A047218 -id:A047218 - cp's OEIS frontend

A143901 Rectangular array R by antidiagonals: R(m,n) = floor((m*n+1)/2).

1, 1, 1, 2, 2, 2, 2, 3, 3, 2, 3, 4, 5, 4, 3, 3, 5, 6, 6, 5, 3, 4, 6, 8, 8, 8, 6, 4, 4, 7, 9, 10, 10, 9, 7, 4, 5, 8, 11, 12, 13, 12, 11, 8, 5, 5, 9, 12, 14, 15, 15, 14, 12, 9, 5, 6, 10, 14, 16, 18, 18, 18, 16, 14, 10, 6, 6, 11, 15, 18, 20, 21, 21, 20, 18, 15, 11, 6, 7, 12, 17, 20, 23, 24, 25, 24

Offset: 1

Clark Kimberling, Sep 04 2008

Old name was: R(m,n) = number of white squares.

			Northwest corner:
1 1 2 2 3 3 4 4
1 2 3 4 5 6 7 8
2 3 5 6 8 9 11 12
2 4 6 8 10 12 14 16

Cf. A143902.
Antidiagonal sums: (1,2,6,10,19,...)=A005993.
Diagonals: A000982, A000217, A007590, A000096, A116940.
Rows and columns: A004526, A000027, A007494, A005843, A047218 et al.

Entry revised by N. J. A. Sloane, Jun 12 2015

A192328 Numbers of the form 20*k+7 which are three times a square.

Original entry on oeis.org

27, 147, 507, 867, 1587, 2187, 3267, 4107, 5547, 6627, 8427, 9747, 11907, 13467, 15987, 17787, 20667, 22707, 25947, 28227, 31827, 34347, 38307, 41067, 45387, 48387, 53067, 56307, 61347, 64827, 70227, 73947, 79707, 83667, 89787, 93987

Offset: 1

Bruno Berselli, Jun 28 2011

Text of the theorem in the paper mentioned in References: The necessary and sufficient condition so that a number of the form 20*k+7 is three times a square is that k is of the form 3*h*(5*h+3)+1 or 3*h*(5*h+7)+7.
A119617 gives the values of k.
A080512*120 gives the first differences.

"Supplemento al Periodico di Matematica", Raffaello Giusti Editore (Livorno) - Mar 1901 - p. 75 (Problem 286 and its generalization, G. Cardoso-Laynes).

Mohammed Yaseen, Table of n, a(n) for n = 1..10000 (first 1000 terms from Bruno Berselli)
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A134153, A134154.

Cf. A020742, A047218.

[m: m in [7..10^5 by 20] | IsSquare(m/3)];

select(t -> issqr(t/3), [seq(20*i+7,i=1..10000,3)]); # Robert Israel, Apr 28 2023

Select[20 Range[5000] + 7, IntegerQ[Sqrt[#/3]] &] (* or *) LinearRecurrence[{1,2,-2,-1,1}, {27,147,507,867,1587}, 40] (* Harvey P. Dale, Jul 06 2011 *)
CoefficientList[Series[3 (9 + 40 x + 102 x^2 + 40 x^3 + 9 x^4) / ((1 + x)^2 (1 - x)^3), {x, 0, 35}], x] (* Vincenzo Librandi, Aug 19 2013 *)

for(k=0, 5*10^3, m=20*k+7; if(issquare(m/3), print1(m",")));

a(n)=my(m=n--\4); 1200*m^2+[360*m+27, 840*m+147, 1560*m+507, 2040*m+867][n%4+1] \\ Charles R Greathouse IV, Jun 29 2011

G.f.: 3*x*(9 + 40*x + 102*x^2 + 40*x^3 + 9*x^4)/((1 + x)^2*(1 - x)^3).
a(n) = 3*((10*(n-1) + (-1)^(n-1) + 5)/2)^2.
a(n) = a(-n-1) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
Sum_{i=1..n} a(i) = n*(50*(n-1)*(n+1) + 15*(-1)^(n-1) + 39)/2.
a(n) = 3*A020742(A047218(n))^2.

