A028563 -id:A028563 - cp's OEIS frontend

A132769 a(n) = n*(n + 27).

0, 28, 58, 90, 124, 160, 198, 238, 280, 324, 370, 418, 468, 520, 574, 630, 688, 748, 810, 874, 940, 1008, 1078, 1150, 1224, 1300, 1378, 1458, 1540, 1624, 1710, 1798, 1888, 1980, 2074, 2170, 2268, 2368, 2470, 2574, 2680, 2788, 2898, 3010, 3124, 3240, 3358, 3478

Offset: 0

Omar E. Pol, Aug 28 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566.
Cf. A028569, A098849, A098850, A098603, A098847, A098848, A102928, A120071, A132759.
Cf. A132760, A132761, A132762, A132763, A132764, A132765, A132766, A132767, A132768.
Cf. A132756.

Table[n(n+27),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,28,58},50] (* Harvey P. Dale, Oct 14 2012 *)

a(n)=n*(n+27) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2*n + a(n-1) + 26, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=28, a(2)=58; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Oct 14 2012
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(27)/27 = A001008(27)/A102928(27) = 312536252003/2168462696400, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/27 - 57128792093/2168462696400. (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: 2*x*(14 - 13*x)/(1 - x)^3.
E.g.f.: exp(x)*x*(28 + x).
a(n) = 2*A132756(n). (End)

A116283 k times k+7 gives the concatenation of two numbers m and m-1.

Original entry on oeis.org

7, 30, 64, 42753, 57241, 75423, 425072, 574922, 979528, 4301393, 5698601, 7028666, 4925000747, 5074999247, 7748266574, 8511881484, 8814851184, 7059602159672, 7106167933828, 7439286611621, 7485852385777

Offset: 1

Giovanni Resta, Feb 06 2006

Cf. A001704, A028563.

Cf. A116152, A116269, A116282, A116284, A116291.

def ok(n):
    s = str(n*(n+7)); h = (len(s)+1)//2; return int(s[:h])-1 == int(s[h:])
print(list(filter(ok, range(2, 10**6)))) # Michael S. Branicky, Jul 30 2021

A164008 Zero together with row 8 of the array in A163280.

Original entry on oeis.org

0, 17, 34, 39, 68, 65, 102, 98, 128, 153, 170, 198, 228, 260, 294, 330, 368, 408, 450, 494, 540, 588, 638, 690, 744, 800, 858, 918, 980, 1044, 1110, 1178, 1248, 1320, 1394, 1470, 1548, 1628, 1710, 1794, 1880, 1968, 2058, 2150, 2244, 2340, 2438, 2538, 2640

Offset: 0

Omar E. Pol, Aug 08 2009

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Cf. A008578, A161344, A161345, A163280, A164000, A164007, A164009.

A033676 := proc(n) local dvs; dvs := sort(convert(numtheory[divisors](n), list)) ; op(floor((nops(dvs)+1)/2) , dvs) ; end: A163280 := proc(n, k) local r, T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: printf("0,") ; for n from 1 to 70 do printf("%d,",A163280(8,n)) ; end do ; # R. J. Mathar, Feb 05 2010

LinearRecurrence[{3,-3,1},{0,17,34,39,68,65,102,98,128,153,170,198,228},50] (* Harvey P. Dale, Dec 25 2022 *)

Conjecture: a(n) = A028563(n), n > 9. [R. J. Mathar, Jul 31 2010]

Terms beyond 228 from R. J. Mathar, Feb 05 2010

A132770 a(n) = n*(n + 28).

Original entry on oeis.org

0, 29, 60, 93, 128, 165, 204, 245, 288, 333, 380, 429, 480, 533, 588, 645, 704, 765, 828, 893, 960, 1029, 1100, 1173, 1248, 1325, 1404, 1485, 1568, 1653, 1740, 1829, 1920, 2013, 2108, 2205, 2304, 2405, 2508, 2613, 2720, 2829, 2940, 3053, 3168, 3285, 3404, 3525

Offset: 0

Omar E. Pol, Aug 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..5000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569.
Cf. A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132759, A132760, A132761.
Cf. A132762, A132763, A132764, A132765, A132766, A132767, A132768, A132769.

#(#+28)&/@Range[0,50] (* or *) LinearRecurrence[{3,-3,1},{0,29,60},50] (* Harvey P. Dale, Apr 30 2018 *)

a(n)=n*(n+28) \\ Charles R Greathouse IV, Jun 17 2017

[n*(n+28) for n in (0..50)] # G. C. Greubel, Mar 13 2022

a(n) = 2*n + a(n-1) + 27, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(28)/28 = A001008(28)/A102928(28) = 315404588903/2248776129600, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 7751493599/321253732800. (End)
G.f.: x*(29 - 27*x)/(1-x)^3. - Harvey P. Dale, Aug 03 2021
E.g.f.: x*(29 + x)*exp(x). - G. C. Greubel, Mar 13 2022

A132771 a(n) = n*(n + 29).

