A317314 -id:A317314 - cp's OEIS frontend

A274979 Integers of the form m*(m + 7)/8.

0, 1, 15, 18, 46, 51, 93, 100, 156, 165, 235, 246, 330, 343, 441, 456, 568, 585, 711, 730, 870, 891, 1045, 1068, 1236, 1261, 1443, 1470, 1666, 1695, 1905, 1936, 2160, 2193, 2431, 2466, 2718, 2755, 3021, 3060, 3340, 3381, 3675, 3718, 4026, 4071, 4393, 4440, 4776, 4825

Offset: 1

Bruno Berselli, Jul 15 2016

Nonnegative values of m are listed in A047393.
Also, numbers h such that 32*h + 49 is a square.
Equivalently, numbers of the form i*(8*i + 7) with i = 0, -1, 1, -2, 2, -3, 3, ...
Infinitely many squares belong to this sequence.
The first bisection is A139278, and 0 followed by the second bisection gives A051870.
Generalized 18-gonal (or octadecagonal) numbers (see the third comment). - Omar E. Pol, Jun 06 2018
Partial sums of A317314. - Omar E. Pol, Jul 28 2018
Exponents in expansion of Product_{n >= 1} (1 + x^(16*n-15))*(1 + x^(16*n-1))*(1 - x^(16*n)) = 1 + x + x^15 + x^18 + x^46 + .... - Peter Bala, Dec 10 2020
Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, k >= 5. They are also the partial sums of the sequence formed by the multiples of (k - 4) and the odd numbers (A005408) interleaved, k >= 5. In this case k = 18. - Omar E. Pol, Apr 25 2021

			100 is in the sequence because 100 = 25*(25+7)/8 or also 100 = 4*(8*4-7).
From _Omar E. Pol_, Apr 24 2021: (Start)
Illustration of initial terms as vertices of a rectangular spiral:
        46_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _18
         |                                                       |
         |                           0                           |
         |                           |_ _ _ _ _ _ _ _ _ _ _ _ _ _|
         |                           1                           15
         |
        51
More generally, all generalized k-gonal numbers can be represented with this kind of spirals, k >= 5. In this case  k = 18. (End)

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

Cf. sequences of the form m*(m+k)/(k+1) listed in A274978.
Cf. similar sequences listed in A299645.
Cf. A317314.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), this sequence (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

[t: m in [0..200] | IsIntegral(t) where t is m*(m+7)/8];

Select[m = Range[0, 200]; m (m + 7)/8, IntegerQ] (* Jean-François Alcover, Jul 21 2016 *)
Select[Table[(m(m+7))/8,{m,0,200}],IntegerQ] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{0,1,15,18,46},50] (* Harvey P. Dale, May 07 2019 *)

def A274979(n): return (n>>1)*((n<<2)+(3 if n&1 else -7)) # Chai Wah Wu, Mar 11 2025

def A274979_list(len):
    h = lambda m: m*(m+7)/8
    return [h(m) for m in (0..len) if h(m) in ZZ]
print(A274979_list(199)) # Peter Luschny, Jul 18 2016

O.g.f.: x^2*(1 + 14*x + x^2)/((1 + x)^2*(1 - x)^3).
E.g.f.: (3*(2*x + 1)*exp(-x) + (8*x^2 - 3)*exp(x))/4.
a(n) = (8*(n-1)*n - 3*(2*n-1)*(-1)^n - 3)/4.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n >= 6. - Wesley Ivan Hurt, Dec 18 2020
From Amiram Eldar, Feb 28 2022: (Start)
Sum_{n>=2} 1/a(n) = (8 + 7*(sqrt(2)+1)*Pi)/49.
Sum_{n>=2} (-1)^n/a(n) = 8*log(2)/7 + 2*sqrt(2)*log(sqrt(2)+1)/7 - 8/49. (End)
a(n) = (n-1)*(4*n+3)/2 if n is odd and a(n) = n*(4*n-7)/2 if n is even. - Chai Wah Wu, Mar 11 2025

A195151 Square array read by antidiagonals upwards: T(n,k) = n((k-2)(-1)^n+k+2)/4, n >= 0, k >= 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 3, 1, 1, 0, 0, 3, 2, 1, 0, 5, 2, 3, 3, 1, 0, 0, 5, 4, 3, 4, 1, 0, 7, 3, 5, 6, 3, 5, 1, 0, 0, 7, 6, 5, 8, 3, 6, 1, 0, 9, 4, 7, 9, 5, 10, 3, 7, 1, 0, 0, 9, 8, 7, 12, 5, 12, 3, 8, 1, 0, 11, 5, 9, 12, 7, 15, 5, 14, 3, 9, 1, 0, 0, 11, 10, 9, 16, 7

Offset: 0

Omar E. Pol, Sep 14 2011

Also square array T(n,k) read by antidiagonals in which column k lists the multiples of k and the odd numbers interleaved, n>=0, k>=0. Also square array T(n,k) read by antidiagonals in which if n is even then row n lists the multiples of (n/2), otherwise if n is odd then row n lists a constant sequence: the all n's sequence. Partial sums of the numbers of column k give the column k of A195152. Note that if k >= 1 then partial sums of the numbers of the column k give the generalized m-gonal numbers, where m = k + 4.

