A105635 -id:A105635 - 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

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

A111955 a(n) = A078343(n) + (-1)^n.

Original entry on oeis.org

0, 1, 4, 7, 20, 45, 112, 267, 648, 1561, 3772, 9103, 21980, 53061, 128104, 309267, 746640, 1802545, 4351732, 10506007, 25363748, 61233501, 147830752, 356895003, 861620760, 2080136521, 5021893804, 12123924127, 29269742060, 70663408245

Offset: 0

Creighton Dement, Aug 25 2005

This sequence is a companion sequence to A111954 (compare formula / program code). Three other companion sequences (i.e., they are generated by the same floretion given in the program code) are A105635, A097076 and A100828.

Floretion Algebra Multiplication Program, FAMP Code: 4kbasejseq[J*D] with J = - .25'i + .25'j + .5'k - .25i' + .25j' + .5k' - .5'kk' - .25'ik' - .25'jk' - .25'ki' - .25'kj' - .5e and D = + .5'i - .25'j + .25'k + .5i' - .25j' + .25k' - .5'ii' - .25'ij' - .25'ik' - .25'ji' - .25'ki' - .5e. (an initial term 0 was added to the sequence)

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

Cf. A078343, A000129, A001333, A111954, A111956, A007070, A077995, A100828, A097076, A105635, A048655.

LinearRecurrence[{1,3,1},{0,1,4},40] (* Harvey P. Dale, Mar 12 2015 *)

a(n) + a(n+1) = A048655(n).
a(n) = a(n-1) + 3*a(n-2) + a(n-3), n >= 3; a(n) = (-1/4*sqrt(2)+1)*(1-sqrt(2))^n + (1/4*sqrt(2)+1)*(1+sqrt(2))^n - (-1)^n;
G.f.: -x*(1+3*x) / ( (1+x)*(x^2+2*x-1) ). - R. J. Mathar, Oct 02 2012
E.g.f.: cosh(x) - exp(x)*cosh(sqrt(2)*x) - sinh(x) + 3*exp(x)*sinh(sqrt(2)*x)/sqrt(2). - Stefano Spezia, May 26 2024

A251867 Numbers n such that n^2 + (n+1)^2 is equal to the sum of the hexagonal numbers H(m) and H(m+1) for some m.

Original entry on oeis.org

0, 14, 492, 16730, 568344, 19306982, 655869060, 22280241074, 756872327472, 25711378892990, 873430010034204, 29670908962269962, 1007937474707144520, 34240203231080643734, 1163158972382034742452, 39513164857758100599650, 1342284446191393385645664

Offset: 1

Colin Barker, Dec 10 2014

Also nonnegative integers y in the solutions to 4*x^2-2*y^2+2*x-2*y = 0, the corresponding values of x being A220185.

			14 is in the sequence because 14^2+15^2 = 196+225 = 421 = 190+231 = H(10)+H(11).

Colin Barker, Table of n, a(n) for n = 1..654
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A000290, A000384, A220185, A105635.

I:=[0,14,492]; [n le 3 select I[n] else 35*Self(n-1)-35*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Sep 06 2015

LinearRecurrence[{35, -35, 1}, {0, 14, 492}, 20] (* Vincenzo Librandi, Sep 06 2015 *)

concat(0, Vec(-2*x^2*(x+7)/((x-1)*(x^2-34*x+1)) + O(x^100)))

a(n) = 35*a(n-1)-35*a(n-2)+a(n-3).
G.f.: -2*x^2*(x+7) / ((x-1)*(x^2-34*x+1)).
a(n) = A220185(n) + A001542(n-1)^2. - Alexander Samokrutov, Sep 05 2015
a(n) = (-4+(10+7*sqrt(2))*(17+12*sqrt(2))^(-n)+(10-7*sqrt(2))*(17+12*sqrt(2))^n)/8. - Colin Barker, Mar 02 2016
a(n) = A105635(4*n-4). - Greg Dresden, Aug 30 2021

A157901 Triangle read by rows: A000012 * A157898.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 4, 4, 4, 3, 6, 10, 8, 8, 3, 9, 16, 24, 16, 16, 4, 12, 28, 40, 56, 32, 32, 4, 16, 40, 80, 96, 128, 64, 64, 5, 20, 60, 120, 216, 224, 288, 128, 128, 5, 25, 80, 200, 336, 560, 512, 640, 256, 256, 6, 30, 110, 280, 616, 896, 1408, 1152, 1408, 512, 512

Offset: 0

Gary W. Adamson & Roger L. Bagula, Mar 08 2009

Multiplication of the lower triangular matrix A157898 from the left by A000012 means: these are partial column sums of A157898.

			First few rows of the triangle, n>=0:
  1;
  1,  1;
  2,  2,  2;
  2,  4,  4,   4;
  3,  6, 10,   8,   8;
  3,  9, 16,  24,  16,  16;
  4, 12, 28,  40,  56,  32,  32;
  4, 16, 40,  80,  96, 128,  64,  64;
  5, 20, 60, 120, 216, 224, 288, 128, 128;
  5, 25, 80, 200, 336, 560, 512, 640, 256, 256;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000012, A105635, A157898.
Columns: A004526 (k=0), A002620 (k=1), A006584 (k=2), 4*A096338 (k=3), 8*A177747 (k=4), 16*A299337 (k=5), 32*A178440 (k=6).
Sums include: A105635(n+1) (row), A166486(n+1) (alternating sign diagonal), A232801(n+1) (diagonal).

A011782:= func< n | n eq 0 select 1 else 2^(n-1) >;
function t(n, k) // t = A059576
  if k eq 0 or k eq n then return A011782(n);
  else return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1);
  end if; return t;
end function;
A157898:= func< n, k | (&+[(-1)^(n-j)*Binomial(n, j)*t(j, k): j in [k..n]]) >;
A157071:= func< n,k | (&+[A157898(j+k,k): j in [0..n-k]]) >;
[A157071(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 27 2025

t[n_, k_]:= t[n,k]= If[k==0 || k==n, 2^(n-1) +Boole[n==0]/2, 2*t[n-1,k-1] +2*t[n-1,k] - (2 -Boole[n==2])*t[n-2,k-1]]; (* t = A059576 *)
A157898[n_, k_]:= A157898[n,k]= Sum[(-1)^(n-j)*Binomial[n,j]*t[j,k], {j,k,n}];
A157901[n_, k_]:= A157901[n,k]= Sum[A157898[j+k,k], {j,0,n-k}];
Table[A157901[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 27 2025 *)

@CachedFunction
def t(n, k): # t = A059576
    if (k==0 or k==n): return bool(n==0)/2 + 2^(n-1) # A011782
    else: return 2*t(n-1, k-1) + 2*t(n-1, k) - (2 - 0^(n-2))*t(n-2, k-1)
def A157898(n, k): return sum((-1)^(n+k-j)*binomial(n, j+k)*t(j+k, k) for j in range(n-k+1))
def A157071(n,k): return sum(A157898(j+k,k) for j in range(n-k+1))
print(flatten([[A157071(n,k) for k in range(n+1)] for n in range(10)])) # G. C. Greubel, Aug 27 2025

T(n,k) = Sum_{j=0..n} A157898(j,k).

Edited by the Associate Editors of the OEIS, Apr 10 2009

More terms from G. C. Greubel, Aug 27 2025

cp's OEIS Frontend

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

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

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A111955 a(n) = A078343(n) + (-1)^n.

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A251867 Numbers n such that n^2 + (n+1)^2 is equal to the sum of the hexagonal numbers H(m) and H(m+1) for some m.

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A157901 Triangle read by rows: A000012 * A157898.

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