A151798 -id:A151798 - cp's OEIS frontend

A131098 Partial sums of A151798.

1, 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99, 103, 107, 111, 115, 119, 123, 127, 131, 135, 139, 143, 147, 151, 155, 159, 163, 167, 171, 175, 179, 183, 187, 191, 195, 199, 203, 207, 211, 215, 219, 223, 227, 231, 235, 239

Offset: 1

Hans Isdahl, Sep 24 2007

1 together with A004767. - Omar E. Pol, Feb 23 2014

			g.f. = x + 3*x^2 + 7*x^3 + 11*x^4 + 15*x^5 + 19*x^6 + 23*x^7 + 27*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
David Applegate, The movie version
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A151798, A004767.

I:=[1,3,7]; [n le 3 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..70]]; // Vincenzo Librandi, Feb 25 2014

CoefficientList[Series[(x + 2 x^2 + 1)/(x - 1)^2, {x, 0, 80}], x] (* Vincenzo Librandi, Feb 25 2014 *)
LinearRecurrence[{2,-1},{1,3,7},70] (* Harvey P. Dale, Jan 03 2023 *)

A131098(n)=abs(4*n-5) \\ M. F. Hasler, Apr 27 2018

a(1) = 1, a(n) = 4*n - 5 for n >= 2. - Jaroslav Krizek, Aug 15 2009
G.f.: x*(x+2*x^2+1)/(x-1)^2. - R. J. Mathar, Dec 08 2010
E.g.f.: exp(x)*(4*x - 5) + 5 + 2*x. - Stefano Spezia, Mar 21 2025

Edited by N. J. A. Sloane, Jun 29 2009

A174723 a(n) = n(4n^2 - 3*n + 5)/6.

Original entry on oeis.org

1, 5, 16, 38, 75, 131, 210, 316, 453, 625, 836, 1090, 1391, 1743, 2150, 2616, 3145, 3741, 4408, 5150, 5971, 6875, 7866, 8948, 10125, 11401, 12780, 14266, 15863, 17575, 19406, 21360, 23441, 25653, 28000, 30486, 33115, 35891, 38818, 41900, 45141

Offset: 1

Michel Lagneau, Mar 28 2010

We prove that a(n) = Sum_{k=1..n^2} floor(sqrt(k)): a(n) = Sum_{k=1..3} 1 + Sum_{k=4..8} 2 + ... + Sum_{k=(n-1)^2..n^2 - 1} (n-1) + n = 3*1 + 5*2 + 7*3 + ... + (2n-1)(n-1)+ n = Sum_{k=1..n} (2k-1)*(k-1) + n = 2*Sum_{k=1..n} k^2 - 3*Sum_{k=1..n} k + 2n = 2n(n+1)(2n+1)/6 - 3n(n+1)/2 + 2n = n*(4n^2 - 3n + 5) / 6.

Notice that a(4) = 4 + 3*5 + 2*6 + 1*7 and a(8) = 8 + 7*9 + 6*10 + 5*11 + 4*12 + 3*13 + 2*14 + 1*15. In general, a(n) = n + Sum_{k=1..n-1} (n-k)*(n+k). - J. M. Bergot, Jul 31 2013

			From _Bruno Berselli_, Feb 17 2015: (Start)
Third differences:  1, 2,  4,  4,   4,   4,   4, (repeat 4) ... (A151798)
Second differences: 1, 3,  7, 11,  15,  19,  23,  27,   31, ... (A131098)
First differences:  1, 4, 11, 22,  37,  56,  79, 106,  137, ... (A084849)
-------------------------------------------------------------------------
This sequence:      1, 5, 16, 38,  75, 131, 210, 316,  453, ...
-------------------------------------------------------------------------
Partial sums:       1, 6, 22, 60, 135, 266, 476, 792, 1245, ... (A071239)
(End)

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford Univ. Press, 1954.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000196, A071239, A084849, A131098, A151798, A022554.

I:=[1, 5, 16, 38]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jul 04 2012

A174723 := proc(n)
        n*(4*n^2-3*n+5)/6 ;
end proc:
seq( A174723(n),n=1..20) ; # R. J. Mathar, Nov 07 2011

Table[n (4n^2-3n+5)/6,{n,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,5,16,38},50] (* Harvey P. Dale, Jan 16 2012 *)

a(n)=n*(4*n^2-3*n+5)/6 \\ Charles R Greathouse IV, Oct 07 2015

G.f. x*(1 + x + 2*x^2) / (x-1)^4. - R. J. Mathar, Nov 07 2011
a(1)=1, a(2)=5, a(3)=16, a(4)=38; for n > 4, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Jan 16 2012
a(n) = A022554(n^2). - Ridouane Oudra, Jun 13 2025

A255176 a(n) = H_n(2,2) where H_n is the n-th hyperoperator.