Offset corrected by Mohammed Yaseen, Apr 27 2023

A332102 Least m > 0 such that 2*m^n <= Sum_{k < m} k^n.

Original entry on oeis.org

3, 5, 8, 10, 13, 15, 18, 20, 23, 25, 28, 30, 33, 35, 38, 40, 42, 45, 47, 50, 52, 55, 57, 60, 62, 65, 67, 70, 72, 75, 77, 79, 82, 84, 87, 89, 92, 94, 97, 99, 102, 104, 107, 109, 112, 114, 116, 119, 121, 124, 126, 129, 131, 134, 136, 139, 141, 144, 146, 149, 151, 153, 156, 158, 161, 163, 166

Offset: 0

M. F. Hasler, Apr 18 2020

nonn

Obviously a(n) is a lower limit for any s solution to 2*s^n = Sum_{x in S} x^n, S subset of {1, ..., s-1}.

First differences are (2, 3, 2, 3, ...) except for a duplicated 2 in positions {16, 31, 46, 61, 76, 91; 104, 119, 134, 149, 164, 179, 194, 209, 224, 239, 254, 269; 282, 297, ...}: Here the first differences are always 15 except for a 13 after the 6th, 18th, ... term.

			For n=0, 2*m^0 = 2 > Sum_{k<m} k^0 = m - 1 <=> 3 > m, so a(0) = 3.
For n=1, 2*m^1 > Sum_{k<m} k^1 = m(m-1)/2 <=> 4 > m - 1, so a(1) = 5.

Cf. A332101 (same without factor 2 in definition).

Cf. A195168, A047218, A029919 (all have common initial terms but differ later and only remain lower resp. upper bounds).

Table[Block[{m = 1, s = 0}, While[2 m^n > s, s = s + m^n; m++]; m], {n, 0, 66}] (* Michael De Vlieger, Apr 30 2020 *)

apply( A332102(n, s)=for(m=1, oo, s<2*m^n||return(m); s+=m^n), [0..66])

a(n) >= A195168(n+1) with equality for n not in {13, 15; 26, 28, 30; 39, 41, 43, 45; 52, 54, ..., 60; 65, 67, ..., 75, 78, 80, ..., 90; 89, 91, ..., 103; 102, 104, ..., 114, 115, ...} \ {120, 122, 124, 126, 135, 137, 139, 150, 152, 165}.
a(n) <= A047218(n+2) with equality for n <= 17 and even n <= 34.
Conjecture: a(n) = round(n/log(3/2) + 3).

A063093 Dimension of the space of weight 2n cusp forms for Gamma_0( 25 ).

Original entry on oeis.org

0, 5, 9, 15, 19, 25, 29, 35, 39, 45, 49, 55, 59, 65, 69, 75, 79, 85, 89, 95, 99, 105, 109, 115, 119, 125, 129, 135, 139, 145, 149, 155, 159, 165, 169, 175, 179, 185, 189, 195, 199, 205, 209, 215, 219, 225, 229, 235, 239, 245

Offset: 1

N. J. A. Sloane, Jul 08 2001

If b(n) is the sequence of integers congruent to {0,3} (mod 5) and c(n) is the sequence of integers congruent to {2,4}(mod 5). Then a(n) = b(n) + c(n). Equivalently a(n) = A047218(n+1) + A047211(n). - Anthony Hernandez, Aug 16 2016

William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001622, A047211, A047218.

