} Pell(n) *x^n. (End)
a(n) = A105635(n-1) + A105635(n+1). - R. J. Mathar, Mar 23 2023

A113281 Self-convolution equals A113224.

Original entry on oeis.org

1, 1, 3, 5, 11, 23, 51, 113, 255, 579, 1325, 3047, 7039, 16319, 37951, 88489, 206799, 484255, 1135969, 2668951, 6279509, 14793169, 34889553, 82372723, 194664283, 460436551, 1089943229, 2582033519, 6120967411, 14519686915, 34463104999

Offset: 0

Paul D. Hanna, Oct 22 2005

nonn

Convolution of bisections A113283 and A113284 yields A082639, which is a bisection of the self-convolution of this sequence. Terms of the logarithmic derivative A113282 are related to the self-convolution A113224 by: A113282(2*n) = A113224(2*n), A113282(4*n+1) = 3 + A113224(4*n+1), A113282(4*n+3) = 1 + A113224(4*n+3).

Cf. A113282, A113283, A113284.

G.f.: A(x) = ( (1+x^2)/(1-x^2)/(1-2*x-x^2) )^(1/2).

a(n) ~ (1 + sqrt(2))^(n + 1/2) / sqrt(2*Pi*n). - Vaclav Kotesovec, Oct 18 2020

A113225 a(2n) = A011900(n), a(2n+1) = A001109(n+1).

Original entry on oeis.org

1, 1, 3, 6, 15, 35, 85, 204, 493, 1189, 2871, 6930, 16731, 40391, 97513, 235416, 568345, 1372105, 3312555, 7997214, 19306983, 46611179, 112529341, 271669860, 655869061, 1583407981, 3822685023, 9228778026, 22280241075, 53789260175

Offset: 0

Creighton Dement, Oct 18 2005

a(n+1) - a(n) = A097075(n+1), a(n) + a(n+1) = A024537(n+1), a(n+2) - a(n+1) - a(n) = A105635(n+1).
For n >= 1, a(n) is also the edge cover number and edge cut count of the n-Pell graph. - Eric W. Weisstein, Aug 01 2023
Also the independence number, Lovasz number, and Shannon capacity of the n-Pell graph. - Eric W. Weisstein, Aug 01 2023
Floretion Algebra Multiplication Program, FAMP Code: -2jbasejseq[B*C], B = - .5'i + .5'j - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj'; C = + .5'i + .5i' + .5'ii' + .5e

C. Dement, Floretion Integer Sequences (work in progress).

Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See p. 19.
Eric Weisstein's World of Mathematics, Edge Cover Number.
Eric Weisstein's World of Mathematics, Edge Cut.
Eric Weisstein's World of Mathematics, Independence Number.
Eric Weisstein's World of Mathematics, Lovasz Number.
Eric Weisstein's World of Mathematics, Pell Graph.
Eric Weisstein's World of Mathematics, Shannon Capacity.
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).

Cf. A113224, A002315, A082639, A100828.

seq(iquo(fibonacci(n,2),1)-iquo(fibonacci(n,2),2),n=1..30); # Zerinvary Lajos, Apr 20 2008
with(combinat):seq(ceil(fibonacci(n,2)/2), n=1..30); # Zerinvary Lajos, Jan 12 2009

Ceiling[Fibonacci[Range[20], 2]/2]
Table[(1 + (-1)^n + 2 Fibonacci[n + 1, 2])/4, {n, 0, 20}] // Expand
CoefficientList[Series[-(-1 + x + x^2)/(1 - 2 x - 2 x^2 + 2 x^3 + x^4), {x, 0, 20}], x]
LinearRecurrence[{2, 2, -2, -1}, {1, 1, 3, 6}, 20]

{a(n)=local(y); if(n<0, 0, n++; y=x/(x^2+x-1)+x*O(x^n); polcoeff( y/(y^2-1), n))} /* Michael Somos, Sep 09 2006 */

G.f.: y/(y^2-1) where y=x/(x^2+x-1) if offset=1. - Michael Somos, Sep 09 2006
G.f.: (-1+x+x^2)/((1-x)*(x+1)*(x^2+2*x-1)).
Diagonal sums of A119468. - Paul Barry, May 21 2006
a(n) = (1 + (-1)^n + 2 A000129(n+1))/4. - Eric W. Weisstein, Aug 01 2023
a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4). - Eric W. Weisstein, Aug 01 2023

A269499 Nontrivial integer solutions s to the equations Sum_{i} ((-1)^i)binomial(m,i)binomial(s-m,t-i) = 0 listed in increasing order.

