A160790 -id:A160790 - cp's OEIS frontend

A160791 a(n) = binomial(N, n - N) where N = 1 + floor(n/2).

0, 1, 1, 2, 3, 3, 6, 4, 10, 5, 15, 6, 21, 7, 28, 8, 36, 9, 45, 10, 55, 11, 66, 12, 78, 13, 91, 14, 105, 15, 120, 16, 136, 17, 153, 18, 171, 19, 190, 20, 210, 21, 231, 22, 253, 23, 276, 24, 300, 25, 325, 26, 351, 27, 378, 28, 406, 29, 435, 30, 465

Offset: 0

Omar E. Pol, May 29 2009

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

First differences of A160790.

Cf. A160791, A160792, A160793.

[(n^2+6*n+4+(n^2-2*n-4)*(-1)^n)/16: n in [0..70]]; // Vincenzo Librandi, Apr 02 2015

a := proc(n) 1 + floor(n/2); binomial(%, n - %) end:
seq(a(n), n = 0..60);  # Peter Luschny, Jul 02 2024

Join[{0}, Riffle[Range[30], Range[30] (Range[30] + 1)/2]] (* Bruno Berselli, Jul 15 2013 *)
LinearRecurrence[{0, 3, 0, -3, 0, 1}, {0, 1, 1, 2, 3, 3, 6}, 60] (* Vincenzo Librandi, Apr 02 2015 *)

Vec(x*(1+x-x^2)/(1-x^2)^3 + O(x^80)) \\ Michel Marcus, Apr 01 2015

From R. J. Mathar, Feb 09 2010: (Start)
a(2n+1) = n+1 and a(2n) = A000217(n) with a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).
G.f.: x*(1+x-x^2)/(1-x^2)^3. (End)
a(n) = (n^2+6*n+4+(n^2-2*n-4)*(-1)^n)/16. - Luce ETIENNE, Mar 31 2015
E.g.f.: (x*(x+4)*cosh(x) + (3*x+4)*sinh(x))/8. - G. C. Greubel, Apr 26 2018

a(0) = 0 prepended and new name by Peter Luschny, Jul 02 2024

A160792 Vertex number of a rectangular spiral related to prime numbers. The distances between nearest edges of the spiral that are parallel to the initial edge are the prime numbers, while the distances between nearest edges perpendicular to the initial edge are all one.

Original entry on oeis.org

0, 1, 3, 5, 10, 13, 23, 27, 44, 49, 77, 83, 124, 131, 189, 197, 274, 283, 383, 393, 522, 533, 693, 705, 902, 915, 1153, 1167, 1448, 1463, 1791, 1807, 2188, 2205, 2645, 2663, 3164, 3183, 3751, 3771, 4410, 4431, 5143, 5165, 5956, 5979, 6853, 6877, 7840, 7865

Offset: 0

Omar E. Pol, May 29 2009

First differences give A160793. - Omar E. Pol, Oct 31 2011

Nathaniel Johnston, Table of n, a(n) for n = 0..5000

Cf. A000040, A160790, A160791, A160793.

A160792 := proc(n) option remember: if(n<=1)then return n: fi: if(n mod 2 = 0)then return procname(n-1)+add(ithprime(j),j=1..n/2): fi: return procname(n-1)+ceil(n/2): end: seq(A160792(n),n=0..49); # Nathaniel Johnston, Jun 16 2011

a(2n) = a(2n-1) + Sum_{j=1..n} prime(j); a(2n+1) = a(2n) + n + 1 for n >= 1. - Nathaniel Johnston, Jun 16 2011

Terms after a(10) and edited by Nathaniel Johnston, Jun 16 2011

A160793 Natural numbers and the sum of first n primes interleaved.

Original entry on oeis.org

1, 2, 2, 5, 3, 10, 4, 17, 5, 28, 6, 41, 7, 58, 8, 77, 9, 100, 10, 129, 11, 160, 12, 197, 13, 238, 14, 281, 15, 328, 16, 381, 17, 440, 18, 501, 19, 568, 20, 639, 21, 712, 22, 791, 23, 874, 24, 963, 25, 1060, 26, 1161, 27, 1264, 28, 1371, 29, 1480

Offset: 1

Omar E. Pol, May 29 2009

Also first differences of A160792: length of edges of a square spiral related to prime numbers A000040 whose vertices are the numbers A160792.

A000027 and A007504 interleaved. - Omar E. Pol, Oct 31 2011

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A160790, A160791, A160792.