Original entry on oeis.org

0, 30, 62, 96, 132, 170, 210, 252, 296, 342, 390, 440, 492, 546, 602, 660, 720, 782, 846, 912, 980, 1050, 1122, 1196, 1272, 1350, 1430, 1512, 1596, 1682, 1770, 1860, 1952, 2046, 2142, 2240, 2340, 2442, 2546, 2652, 2760, 2870, 2982, 3096, 3212, 3330, 3450, 3572

Offset: 0

Omar E. Pol, Aug 28 2007

G. C. Greubel, Table of n, a(n) for n = 0..5000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569.
Cf. A098849, A098850, A098603, A098847, A098848, A102928, A120071, A132759, A132760, A132761.
Cf. A132762, A132763, A132764, A132765, A132766, A132767, A132768, A132769, A132770.

Table[n(n+29), {n,0,50}] (* Bruno Berselli, Apr 03 2015 *)
LinearRecurrence[{3,-3,1},{0,30,62},50] (* Harvey P. Dale, Oct 18 2024 *)

a(n)=n*(n+29) \\ Charles R Greathouse IV, Jun 17 2017

[n*(n+29) for n in (0..50)] # G. C. Greubel, Mar 13 2022

a(n) = 2*n + a(n-1) + 28 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(29)/29 = A001008(29)/A102928(29) = 9227046511387/67543597321200, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/29 - 236266661971/9649085331600. (End)
From G. C. Greubel, Mar 13 2022: (Start)
G.f.: 2*(15*x - 14*x^2)/(1-x)^3.
E.g.f.: x*(30 + x)*exp(x). (End)

A132772 a(n) = n*(n + 30).

Original entry on oeis.org

0, 31, 64, 99, 136, 175, 216, 259, 304, 351, 400, 451, 504, 559, 616, 675, 736, 799, 864, 931, 1000, 1071, 1144, 1219, 1296, 1375, 1456, 1539, 1624, 1711, 1800, 1891, 1984, 2079, 2176, 2275, 2376, 2479, 2584, 2691, 2800, 2911, 3024, 3139, 3256, 3375, 3496, 3619

Offset: 0

Omar E. Pol, Aug 28 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566.
Cf. A028569, A067079, A098849, A098850, A098603, A098847, A098848, A102928, A120071.
Cf. A132759, A132760, A132761, A132762, A132763, A132764, A132765, A132766, A132767.
Cf. A132768, A132769, A132770, A132771.

Table[n(n+30),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,31,64},50] (* Harvey P. Dale, Mar 06 2015 *)

a(n)=n*(n+30) \\ Charles R Greathouse IV, Jun 17 2017

[n*(n+30) for n in (0..50)] # G. C. Greubel, Mar 13 2022

G.f.: x*(31-29*x)/(1-x)^3. - R. J. Mathar, Nov 14 2007
a(n) = 2*n + a(n-1) + 29 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=31, a(2)=64, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Mar 06 2015
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(30)/30 = A001008(30)/A102928(30) = 9304682830147/69872686884000, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 225175759291/9981812412000. (End)
E.g.f.: x*(31 + x)*exp(x). - G. C. Greubel, Mar 13 2022

A020744 Pisot sequences P(8,10), T(8,10).

Original entry on oeis.org

8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138

Offset: 0

David W. Wilson

Conjecturally, even sums of four primes. - Charles R Greathouse IV, Feb 16 2012

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

Subsequence of A005843, A020739. See A008776 for definitions of Pisot sequences.

Cf. A020705, A028563.

LinearRecurrence[{2,-1},{8,10},70] (* Harvey P. Dale, Jul 19 2015 *)
P[x_, y_, z_] := Block[{a}, a[0] = x; a[1] = y; a[n_] := a[n] = Ceiling[a[n - 1]^2/a[n - 2] - 1/2]; Table[a[n], {n, 0, z}]]; P[8, 10, 65] (* or *)
T[x_, y_, z_] := Block[{a}, a[0] = x; a[1] = y; a[n_] := a[n] = Floor[a[n - 1]^2/a[n - 2]]; Table[a[n], {n, 0, z}]]; T[8, 10, 65] (* Michael De Vlieger, Aug 08 2016 *)

a(n)=2*n+8 \\ Charles R Greathouse IV, Feb 16 2012

pisotP(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = ceil(a[n-1]^2/a[n-2]-1/2));
  a
}
pisotP(50, 8, 10) \\ Colin Barker, Aug 08 2016

a(n) = 2*n + 8.
a(n) = 2*a(n-1) - a(n-2).
From Elmo R. Oliveira, Oct 30 2024: (Start)
G.f.: 2*(4 - 3*x)/(1 - x)^2.
E.g.f.: 2*(4 + x)*exp(x).
a(n) = 2*A020705(n) = A028563(n+1) - A028563(n). (End)

A028564 Numbers k such that k*(k+7) is a palindrome.