Original entry on oeis.org

36, 66, 67, 98, 132, 177, 214, 289, 345, 465, 514, 576, 774, 932, 1029, 1219, 1252

Offset: 1

René Gy, Feb 28 2016

A nontrivial integer solution s to the equations S(m,s,t) = Sum_{i} (((-1)^i)*binomial(m, i)*binomial(s - m, t - i)) = 0 is an integer s such that there exist integers m, t and 3 < m,t < s/2 such that S(m,s,t)=0 and m,s,t are not such that s=8*k+1 and t=2*k or m=2*k for some integer k.
S(m,s,t)=0 iff S(t,s,m)=0 iff S(s-m,s,t)=0 iff S(s-t,s,m)=0.
For t(resp. m)=2, s=(k+2)^2, m(resp. t)=((k + 2)*(k + 1))/2 is an infinite family of trivial solutions.
For t(resp. m)=3, s=3*k^2 + 8*k + 6, m(resp. t)=((k + 1)*(3*k + 2))/2 is another infinite family of trivial solutions.
For t(resp. m)=3, s=3*k^2 + 10*k + 9, m(resp. t)=((k + 1)*(3*k + 4))/2 is another infinite family of trivial solutions.
For t(resp. m)=2*k, s=8*k+1, m(resp. t)=4*k-1 is another infinite family of trivial solutions.
1521, 3193, 3362, 10882, 15043, 19600 also belong to the sequence, but the list has been checked to be complete only up to 1252.
A082639(k) (from k=4) is included in the sequence, because Sum_{i} (((-1)^i)*binomial(m(k), i)*binomial(s(k) - m, t(k) - i)) = 0, with s(k) = A082639(k) and m(k) = (g^k + g^(-k) - 10)/4 with g=3+2*sqrt(2) and t(k) = (h*g^k + 2 h^(-1)*g^(-k) - 4)/8 with h=2-2*sqrt(2). In other words, they are positive integers s of the form s=2*m+4 where (m,t) m>6 is any couple of positive integer solutions to the Diophantine equation m^2 - 4*m*t + 2*t^2 + 3*m - 8*t + 2 = 0 (there are infinitely many).
Nine such infinite subsequences for the present sequence are known.
The eight other similar subsequences are:
- positive integers s of the form s=2*m+5 where (m,t) is a solution to the Diophantine equation 5*m^2 - 10*m*t + 4*t^2 + 25*m - 26*t + 32 = 0 (producing 2 subsequences).
- positive integers s of the form s=2*m+5 where (m,t) m>10 is a solution to the Diophantine equation m^2 - 6*m*t + 4*t^2 + 3*m - 14*t + 2 = 0 (producing 2 subsequences).
- positive integers s of the form s=2*m+6 where (m,t) is a solution to the Diophantine equation m^2 - 8*m*t + 4*t^2 + 3*m - 24*t + 2 = 0 (producing 2 subsequences).
- positive integers s of the form s=2*m+8 where (m,t) m>7 is a solution to the Diophantine equation m^2 - 4*m*t + 2*t^2 + 7*m - 16*t + 16 = 0 (producing 2 subsequences).
A finite number of nontrivial (sporadic) terms of the sequence (not belonging to one of the above nine subsequences) are also known: 67, 289, 345, 1029, 1521, 10882, 15043, 48324.

			36=14+22 belongs to the sequence because Sum_{i=0..5} (((-1)^i)*binomial(14, i)*binomial(22,5-i)) = 0, both 5 and 14 are less than 18 and (14,36,5) is not in one of the above trivial families.

René Gy, Trying to solve the equation Sum_{i}(-1)^i*binomial(m,i)*binomial(n-m,t-i)=0 for non-negative integers m,n,t, Math StackExchange.
L. Habsieger and D. Stanton, More Zeros of Krawtchouk Polynomials, IMA Preprint Series #441, August 1988.
I. Krasikov, On Integral Zeros of Krawtchouk Polynomials, Journal of Combinatorial Theory, Series A 74, 71-99 (1996).