Module[{nn=30,sp},sp=Accumulate[Prime[Range[nn]]];Riffle[Range[nn],sp]] (* Harvey P. Dale, May 22 2018 *)

a(2n-1) = n, a(2n) = A007504(n). - Omar E. Pol, Oct 31 2011

More terms from Sean A. Irvine, Oct 31 2011

New definition from Omar E. Pol, Oct 31 2011

A300522 a(n) = (5n + 3)(5n + 4)(5*n + 5)/6.

Original entry on oeis.org

10, 120, 455, 1140, 2300, 4060, 6545, 9880, 14190, 19600, 26235, 34220, 43680, 54740, 67525, 82160, 98770, 117480, 138415, 161700, 187460, 215820, 246905, 280840, 317750, 357760, 400995, 447580, 497640, 551300, 608685, 669920, 735130, 804440, 877975, 955860, 1038220, 1125180

Offset: 0

Bruno Berselli, Mar 08 2018

Al-Saedi has discovered that p(10*n+2,4) + p(10*n+3,4) + p(10*n+4,4) == 0 (mod 5), where p(m,k) denote the number of partitions of m into at most k parts [see Theorem 3.6, formula 24, in Links and References sections].
Hirschhorn showed that p(10*n+2,4) + p(10*n+3,4) + p(10*n+4,4) = (5*n+3)*(5*n+4)*(5*n+5)/6 [see References section: paragraph 3, "Proofs of (1.5)-(1.8)"].
The sequence binomial(5*m,3), m>=0, begins 0, 0, 0, 10, 120, 455, 1140, 2300, 4060, ... - N. J. A. Sloane, Jun 13 2020

Ali H. Al-Saedi, Using Periodicity to Obtain Partition Congruences, Journal of Number Theory, Vol. 178, 2017, pages 158-178.
Michael D. Hirschhorn, Congruences modulo 5 for partitions into at most four parts, The Fibonacci Quarterly, Vol. 56, Number 1, 2018, pages 34-37.

Colin Barker, Table of n, a(n) for n = 0..1000
Ali H. Al-Saedi, Using Periodicity to Obtain Partition Congruences, arXiv:1609.03633 [math.NT], 2017, pages 12-13.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Subsequence of A160790.

Cf. A000292, A007318, A300523.

List([0..40], n -> (5*n+3)*(5*n+4)*(5*n+5)/6);

[div((5*n+3)*(5*n+4)*(5*n+5), 6) for n in 0:40] |> println

[(5*n+3)*(5*n+4)*(5*n+5)/6: n in [0..40]];

Table[(5 n + 3) (5 n + 4) (5 n + 5)/6, {n, 0, 40}]

makelist((5*n+3)*(5*n+4)*(5*n+5)/6, n, 0, 40);

vector(40, n, n--; (5*n+3)*(5*n+4)*(5*n+5)/6)

[(5*n+3)*(5*n+4)*(5*n+5)/6 for n in range(40)]

[(5*n+3)*(5*n+4)*(5*n+5)/6 for n in (0..40)]

O.g.f.: 5*(2 + 16*x + 7*x^2)/(1 - x)^4 [formula 4.1 in Hirschhorn's paper].
E.g.f.: 5*(12 + 132*x + 135*x^2 + 25*x^3)*exp(x)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(-n) = -A300523(n-2).
Sum_{n>=0} 1/a(n) = 3*sqrt(5+2/sqrt(5))*Pi/10 - 9*sqrt(5)*log(phi)/10 - 3*log(5)/4. - Amiram Eldar, Jan 04 2022

A300523 a(n) = (5n + 5)(5n + 6)(5*n + 7)/6.

Original entry on oeis.org

35, 220, 680, 1540, 2925, 4960, 7770, 11480, 16215, 22100, 29260, 37820, 47905, 59640, 73150, 88560, 105995, 125580, 147440, 171700, 198485, 227920, 260130, 295240, 333375, 374660, 419220, 467180, 518665, 573800, 632710, 695520, 762355, 833340, 908600, 988260, 1072445

Offset: 0

Bruno Berselli, Mar 08 2018

Al-Saedi has discovered that p(10*n+6,4) + p(10*n+7,4) + p(10*n+8,4) == 0 (mod 5), where p(m,k) denote the number of partitions of m into at most k parts [see Theorem 3.6, formula 23, in Links and References sections].