Original entry on oeis.org

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, 4

Offset: 0

Natan Arie Consigli, Feb 25 2015

See A054871 for definitions and key links.
Also, decimal expansion of 31/90. - Bruno Berselli, Mar 18 2015
Essentially the same as A010709, A040002, A113311, A123932, and A151798. - R. J. Mathar, Mar 20 2015
Remainder of the Euclidean division when 10^(10^n) is divided by 7 (proof by induction for n >= 1) [see reference Julien Freslon & Jérôme Poineau]; example: 10^(10^1) = 1428571428 * 7 + 4. - Bernard Schott, Aug 28 2020

			a(0) = H_0(2,2) = 2+1 = 3.
a(1) = H_1(2,2) = 2+2 = 4.
a(2) = H_2(2,2) = 2*2 = 4.
a(3) = H_3(2,2) = 2^2 = 4.
a(n) = H_n(2,2) = H_{n-1}(2,H_n(2,1)) = H_{n-1}(2,2) = 4, for n>1.

Julien Freslon & Jérôme Poineau, Les 100 exercices-types de mathématiques: MPSI/PCSI/PTSI, EdiScience, 2007, Exercice 11.2, page 242.

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

Cf. A054871.

Cf. A040000, A157532.

G.f.: (3 + x)/(1 - x). - Bruno Berselli, Mar 18 2015

a(n) = 10^(10^n) mod 7. - Bernard Schott, Aug 28 2020

Edited by Danny Rorabaugh, Oct 20 2015

A322419 Number of n-step self-avoiding walks on L-lattice.

Original entry on oeis.org

1, 2, 4, 8, 12, 20, 32, 52, 84, 136, 220, 356, 564, 904, 1448, 2320, 3684, 5872, 9376, 14960, 23688, 37652, 59912, 95316, 150744, 239080, 379528, 602424, 951788, 1507136, 2388252, 3784344, 5973988, 9447880, 14950796, 23658540, 37321752, 58965260, 93206864, 147333080, 232286272

Offset: 0

Robert FERREOL, Dec 07 2018

The L-lattice is an oriented square lattice in which each step must be followed by a step perpendicular to the preceding one.

			a(1) = 2 because there are only two possible directions at each intersection; for the same reason a(2) = 2*2 and a(3) = 2*4 ; but a(4) = 12 (not 16) because four paths return to the starting point and are not self-avoiding. See the 12 paths under "links".

Sean A. Irvine, Table of n, a(n) for n = 0..50
Robert FERREOL, The a(4)=12 walks in L-lattice
Keh-Ying Lin and Yee-Mou Kao, Universal amplitude combinations for self-avoiding walks and polygons on directed lattices, J. Phys. A: Math. Gen. 32 (1999), page 6929.
A. Malakis, Self-avoiding walks on oriented square lattices, Journal of Physics A: Mathematical and General, Volume 8, Number 12 (1975), page 1890.
Wikipedia, Connective constant

Cf. A001411 (square lattice), A117633 (Manhattan lattice), A189722, A004277 (coordination sequence), A151798.

walks:=proc(n)
    option remember;
    local i,father,End,X,walkN,dir,u,x,y;
    if n=1 then [[[0,0]]] else
         father:=walks(n-1):
         walkN:=NULL:
         for i to nops(father) do
            u:=father[i]:End:=u[n-1]:if n mod 2 = 0 then
            dir:=[[1,0], [-1, 0]] else dir := [[0,1], [0, -1]] fi:
            for X in dir do
             if not(member(End+X,u)) then walkN:=walkN,[op(u),End+X] fi;
             od od:
         [walkN] fi end:
n:=5:L:=walks(n):N:=nops(L);
# This program explicitly gives the a(n) walks.

mo = {{1, 0}, {-1, 0}}; moo = {{0, 1}, {0, -1}}; a[0] = 1;
a[tg_, p_: {{0, 0}}] := Module[{e, mv},
If[Mod[tg, 2] == 0, mv = Complement[Last[p] + # & /@ mo, p],
mv = Complement[Last[p] + # & /@ moo, p]];
If[tg == 1, Length@mv, Sum[a[tg - 1, Append[p, e]], {e, mv}]]];
a /@ Range[0, 20] (* after the program from Giovanni Resta at A001411 *)

def add(L, x):
    M = [y for y in L]
    M.append(x)
    return M
plus = lambda L, M: [x + y for x, y in zip(L, M)]
mo = [[1, 0], [-1, 0]]
moo = [[0, 1], [0, -1]]
def a(n, P=[[0, 0]]):
    if n == 0:
        return 1
    if n % 2 == 0:
        mv1 = [plus(P[-1], x) for x in mo]
    else:
        mv1 = [plus(P[-1], x) for x in moo]
    mv2 = [x for x in mv1 if x not in P]
    if n == 1:
        return len(mv2)
    else:
        return sum(a(n - 1, add(P, x)) for x in mv2)
[a(n) for n in range(21)]

a(n) = 4*A189722(n) for n >= 2.