Original entry on oeis.org

0, 1, 4, 19, 26, 219, 664, 902, 2109, 2972, 2982, 9002, 21009, 29696, 90002, 210009, 223114, 292967, 669024, 900002, 942997, 2100009, 2109664, 2219779, 8972876, 9000002, 21000009, 21037924, 21117304, 21152824, 21155344, 28867722, 29802897, 29857886, 90000002

Offset: 1

Patrick De Geest

Michael S. Branicky, Table of n, a(n) for n = 1..49
Patrick De Geest, Palindromic Quasipronics of the form n(n+x)
Erich Friedman, What's Special About This Number? (See entries 664, 902, 2109, 2982, 9002.)

Cf. A028563, A028565.

from itertools import count, islice
def ispal(n): s = str(n); return s == s[::-1]
def agen():
    for k in count(0):
        if ispal(k*(k+7)):
            yield k
print(list(islice(agen(), 26))) # Michael S. Branicky, Jan 26 2022

a(32) and beyond from Michael S. Branicky, Jan 26 2022

A028565 Palindromes of form k*(k+7).

Original entry on oeis.org

0, 8, 44, 494, 858, 49494, 445544, 819918, 4462644, 8853588, 8913198, 81099018, 441525144, 882060288, 8100990018, 44105250144, 49781418794, 85831713858, 447597795744, 810009900018, 889249942988, 4410052500144, 4450696960544, 4927434347294, 80512566521508, 81000099000018

Offset: 1

Patrick De Geest

Michael S. Branicky, Table of n, a(n) for n = 1..49
Patrick De Geest, Palindromic Quasipronics of the form n(n+x)

Cf. A028563, A028564.

Select[Table[k(k+7),{k,0,9000100}],PalindromeQ] (* Harvey P. Dale, Dec 17 2024 *)

from itertools import count, islice
def ispal(n): s = str(n); return s == s[::-1]
def agen():
    for k in count(0):
        if ispal(k*(k+7)):
            yield k*(k+7)
print(list(islice(agen(), 28))) # Michael S. Branicky, Jan 26 2022

a(n) = A028564(n) * (A028564(n) + 7). - Michael S. Branicky, Jan 26 2022

a(23) and beyond from Michael S. Branicky, Jan 26 2022

A091435 Array T(n,k) = n*(n+k), read by antidiagonals.

Original entry on oeis.org

0, 1, 0, 4, 2, 0, 9, 6, 3, 0, 16, 12, 8, 4, 0, 25, 20, 15, 10, 5, 0, 36, 30, 24, 18, 12, 6, 0, 49, 42, 35, 28, 21, 14, 7, 0, 64, 56, 48, 40, 32, 24, 16, 8, 0, 81, 72, 63, 54, 45, 36, 27, 18, 9, 0, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 0, 121, 110, 99, 88, 77, 66, 55, 44, 33, 22, 11, 0

Offset: 0

Ross La Haye, Mar 02 2004

			Table begins
   0;
   1,  0;
   4,  2,  0;
   9,  6,  3,  0;
  16, 12,  8,  4,  0;
  25, 20, 15, 10,  5,  0;
  36, 30, 24, 18, 12,  6,  0;
  ...
a(5,3) = 40 because 5 * (5 + 3) = 5 * 8 = 40.

Muniru A Asiru, Table of n, a(n) for n = 0..5150 (rows n = 0..100,flattened)
P. De Geest, Palindromic Quasipronics of the form n(n+x)

Columns: a(n, 0) = A000290(n), a(n, 1) = A002378(n), a(n, 2) = A005563(n), a(n, 3) = A028552(n), a(n, 4) = A028347(n+2), a(n, 5) = A028557(n), a(n, 6) = A028560(n), a(n, 7) = A028563(n), a(n, 8) = A028566(n). Diagonals: a(n, n-4) = A054000(n-1), a(n, n-3) = A014107(n), a(n, n-2) = A046092(n-1), a(n, n-1) = A000384(n), a(n, n) = A001105(n), a(n, n+1) = A014105(n), a(n, n+2) = A046092(n), a(n, n+3) = A014106(n), a(n, n+4) = A054000(n+1), a(n, n+5) = A033537(n). Also note that the sums of the antidiagonals = A002411.

Cf. A056536, A056537, A082156.

Flat(List([0..11],j->List([0..j],i->j*(j-i)))); # Muniru A Asiru, Sep 11 2018

Maple
```
seq(seq((j-i)*j,i=0..j),j=0..14);
```

Table[# (# + k) &[m - k], {m, 0, 11}, {k, 0, m}] // Flatten (* Michael De Vlieger, Oct 15 2018 *)

G.f.: x*(1+x-2*x^2*y)/((1-x*y)^2*(1-x)^3). - Vladeta Jovovic, Mar 05 2004

More terms from Emeric Deutsch, Mar 15 2004

cp's OEIS Frontend

A132769 a(n) = n*(n + 27).

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A116283 k times k+7 gives the concatenation of two numbers m and m-1.

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A164008 Zero together with row 8 of the array in A163280.

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A132770 a(n) = n*(n + 28).

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A132771 a(n) = n*(n + 29).

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A132772 a(n) = n*(n + 30).

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A020744 Pisot sequences P(8,10), T(8,10).

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A028564 Numbers k such that k*(k+7) is a palindrome.

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