Cf. A269563, A082639.

f[n_,m_,t_]:= Sum[(-1)^i*Binomial[m, i]*Binomial[n-m,t-i],{i,0,t}];lim=200; list={}; Do[ Do[Do [If[ Mod[n,8]==1&& t==2*Quotient[n,8],Continue,If[f[n,m,t]==0  ,AppendTo[list,n]]],{t,4,m}] ,{m,4,n/2-1}],{n,10,lim}];Print [Union [list]]

isok(s) = {for (m=4, s\2-1, for (t=4, m, if (!(((s % 8) == 1) && (t == 2*(s\8))), if (sum(i=0, t, (-1)^i*binomial(m, i)*binomial(s-m, t-i)) == 0, return (1)););););} \\ Michel Marcus, Mar 01 2016

a(16)-a(17) from Michel Marcus, Apr 04 2016

A105058 Expansion of g.f. (1+8x-x^2)/((1+x)(1-6*x+x^2)).

Original entry on oeis.org

1, 13, 69, 409, 2377, 13861, 80781, 470833, 2744209, 15994429, 93222357, 543339721, 3166815961, 18457556053, 107578520349, 627013566049, 3654502875937, 21300003689581, 124145519261541, 723573111879673

Offset: 0

Creighton Dement, Apr 04 2005

A floretion-generated sequence relating the squares of the numerators of continued fraction convergents to sqrt(2) to the squares of the denominators of continued fraction convergents to sqrt(2) (Pell numbers).
Floretion Algebra Multiplication Program, FAMP Code:
1dia[J]tesseq[ - .5'j + .5'k - .5j' + .5k' - 2'ii' + 'jj' - 'kk' + .5'ij' + .5'ik' + .5'ji' + 'jk' + .5'ki' + 'kj' + e ]. Identity used: dia[I]tes + dia[J]tes + dia[K]tes = jes + fam + 3tes.

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

Cf. A000129, A001109, A001333, A046729, A077444.

Cf. A077445, A079291, A082639, A084158, A090390.

[Evaluate(DicksonSecond(2*n+1, -1), 2) -(-1)^n: n in [0..30]]; // G. C. Greubel, Aug 21 2022

CoefficientList[ Series[(1+8x-x^2)/((1+x)(1-6x+x^2)), {x,0,30}], x] (* Robert G. Wilson v, Apr 06 2005 *)
LinearRecurrence[{5,5,-1}, {1,13,69}, 30] (* Harvey P. Dale, Jun 03 2017 *)

[lucas_number1(2*n+2,2,-1) -(-1)^n for n in (0..30)] # G. C. Greubel, Aug 21 2022

a(n) = 2 * A001109(n+1) - (-1)^n.
G.f.: G(0)/(1-3*x) - 1/(1+x), where G(k) = 1 + 1/(1 - x*(8*k-9)/( x*(8*k-1) - 3/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 12 2013
From G. C. Greubel, Aug 21 2022: (Start)
a(n) = A000129(2*n+2) - (-1)^n.
E.g.f.: exp(3*x)*( 2*cosh(2*sqrt(2)*x) + (3/sqrt(2))*sinh(2*sqrt(2)*x)) - exp(-x). (End)

cp's OEIS Frontend

A113224 a(2n) = A002315(n), a(2n+1) = A082639(n+1).

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A113281 Self-convolution equals A113224.

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A113225 a(2n) = A011900(n), a(2n+1) = A001109(n+1).

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A269499 Nontrivial integer solutions s to the equations Sum_{i} ((-1)^i)*binomial(m,i)*binomial(s-m,t-i) = 0 listed in increasing order.

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A105058 Expansion of g.f. (1+8*x-x^2)/((1+x)*(1-6*x+x^2)).

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A269499 Nontrivial integer solutions s to the equations Sum_{i} ((-1)^i)binomial(m,i)binomial(s-m,t-i) = 0 listed in increasing order.

A105058 Expansion of g.f. (1+8x-x^2)/((1+x)(1-6*x+x^2)).