A111711 -id:A111711 - cp's OEIS frontend

A111710 Consider the triangle shown below in which the n-th row contains the n smallest numbers greater than those in the previous row such that the arithmetic mean is an integer. Sequence contains the leading diagonal.

1, 4, 7, 13, 18, 27, 34, 46, 55, 70, 81, 99, 112, 133, 148, 172, 189, 216, 235, 265, 286, 319, 342, 378, 403, 442, 469, 511, 540, 585, 616, 664, 697, 748, 783, 837, 874, 931, 970, 1030, 1071, 1134, 1177, 1243, 1288, 1357, 1404, 1476, 1525, 1600, 1651, 1729

Offset: 1

Amarnath Murthy, Aug 24 2005

			The fourth row is 8,9,10 and 13,(8+9+10 +13)/4 = 10.
Triangle begins:
1
2 4
5 6 7
8 9 10 13
14 15 16 17 18
19 20 21 22 23 27
28 29 30 31 32 33 34

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A111711, A111712.

Cf. A085787. - R. J. Mathar, Aug 15 2008

LinearRecurrence[{1, 2, -2, -1, 1}, {1, 4, 7, 13, 18}, 100] (* Paolo Xausa, Feb 09 2024 *)

Vec(x*(1+3*x+x^2)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 26 2016

a(1)=1, a(2n) = a(2n-1)+3n, a(2n+1)=a(2n)+2n+1. - Franklin T. Adams-Watters, May 01 2006
G.f.: -x*(1+3*x+x^2) / ( (1+x)^2*(x-1)^3 ). a(n+1)-a(n) = A080512(n+1). - R. J. Mathar, May 02 2013
From Colin Barker, Jan 26 2016: (Start)
a(n) = (10*n^2+2*(-1)^n*n+10*n+(-1)^n-1)/16.
a(n) = (5*n^2+6*n)/8 for n even.
a(n) = (5*n^2+4*n-1)/8 for n odd. (End)

More terms from Franklin T. Adams-Watters, May 01 2006

A111712 Arithmetic mean of the n-th row of triangle mentioned in A111710.

Original entry on oeis.org

1, 3, 6, 10, 16, 22, 31, 39, 51, 61, 76, 88, 106, 120, 141, 157, 181, 199, 226, 246, 276, 298, 331, 355, 391, 417, 456, 484, 526, 556, 601, 633, 681, 715, 766, 802, 856, 894, 951, 991, 1051, 1093, 1156, 1200, 1266, 1312, 1381, 1429, 1501, 1551, 1626, 1678

Offset: 1

Amarnath Murthy, Aug 24 2005

			a(4) = (8+9+10+13)/4 =10

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A111710, A111711.

Partial sums of A195013 prepended with 1.

a111712 n = a111712_list !! (n-1)
a111712_list = scanl (+) 1 a195013_list
-- Reinhard Zumkeller, Apr 06 2015

With[{r = Range[50]}, Accumulate[Join[{1}, Riffle[2*r, 3*r]]]] (* or *)
LinearRecurrence[{1, 2, -2, -1, 1}, {1, 3, 6, 10, 16}, 100] (* Paolo Xausa, Feb 09 2024 *)

a(1)=1, a(2n) = a(2n-1)+2n, a(2n+1)=a(2n)+3n. a(n) = A111711(n)+floor(n/2). - Franklin T. Adams-Watters, May 01 2006

More terms from Franklin T. Adams-Watters, May 01 2006

A257143 a(2n) = 3n if n>0, a(2n + 1) = 2n + 1, a(0) = 1.

Original entry on oeis.org

1, 1, 3, 3, 6, 5, 9, 7, 12, 9, 15, 11, 18, 13, 21, 15, 24, 17, 27, 19, 30, 21, 33, 23, 36, 25, 39, 27, 42, 29, 45, 31, 48, 33, 51, 35, 54, 37, 57, 39, 60, 41, 63, 43, 66, 45, 69, 47, 72, 49, 75, 51, 78, 53, 81, 55, 84, 57, 87, 59, 90, 61, 93, 63, 96, 65, 99

Offset: 0

Michael Somos, Apr 16 2015

			G.f. = 1 + x + 3*x^2 + 3*x^3 + 6*x^4 + 5*x^5 + 9*x^6 + 7*x^7 + 12*x^8 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A080512, A111711 (partial sums), A188626.

Cf. A005408, A008486.

import Data.List (transpose)
a257143 n = a257143_list !! n
a257143_list = concat $ transpose [a008486_list, a005408_list]
-- Reinhard Zumkeller, Apr 17 2015

a[ n_] := Which[ n < 1, Boole[n == 0], OddQ[n], n, True, 3 n/2];
a[ n_] := SeriesCoefficient[ (1 + x + x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4), {x, 0, n}];

{a(n) = if( n<1, n==0, n%2, n, 3*n/2)};

{a(n) = if( n<0, 0, polcoeff( (1 - x^5) / ((1 - x) * (1 - x^2)^2) + x * O(x^n), n))};

a(n) is multiplicative with a(2^e) = 3 * 2^(e-1) if e>0, a(p^e) = p^e otherwise and a(0) = 1.
Euler transform of length 5 sequence [ 1, 2, 0, 0, -1].
G.f.: (1 - x^5) / ((1 - x) * (1 - x^2)^2).
G.f.: (1 + x + x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4).
a(n) = A080512(n) if n>0.
First difference of A111711.
A188626(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n.
From Amiram Eldar, Jan 03 2023: (Start)
Dirichlet g.f.: zeta(s-1)*(1+1/2^s).
Sum_{k=1..n} a(k) ~ (5/8) * n^2. (End)

cp's OEIS Frontend

A111710 Consider the triangle shown below in which the n-th row contains the n smallest numbers greater than those in the previous row such that the arithmetic mean is an integer. Sequence contains the leading diagonal.

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A111712 Arithmetic mean of the n-th row of triangle mentioned in A111710.

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Haskell

Mathematica

Formula

Extensions

A257143 a(2n) = 3n if n>0, a(2n + 1) = 2n + 1, a(0) = 1.

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cp's OEIS Frontend

A111710 Consider the triangle shown below in which the n-th row contains the n smallest numbers greater than those in the previous row such that the arithmetic mean is an integer. Sequence contains the leading diagonal.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A111712 Arithmetic mean of the n-th row of triangle mentioned in A111710.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

Extensions

A257143 a(2*n) = 3*n if n>0, a(2*n + 1) = 2*n + 1, a(0) = 1.

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Author

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Programs

Haskell

Mathematica

PARI

PARI

Formula

A257143 a(2n) = 3n if n>0, a(2n + 1) = 2n + 1, a(0) = 1.