It is proved that a(n)^(1/n) has a limit mu called the "connective constant" of the L-lattice; approximate value of mu: 1.5657. It is only conjectured that a(n + 1) ~ mu * a(n).

A172090 Triangle T(n, k) = f(n-k) + f(k) - f(n), where f(n) = -3*n with f(0) = 1, f(1) = -2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 0

Roger L. Bagula, Jan 25 2010

			Triangle begins as:
  1;
  1, 1;
  1, 2, 1;
  1, 1, 1, 1;
  1, 1, 0, 1, 1;
  1, 1, 0, 0, 1, 1;
  1, 1, 0, 0, 0, 1, 1;
  1, 1, 0, 0, 0, 0, 1, 1;
  1, 1, 0, 0, 0, 0, 0, 1, 1;
  1, 1, 0, 0, 0, 0, 0, 0, 1, 1;
  1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1;

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

Row sums are A151798.

(* First program *)
f[n_]:= f[n]= If[n < 2, (-1)^n*(n+1), -3*n];
T[n_, k_]:= f[n-k] +f[k] -f[n];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 29 2021 *)
(* Second program *)
T[n_, k_]:= If[n<3, Binomial[n, k], If[n==3 || k<2 || k>n-2, 1, 0]];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 29 2021 *)

def f(n): return (-1)^n*(n+1) if (n<2) else -3*n
def T(n,k): return f(n-k) + f(k) - f(n)
flatten([[T(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 29 2021

T(n, k) = f(n-k) + f(k) - f(n), where f(n) = -3*n with f(0) = 1, f(1) = -2.
From G. C. Greubel, Apr 29 2021: (Start)
T(n, k) is defined by T(n, 0) = T(n, 1) = T(n, n-1) = T(n, n) = T(3, k) = 1, T(2, 1) = 2 and 0 otherwise.
Sum_{k=0..n} T(n,k) = A151798(n). (End)

Edited by G. C. Greubel, Apr 29 2021

A267649 a(0) = a(1) = 2 then a(n) = 4 for n>=2.

Original entry on oeis.org

2, 2, 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

Natan Arie Consigli, Jan 19 2016

Decimal expansion of 101/450.
Also list of smallest n-composites.
A hyperoperator aggregation b[n]c is n-composite if b,c are positive non-right-identity elements.
The identity elements are:
Hyper-0 (zeration): none.
Hyper-1 (addition): 0.
Hyper-2 (multiplication): 1.
Hyper-3 (exponentiation): 1.
Hyper-n (n>2): 1.
For more information on hyperoperations see A054871.
Essentially the same as A255176, A151798, A123932, A113311, A040002 and A010709. - R. J. Mathar, May 25 2023
Continued fraction expansion of 2 + sqrt(1/5) = 2 + sqrt(5)/5. - Elmo R. Oliveira, Aug 06 2024

			a(0) = 2 because 1 is the smallest non-identity element in zeration and 1[0]1=2;
a(1) = 2 because 1 is the smallest non-identity element in addition and 1[1]1=2;
a(2) = 4 because 2 is the smallest non-identity element in multiplication and 2[2]2=4;
a(3) = 4 because 2 is the smallest non-identity element in exponentiation and 2[2]2=4;
a(4) = 4 because 2 is the smallest non-identity element in titration and 2[2]2=4;
Etc.

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

Cf. A000027 (1-composites), A002808 (composites), A267647 (3-composites), A097374 (4-composites).

a(n) = a[n]b where a,b are the positive smallest non-right-identity elements.
From Elmo R. Oliveira, Aug 06 2024: (Start)
G.f.: 4/(1 - x) - 2*(1 + x).
E.g.f.: 4*exp(x) - 2*(1 + x). (End)

cp's OEIS Frontend

A131098 Partial sums of A151798.

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A174723 a(n) = n*(4*n^2 - 3*n + 5)/6.

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A255176 a(n) = H_n(2,2) where H_n is the n-th hyperoperator.

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A322419 Number of n-step self-avoiding walks on L-lattice.

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A172090 Triangle T(n, k) = f(n-k) + f(k) - f(n), where f(n) = -3*n with f(0) = 1, f(1) = -2, read by rows.

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A267649 a(0) = a(1) = 2 then a(n) = 4 for n>=2.

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A174723 a(n) = n(4n^2 - 3*n + 5)/6.