Hirschhorn showed that p(10*n+6,4) + p(10*n+7,4) + p(10*n+8,4) = (5*n+5)*(5*n+6)*(5*n+7)/6 [see References section: paragraph 3, "Proofs of (1.5)-(1.8)"].

Ali H. Al-Saedi, Using Periodicity to Obtain Partition Congruences, Journal of Number Theory, Vol. 178, 2017, pages 158-178.
Michael D. Hirschhorn, Congruences modulo 5 for partitions into at most four parts, The Fibonacci Quarterly, Vol. 56, Number 1, 2018, pages 34-37.

Colin Barker, Table of n, a(n) for n = 0..1000
Ali H. Al-Saedi, Using Periodicity to Obtain Partition Congruences, arXiv:1609.03633 [math.NT], 2017, pages 12-13.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Subsequence of A160790.

Cf. A000292, A300522.

List([0..40], n -> (5*n+5)*(5*n+6)*(5*n+7)/6);

[div((5*n+5)*(5*n+6)*(5*n+7), 6) for n in 0:40] |> println

[(5*n+5)*(5*n+6)*(5*n+7)/6: n in [0..40]];

Table[(5 n + 5) (5 n + 6) (5 n + 7)/6, {n, 0, 40}]
Table[Times@@(5n+{5,6,7})/6,{n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{35,220,680,1540},40] (* Harvey P. Dale, Oct 22 2019 *)

makelist((5*n+5)*(5*n+6)*(5*n+7)/6, n, 0, 40);

vector(40, n, n--; (5*n+5)*(5*n+6)*(5*n+7)/6)

[(5*n+5)*(5*n+6)*(5*n+7)/6 for n in range(40)]

[(5*n+5)*(5*n+6)*(5*n+7)/6 for n in (0..40)]

O.g.f.: 5*(7 + 16*x + 2*x^2)/(1 - x)^4 [formula 4.2 in Hirschhorn's paper].
E.g.f.: 5*(42 + 222*x + 165*x^2 + 25*x^3)*exp(x)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(-n) = -A300522(n-2).

A160794 Vertex number of a rectangular spiral related to Fibonacci numbers and prime numbers. The distances between nearest edges of the spiral that are parallel to the initial edge are the Fibonacci numbers, while the distances between nearest edges perpendicular to the initial edge are the prime numbers.

Original entry on oeis.org

0, 1, 2, 5, 7, 13, 17, 28, 35, 53, 65, 94, 114, 156, 189, 248, 302, 380, 468, 569, 712, 842, 1074, 1235, 1611, 1809, 2418, 2657, 3643, 3925, 5521, 5850, 8433, 8815, 12995, 13436, 20200, 20702, 31647, 32216, 49926, 50566, 79222, 79935, 126302, 127094, 202118

Offset: 0

Omar E. Pol, May 29 2009

Nathaniel Johnston, Table of n, a(n) for n = 0..1000

Cf. A000040, A000045, A160790, A160792.

A160794 := proc(n) option remember: if(n<=1)then return n: fi: if(n mod 2 = 0)then return procname(n-1)+add(combinat[fibonacci](j), j=1..n/2): fi: return procname(n-1)+add(ithprime(j), j=1..floor(n/2))+1: end: seq(A160794(n), n=0..46); # Nathaniel Johnston, Jun 16 2011

a(2n) = a(2n-1) + Sum_{j=1..n} Fibonacci(j); a(2n+1) = a(2n) + 1 + Sum_{j=1..n} prime(j) for n >= 1. - Nathaniel Johnston, Jun 16 2011

Terms after a(12) and edited by Nathaniel Johnston, Jun 16 2011

cp's OEIS Frontend

A160791 a(n) = binomial(N, n - N) where N = 1 + floor(n/2).

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A160792 Vertex number of a rectangular spiral related to prime numbers. The distances between nearest edges of the spiral that are parallel to the initial edge are the prime numbers, while the distances between nearest edges perpendicular to the initial edge are all one.

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Maple

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A160793 Natural numbers and the sum of first n primes interleaved.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A300522 a(n) = (5*n + 3)*(5*n + 4)*(5*n + 5)/6.

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A300523 a(n) = (5*n + 5)*(5*n + 6)*(5*n + 7)/6.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Julia

Magma

Mathematica

Maxima

PARI

Python

Sage

Formula

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A300522 a(n) = (5n + 3)(5n + 4)(5*n + 5)/6.

A300523 a(n) = (5n + 5)(5n + 6)(5*n + 7)/6.