Rest@ CoefficientList[Series[x^2*(5 + 4 x + x^2)/((1 - x)^2*(1 + x)), {x, 0, 50}], x] (* Michael De Vlieger, Aug 26 2016 *)
LinearRecurrence[{1,1,-1},{0,5,9,15},50] (* Harvey P. Dale, Apr 09 2019 *)

a(n) = 10*n - a(n-1) - 16 for n>2, with a(1)=0, a(2)=5. - Vincenzo Librandi, Aug 07 2010
From Colin Barker, Sep 26 2012: (Start)
a(n) = ((-1)^n + 10*n - 11)/2 for n>1.
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
G.f.: x^2*(5+4*x+x^2)/((1-x)^2*(1+x)). (End)
Sum_{n>=2} (-1)^n/a(n) = sqrt(5+2*sqrt(5))*Pi/20 - 3*sqrt(5)*log(phi)/20 - log(5)/8, where phi is the golden ratio (A001622). - Amiram Eldar, Apr 15 2023

A063281 Dimension of the space of weight n cuspidal newforms for Gamma_1( 8 ).

Original entry on oeis.org

-1, 0, 1, 3, 3, 5, 5, 8, 7, 10, 9, 13, 11, 15, 13, 18, 15, 20, 17, 23, 19, 25, 21, 28, 23, 30, 25, 33, 27, 35, 29, 38, 31, 40, 33, 43, 35, 45, 37, 48, 39, 50, 41, 53, 43, 55, 45, 58, 47, 60, 49, 63, 51, 65, 53, 68, 55, 70, 57, 73, 59, 75, 61, 78, 63, 80

Offset: 2

N. J. A. Sloane, Jul 14 2001

sign

William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_1(N))
William A. Stein, The modular forms database
Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 1, 0, -1).

Cf. A005408 (bisection), A047218 (bisection).

G.f.: x^2*(2*x^5 +3*x^4 +3*x^3 +2*x^2 -1)/((x -1)^2*(x +1)^2*(x^2 +1)). [Colin Barker, Sep 28 2012]

A328986 The sequence C(n) defined in the comments (A and B smallest missing numbers, offset 1).

Original entry on oeis.org

4, 10, 16, 21, 28, 33, 39, 45, 51, 57, 62, 68, 74, 80, 86, 91, 98, 103, 109, 115, 120, 127, 132, 138, 144, 150, 156, 161, 168, 173, 179, 185, 190, 197, 202, 208, 214, 220, 226, 231, 237, 243, 249, 255, 260, 267, 272, 278, 284, 290, 296, 301, 307, 313, 319

Offset: 1

N. J. A. Sloane, Nov 07 2019

nonn

Define a triple of sequences A,B,C by A[1]=1, B[1]=2, C[1]=4; for n>=2, A[n] = smallest missing number from the terms of A,B,C defined so far; B[n] = = smallest missing number from the terms of A,B,C defined so far; C[n] = n+A[n]+B[n].
Then A = A286660, B = A080652, C = the present sequence.
Inspired by the triples [A003144, A003145, A004146] and [A298468, A298469, A047218].

			The initial terms are:
n: 1, 2, 3, 4,  5,  6,  7,  8.  9. 10. 11, 12, ...
A: 1, 3, 6, 8, 11, 13, 15, 18, 20, 23, 25, 27, ...
B: 2, 5, 7, 9, 12, 14, 17, 19, 22, 24, 26, 29, ...
C: 4, 10, 16, 21, 28, 33, 39, 45, 51, 57, 62, 68, ...

Cf. A286660, A080652; A003144, A003145, A003146; A298468, A298469, A047218.

cp's OEIS Frontend

A143901 Rectangular array R by antidiagonals: R(m,n) = floor((m*n+1)/2).

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A192328 Numbers of the form 20*k+7 which are three times a square.

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Magma

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A332102 Least m > 0 such that 2*m^n <= Sum_{k < m} k^n.

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Mathematica

PARI

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A063093 Dimension of the space of weight 2n cusp forms for Gamma_0( 25 ).

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Mathematica

Formula

A063281 Dimension of the space of weight n cuspidal newforms for Gamma_1( 8 ).

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Formula

A328986 The sequence C(n) defined in the comments (A and B smallest missing numbers, offset 1).

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