All columns are multiplicative. - Andrew Howroyd, Jul 23 2018

			Array begins:
.  0,   0,   0,   0,   0,   0,   0,   0,   0,   0,...
.  1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
.  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,...
.  3,   3,   3,   3,   3,   3,   3,   3,   3,   3,...
.  0,   2,   4,   6,   8,  10,  12,  14,  16,  18,...
.  5,   5,   5,   5,   5,   5,   5,   5,   5,   5,...
.  0,   3,   6,   9,  12,  15,  18,  21,  24,  27,...
.  7,   7,   7,   7,   7,   7,   7,   7,   7,   7,...
.  0,   4,   8,  12,  16,  20,  24,  28,  32,  36,...
.  9,   9,   9,   9,   9,   9,   9,   9,   9,   9,...
.  0,   5,  10,  15,  20,  25,  30,  35,  40,  45,...
...

Rows: A000004, A000012, A001477, A010701, A005843, A010716, A008585, A010727, A008586, A010734, A008587.

Columns k: A026741 (k=1), A001477 (k=2), zero together with A080512 (k=3), A022998 (k=4), A195140 (k=5), zero together with A165998 (k=6), A195159 (k=7), A195161 (k=8), A195312 k=(9), A195817 (k=10), A317311 (k=11), A317312 (k=12), A317313 (k=13), A317314 k=(14), A317315 (k=15), A317316 (k=16), A317317 (k=17), A317318 (k=18), A317319 k=(19), A317320 (k=20), A317321 (k=21), A317322 (k=22), A317323 (k=23), A317324 k=(24), A317325 (k=25), A317326 (k=26).

T(n,k) = n*((k-2)*(-1)^n+k+2)/4 \\ Andrew Howroyd, Jul 23 2018

A317326 Multiples of 26 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 26, 3, 52, 5, 78, 7, 104, 9, 130, 11, 156, 13, 182, 15, 208, 17, 234, 19, 260, 21, 286, 23, 312, 25, 338, 27, 364, 29, 390, 31, 416, 33, 442, 35, 468, 37, 494, 39, 520, 41, 546, 43, 572, 45, 598, 47, 624, 49, 650, 51, 676, 53, 702, 55, 728, 57, 754, 59, 780, 61, 806, 63, 832, 65, 858, 67, 884, 69

Offset: 0

Omar E. Pol, Jul 25 2018

a(n) is the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 30-gonal numbers (A316729).
Partial sums give the generalized 30-gonal numbers.
More generally, the partial sums of the sequence formed by the multiples of m and the odd numbers interleaved, give the generalized k-gonal numbers, with m >= 1 and k = m + 4.
From Bruno Berselli, Jul 27 2018: (Start)
Also, this type of sequence is characterized by:
O.g.f.: x*(1 + m*x + x^2)/(1 - x^2)^2;
E.g.f.: x*(2 - m + (2 + m)*exp(2*x))*exp(-x)/4;
a(n) = -a(-n) = (2 + m - (2 - m)*(-1)^n)*n/4;
a(n) = (m/2)^((1 + (-1)^n)/2)*n;
a(n) = 2*a(n-2) - a(n-4), with signature (0,2,0,-1). (End)

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

Cf. A252994 and A005408 interleaved.
Column 26 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14), A317311 (k=15), A317312 (k=16), A317313 (k=17), A317314 (k=18), A317315 (k=19), A317316 (k=20), A317317 (k=21), A317318 (k=22), A317319 (k=23), A317320 (k=24), A317321 (k=25), A317322 (k=26), A317323 (k=27), A317324 (k=28), A317325 (k=29), this sequence (k=30).
Cf. A316729.

[13^div(1+(-1)^n,2)*n for n in 0:70] |> println # Bruno Berselli, Jul 28 2018

Table[(7 + 6 (-1)^n) n, {n, 0, 70}] (* Bruno Berselli, Jul 27 2018 *)

a(2*n) = 26*n, a(2*n+1) = 2*n + 1.
From Bruno Berselli, Jul 27 2018: (Start)
O.g.f.: x*(1 + 26*x + x^2)/(1 - x^2)^2.
E.g.f.: x*(-6 + 7*exp(2*x))*exp(-x).
a(n) = -a(-n) = (7 + 6*(-1)^n)*n.
a(n) = 13^((1 + (-1)^n)/2)*n.
a(n) = 2*a(n-2) - a(n-4). (End)
Multiplicative with a(2^e) = 13*2^e, and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 3*2^(3-s)). - Amiram Eldar, Oct 26 2023

cp's OEIS Frontend

A274979 Integers of the form m*(m + 7)/8.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Python

Sage

Formula

A195151 Square array read by antidiagonals upwards: T(n,k) = n((k-2)(-1)^n+k+2)/4, n >= 0, k >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A317326 Multiples of 26 and odd numbers interleaved.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Julia

Mathematica

Formula

cp's OEIS Frontend

A274979 Integers of the form m*(m + 7)/8.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Python

Sage

Formula

A195151 Square array read by antidiagonals upwards: T(n,k) = n*((k-2)*(-1)^n+k+2)/4, n >= 0, k >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A317326 Multiples of 26 and odd numbers interleaved.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Julia

Mathematica

Formula

A195151 Square array read by antidiagonals upwards: T(n,k) = n((k-2)(-1)^n+k+2)/4, n >= 0, k >= 0.