A047215 -id:A047215 - cp's OEIS frontend

A047379 Numbers that are congruent to {0, 2, 4, 5} mod 7.

0, 2, 4, 5, 7, 9, 11, 12, 14, 16, 18, 19, 21, 23, 25, 26, 28, 30, 32, 33, 35, 37, 39, 40, 42, 44, 46, 47, 49, 51, 53, 54, 56, 58, 60, 61, 63, 65, 67, 68, 70, 72, 74, 75, 77, 79, 81, 82, 84, 86, 88, 89, 91, 93, 95, 96

Offset: 1

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047215, A057353.

[Floor((7*n-5)/4) : n in [1..100]]; // Wesley Ivan Hurt, Dec 03 2014

A047379:=n->floor((7*n-5)/4): seq(A047379(n), n=1..100); # Wesley Ivan Hurt, Dec 03 2014

Select[Range[0,100],MemberQ[{0,2,4,5},Mod[#,7]]&] (* Harvey P. Dale, Apr 10 2014 *)

a(n) = floor(floor((7n + 2)/2)/2).
a(n) = floor((7n-5)/4). - Gary Detlefs, Mar 07 2010
G.f.: x^2*(2+2*x+x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
From Wesley Ivan Hurt, Dec 03 2014: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5), n>5;
a(n) = (14*n-13-(-1)^n+2*i^n*(-1)^((3+(-1)^n)/4))/8, where i = sqrt(-1);
a(n) = A047215(n-1) - A057353(n-1). (End)

A117793 Pentagonal numbers divisible by 5.

Original entry on oeis.org

0, 5, 35, 70, 145, 210, 330, 425, 590, 715, 925, 1080, 1335, 1520, 1820, 2035, 2380, 2625, 3015, 3290, 3725, 4030, 4510, 4845, 5370, 5735, 6305, 6700, 7315, 7740, 8400, 8855, 9560, 10045, 10795, 11310, 12105, 12650, 13490, 14065, 14950, 15555, 16485

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 29 2006

Intersection of A008587 and A000326. Their indices are given by A047215. - Michel Marcus, Feb 27 2014

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000326, A008587, A047215.

Select[Table[PolygonalNumber[5, n], {n, 0, 100}], Divisible[#, 5] &] (* Amiram Eldar, Mar 22 2021 *)
Select[PolygonalNumber[5,Range[0,200]],Mod[#,5]==0&] (* Harvey P. Dale, Sep 08 2024 *)

isok(n) = ispolygonal(n, 5) && !(n % 5); \\ Michel Marcus, Feb 27 2014

Empirical G.f.: x^2*(5+30*x+25*x^2+15*x^3)/(1-x)^3/(1+x)^2. - Colin Barker, Feb 14 2012

a(2*n+1) = 5*n*(15*n-1)/2, a(2*n+2) = 5*(5*n+2)*(3*n+1)/2. - Robert Israel, Jun 01 2014

A128316 Triangle read by rows: A000012 * A128315 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 3, -1, 1, 2, 3, -2, 1, 4, -1, 4, -3, 1, 4, 3, -5, 7, -4, 1, 6, -3, 10, -13, 11, -5, 1, 4, 8, -14, 23, -24, 16, -6, 1, 7, -2, 15, -33, 46, -40, 22, -7, 1, 7, 4, -15, 47, -79, 86, -62, 29, -8, 1, 9, -6, 30, -73, 131, -166, 148, -91, 37, -9, 1, 7, 12, -37, 103, -204, 297, -314, 239, -128, 46, -10, 1

Offset: 1

Gary W. Adamson, Feb 25 2007

A128316 * [1,2,3...] = A000034: [1,2,1,2,...].

			First few rows of the triangle:
  1;
  1,  1;
  3, -1,   1;
  2,  3   -2,   1;
  4, -1,   4,  -3,   1;
  4,  3,  -5,   7,  -4,  1;
  6, -3,  10, -13,  11, -5,  1;
  4,  8, -14,  23, -24, 16, -6, 1;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000012, A000034, A014300, A026641, A059851, A128315.

Sums include: A000027 (row), A032766, A047215, A344817 (alternating sign).

A128316:= func< n,k | (&+[(-1)^(j+k)*Floor(n/j)*Binomial(j-1,k-1): j in [k..n]]) >;
[A128316(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 23 2024

T[n_, k_]:= Sum[(-1)^(j+k)*Floor[n/j]*Binomial[j-1,k-1], {j,k,n}];
Table[T[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jun 23 2024 *)

def A128316(n,k): return sum((-1)^(j+k)*int(n//j)*binomial(j-1,k-1) for j in range(k,n+1))
flatten([[A128316(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jun 23 2024

Sum_{k=1..n} T(n, k) = A000027(n) (row sums).
T(n, 1) = A059851(n).
From G. C. Greubel, Jun 23 2024: (Start)
T(n, k) = A010766(n,k) * AA130595(n-1, k-1) as infinite lower triangular matrices.
T(n, k) = Sum_{j=k..n} (-1)^(j+k) * floor(n/j) * binomial(j-1, k-1).
T(2*n-1, n) = (-1)^(n-1)*A026641(n).
T(2*n-2, n-1) = (-1)^n*A014300(n-1), for n >= 2.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A344817(n).
Sum_{k=1..n} k*T(n, k) = A032766(n-1).
Sum_{k=1..n} (k+1)*T(n, k) = A047215(n). (End)

a(28) = 1 inserted and more terms from Georg Fischer, Jun 06 2023

A075328 Difference between n-th pair in A075325.

Original entry on oeis.org

2, 5, 7, 10, 12, 15, 17, 20, 22, 25, 27, 30, 32, 35, 37, 40, 42, 45, 47, 50, 52, 55, 57, 60, 62, 65, 67, 70, 72, 75, 77, 80, 82, 85, 87, 90, 92, 95, 97, 100, 102, 105, 107, 110, 112, 115, 117, 120, 122, 125, 127, 130, 132, 135, 137, 140, 142, 145, 147, 150, 152, 155

Offset: 0

Amarnath Murthy, Sep 18 2002

nonn

Empirically the partial sums of A010693 (i.e., 2, 3 repeated). - Sean A. Irvine, Jul 12 2022

Cf. A075325, A075326, A075327.

Cf. A010693, A047215.

More terms from David Wasserman, Jan 16 2005

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A047379 Numbers that are congruent to {0, 2, 4, 5} mod 7.

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A117793 Pentagonal numbers divisible by 5.

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A128316 Triangle read by rows: A000012 * A128315 as infinite lower triangular matrices.

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A075328 Difference between n-th pair in A075325.

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